Edinburgh Hodge Institute

The Geometry seminar is held every Tuesday, 3:30-4:30pm in the main seminar room at the Bayes Centre (5th floor, room 5.10), but check the schedule for exceptions (sometimes we will be at _JCMB 5323_). The seminar is organised by all faculty working in geometry, and currently coordinated by Francesca Carocci.

Mailing Lists. Since most of our seminars will occur together with MAXIMALS in a doubleheader format, we have a joint mailing list, that we will use to send unified announcements. To subscribe to the maxi-edge mailing list, send an email to sympa at mlist.is.ed.ac.uk, with subject line: subscribe maxi-edge your_first_name your_last_name. To unsubscribe at any time, send an email to the same with subject unsubscribe maxi-edge.

Follow the same instructions to subscribe to the geometrytopology mailing list, which will have Geometry focused announcements.

The seminar is named after William Edge (1904-1997), who is known for example for his work on finite geometry, and worked at the University of Edinburgh for over 40 years (1932-1975). Related seminars are Topology, MAXIMALS, EMPG, GLEN, COW.

Current Semester

Upcoming talks:

• EDGE: Qingyuan Jiang (University of Edinburgh) - The geometry of (resolutions of) degeneracy loci
  22nd October 2019, 2:00pm to 3:00pm Bayes Center 5.10 -- Show/hide abstract
  Abstract: In this talk we will discuss the geometry -- the derived categories, Chow groups and Hodge structures -- on certain canonical resolution spaces of degeneracy loci of maps between vector bundles. We will mainly focus on the classical examples of symmetric powers of curves, and then explain the general results and the proofs through these examples.
• EDGE: Yin Li (King's College London)-TBA
  29th October 2019, 3:30pm to 4:30pm -- Show/hide abstract
  Abstract: EDGE
• EDGE: Tara Holm (Cornell University) - TBA
  12th November 2019, 3:30pm to 4:30pm Bayes Center 5.10
• EDGE: Tomasz Przzezdieck (University of Edinburgh)-TBA
  26th November 2019, 3:30pm to 4:30pm
• EDGE Navid Nabijou (University of Glasgow)-TBA
  3rd December 2019, 3:30pm to 4:30pm
• Special EDGE: Graeme Wilkin (York University)-TBA
  13th January 2020, 3:30pm to 4:30pm TBC -- Show/hide abstract
  Abstract: EDGE

Previous talks:

• EDGE: Qingyuan Jiang (University of Edinburgh) - The geometry of (resolutions of) degeneracy loci
  22nd October 2019, 2:00pm to 3:00pm Bayes Center 5.10 -- Show/hide abstract
  Abstract: In this talk we will discuss the geometry -- the derived categories, Chow groups and Hodge structures -- on certain canonical resolution spaces of degeneracy loci of maps between vector bundles. We will mainly focus on the classical examples of symmetric powers of curves, and then explain the general results and the proofs through these examples.
• Special EDGE: Victor Przyjalkowski-On mirror P=W conjecture
  11th October 2019, 1:00pm to 2:00pm JCMB, 5323 -- Show/hide abstract
  Abstract: We discuss Katzarkov-Kontsevich-Pantev conjectures which relate Hodge numbers of Fano varieties with Hodge-type numbers of their Landau--Ginzburg models. We observe their proofs in dimensions 2 and 3. We also discuss mirror P=W conjecture which claims more deep relation of mixed Hodge structures of log Calabi--Yau varieties and perverse Leray filtrations of Landau--Ginzburg models.
• EDGE: Johan Martens (University of Edinburgh) - The Hitchin connection in (almost) arbitrary characteristic
  8th October 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstract
  Abstract: The Hitchin connection is a flat projective connection on bundles of non-abelian theta-functions over the moduli space of curves, originally introduced by Hitchin in a Kahler context. We will describe a purely algebra-geometric construction of this connection that also works in (most) positive characteristics. A key ingredient is an alternative to the Narasimhan-Atiyah-Bott Kahler form on the moduli space of bundles on a curve. We will comment on the connection with some related topics, such as the Grothendieck-Katz p-curvature conjecture. This is joint work with Baier, Bolognesi and Pauly.
• Special EDGE: Costya Loginov(Steklov Mathematical Institute)-Semistable degenerations of Fano varieties
  3rd October 2019, 2:00pm to 3:00pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: Let us consider a family of projective varieties over a curve germ. We will focus on so called semistable families. By the semistable reduction theorem any family whose generic fiber is smooth is birational to a semistable family after a finite base change. We say that the special fiber of such family is a semistable degeneration of its generic fiber. A natural invariant of the special fiber is its dual complex. For example, due to Kulikov there is a characterization of semistable degenerations of K3 surfaces in terms of its dual complexes. Analogous result was obtained by Fujita for del Pezzo surfaces. We prove that if every fiber of a semistable family is Fano then the dual complex of the special fiber is a simplex of bounded dimension. We show that, in contrast to the case of K3 surfaces, the monodromy around the special fiber is always trivial for semistable degenerations of del Pezzo surfaces. We also discuss maximal degenerations of Fano varieties and prove that they are unique in dimensions not greater than 3.
• EDGE: Richard Thomas (Imperial College) - Vafa-Witten invariants for projective surfaces
  1st October 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstract
  Abstract: Vafa and Witten proposed new gauge-theoretic invariants of 4-manifolds in 1994. I’ll explain how to define them, refine them, and calculate some of them, when the 4-manifold is a complex projective surface.
• Structure and Symmetry Theme Day
  27th September 2019, 10:00am to 5:00pm Bayes Centre 5.10 -- Show/hide abstract
  Abstract: Afternoon Schedule: @14:30 Speaker: Nick Sheridan (University of Edinburgh) Title: The Gamma and SYZ conjectures: a tropical approach to periods. Abstract: I'll start by explaining a new method of computing asymptotics of period integrals using tropical geometry, via some concrete examples. Then I'll use this method to give a geometric explanation for a strange phenomenon in mirror symmetry, called the Gamma Conjecture, which says that mirror symmetry does not respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the mirror multiplied by a certain transcendental characteristic class called the Gamma class. We find that the appearance of zeta(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. * @16:00 Speaker: Roberto Volpato (University of Padova) Title: Strings on K3: the maths and the physics. Abstract: I will give an overview of recent (and less recent) ideas that have been developed in the study of string theory on K3 manifolds, including topics in conformal field theory, moonshine, black hole physics, automorphic forms, hyperkahler geometry. *
• EDGE: Luca Battistella (Max Planck Institute Bonn) - Reduced relative Gromov-Witten theory in genus one
  24th September 2019, 3:30pm to 4:30pm Bayes Centre 5.10 -- Show/hide abstract
  Abstract: In joint work with N. Nabijou and D. Ranganathan, we produce a logarithmic desingularisation of the main component of the moduli space of genus one stable maps to projective space having prescribed contact order with a hyperplane. Recasting ideas of A. Gathmann in more modern language, we observe that these spaces are nested: requiring a contact order higher by one identifies a divisor, which together with some boundary corrections corresponds to a line bundle of tropical origin and tautological class. A splitting principle governing these boundary terms allows us to reconstruct the restricted reduced Gromov-Witten theory of the divisor (together with the rubber and relative theories) inductively from that of the ambient space and genus zero data.
• EDGE: Laura Pertusi (Università di Milano) - Stability conditions on Gushel-Mukai fourfolds
  17th September 2019, 3:30pm to 4:30pm Bayes Centre 5.02 -- Show/hide abstract
  Abstract: An ordinary Gushel-Mukai fourfold X is a smooth quadric section of a linear section of the Grassmannian G(2,5). Kuznetsov and Perry proved that the bounded derived category of X admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Xiaolei Zhao, where we construct Bridgeland stability conditions on the K3 subcategory of X. Then I will explain some applications concerning the existence of a homological associated K3 surface and hyperkaehler geometry.
• EDGE: JongHae Keum (Korean institute for advanced study)-Algebraic surfaces with minimal Betti numbers
  10th September 2019, 3:30pm to 4:30pm JCMB room 5323 -- Show/hide abstract
  Abstract: These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called $Q$-homology projective planes. We describe recent progress in the study of such surfaces including smooth examples, the fake projective planes. We also discuss open questions on Montgomery-Yang problem.
• EDGE: Michel van Garrel (Warwick University) Log BPS numbers as Euler characteristics
  21st May 2019, 2:00pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: Let (S,E) be the pair of a del Pezzo surface and a smooth anticanonical divisor on it. It is an example of a log K3 surface. We are interested in studying suitably defined rational curves in (S,E), whose behavior is parallel to the theory of rational curves on a K3 surface. I will define a moduli space of sheaves on (S,E), whose Euler characteristic gives the desired number of rational curves. Time permitting, I will comment on how this relates to mirror symmetry. This is joint work with Jinwon Choi, Sheldon Katz and Nobuyoshi Takahashi.
• EDGE: Nikolaos Tziolas (University of Cyprus) Vector fields and moduli of canonically polarized surfaces in positive characteristic
  20th May 2019, 10:30am to 11:30am JCMB 5323 -- Show/hide abstract
  Abstract: The purpose of this talk is to present some results about the geometry of smooth canonically polarized surfaces defined over a field of positive characteristic which have a nontrivial global vector field, equivalently non reduced automorphism scheme, and the implications that the existence of such surfaces has in the moduli problem of canonically polarized surfaces.
• EDGE: John Pardon (Princeton University) Structural results in wrapped Floer theory
  13th May 2019, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: I will discuss results relating different partially wrapped Fukaya categories. These include a K\"unneth formula, a `stop removal' result relating partially wrapped Fukaya categories relative to different stops, and a gluing formula for wrapped Fukaya categories. The techniques also lead to generation results for Weinstein manifolds and for Lefschetz fibrations. The methods are mainly geometric, and the key underlying Floer theoretic fact is an exact triangle in the Fukaya category associated to Lagrangian surgery along a short Reeb chord at infinity. This is joint work with Sheel Ganatra and Vivek Shende.
• EDGE: Dylan Allegretti (University of Sheffield) The monodromy of meromorphic projective structures
  7th May 2019, 2:00pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: The notion of a complex projective structure is fundamental in low-dimensional geometry and topology. The space of projective structures on a surface has the structure of a complex manifold, and there is a holomorphic map from this space to the character variety of the surface, sending a projective structure to its monodromy representation. In this talk, I will describe joint work with Tom Bridgeland in which we introduced the notion of a "meromorphic projective structure" with poles at a discrete set of points. In the case of a meromorphic projective structure, the monodromy can be viewed as a point in a moduli space introduced by Fock and Goncharov in their work on cluster varieties. This appears to be a manifestation of a general relationship between cluster varieties and spaces of stability conditions on 3-Calabi-Yau triangulated categories.
• EDGE: Ana Rita Pires (University of Edinburgh) Symplectic embeddings and infinite staircases
  16th April 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball, and found that if the ellipsoid is close to round, the answer is given by an "infinite staircase" determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into certain specific polydisks and ellipsoids (Frenkel - Muller, Cristofaro-Gardiner - Kleinman, Usher), but it seems to be a rare behaviour. In this talk, we will see how lattice point counts, Ehrhart theory, and the sharpness of ECH capacities can be used to investigate for what other domains infinite staircases exist, and discuss the relationship with toric varieties. This is joint work with Dan Cristofaro-Gardiner, Tara Holm, and Alessia Mandini.
• EDGE: Hongjie Yu (IST Austria) Counting $\ell$-adic local systems over a curve
  2nd April 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: Let $X_{1}$ be a projective, smooth and geometrically connected curve over $\mathbb{F}_{q}$ with $q=p^{n}$ elements where $p$ is a prime number, and let $X$ be its base change to an algebraic closure of $\mathbb{F}_{q}$. The Frobenius element permutes the set of isomorphism classes of irreducible $\ell$-adic local systems ($\ell \neq p$) with a fixed rank on $X$. In 1981, Drinfeld has calculated the number of fixed points of this permutation in the rank 2 case. Curiously, it looks like the number of $\mathbb{F}_q$-points of a variety defined over $\mathbb{F}_q$. In this talk, we generalize Drinfeld's result to higher rank case. Our method is purely automorphic, in fact we do that by using Arthur-Lafforgue's trace formula.
• EDGE: Rafe Mazzeo (Stanford) Prospects for the Kapustin-Witten equations
  26th March 2019, 3:30pm to 4:30pm -- Show/hide abstract
  Abstract: Gaiotto and Witten have conjectured a relationship between some count of solutions of the Kapustin-Witten equations on a 4-manifold containing a knot in its boundary and the Jones polynomial of that knot. This is far from established and there are some considerable analytic difficulties ahead. I will describe progress toward this goal, including work by Taubes, joint work with Witten and other joint work with S. He.
• EDGE: Jochen Heinloth (University of Essen) Existence of good moduli spaces for algebraic stacks
  19th March 2019, 3:30pm to 4:30pm Bayes Centre, Seminar Room 5.10 -- Show/hide abstract
  Abstract: In this talk we explain how two very basic models of actions of one parameter subgroups allow to formulate local criteria for the existence of separated moduli spaces which can be verified rather easily for an interesting class of moduli problems. (This is joint work with Jarod Alper and Daniel Halpern-Leistner). As a byproduct we find a short proof for one of the basic decomposition results for reductive groups.
• EDGE: Costya Shramov (Steklov Institute and HSE) Automorphisms of Severi-Brauer surfaces
  12th March 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: I will discuss finite groups acting by automorphisms and birational automorphisms of Severi-Brauer surfaces. We will see that they are bounded provided that the base field is a function field, and also make some observations on their structure in the general case.
• Special EDGE: Roberto Svaldi (Cambridge University) On the boundedness of elliptic Calabi-Yau varieties
  7th March 2019, 2:30pm to 3:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: One of the main goals in Algebraic Geometry is to classify varieties. The minimal model program (MMP) is an ambitious program that aims to realize this goal, from the point of view of birational geometry, that is, we are free to modify the structure of a given variety along closed subsets to improve its geometric features. According to the MMP, there are 3 building blocks in the birational classification of algebraic varieties: Fano varieties, Calabi-Yau varieties, and varieties of general type. One important question, that is needed to further investigate the classification process, is whether or not varieties in these 3 classes have finitely many deformation types (a property called boundedness). Our understanding of the boundedness of Fano varieties and varieties of general type is quite solid but Calabi-Yau varieties are still quite elusive. In this talk, I will discuss recent results on the boundedness of elliptic Calabi-Yau varieties, which are the most relevant in physics. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. This is joint work with C. Birkar and G. Di Cerbo.
• EDGE: Sven Meinhardt (University of Sheffield) Field and Vertex algebras from geometry and topology
  29th January 2019, 3:30pm to 4:30pm -- Show/hide abstract
  Abstract: I will propose a definition of (generalized) field and vertex algebras in any symmetric monoidal category and provide a way to construct nontrivial examples. To keep the talk accessible, all of this will be illustrated for the category of modules over a commutative ring. As we shall see, every oriented cohomology theory provides examples from (non-commutative) geometry. More specifically, every (small) dg-category gives rise to a field algebra through its moduli stack of representations. Another class of examples can be constructed for twisted perfect complexes on schemes. This work was motivated by a recent construction of vertex algebras by Dominic Joyce and all credits go to him.
• EDGE: Tarig Abdelgadir (University of New South Wales) Moduli of tensor stable points and refined representations
  29th January 2019, 2:00pm to 3:00pm -- Show/hide abstract
  Abstract: Moduli spaces are a fruitful way of studying a given abelian category, take for example the moduli of point-like objects in non-commutative projective spaces or the moduli of representations of the McKay quiver. These examples yield moduli varieties, elliptic curves and minimal resolutions of Kleinian singularities. Sometimes, however, the problem lends itself better to algebraic stacks as in the case of Ringel's canonical algebras and weighted projective lines. Our guiding example in this talk will be a generalisation of these, namely Geigle-Lenzing (G-L) projective spaces as defned by Herschend, Iyama, Minamoto, Opperman. We will start with a stack and a direct sum of line bundles that form a tilting bundle and recover the stack from the corresponding category of quiver representations before applying this technique to G-L spaces. This is joint work with Daniel Chan (UNSW Sydney).
• EDGE: Tudor Padurariu (MIT) K-theoretic Hall algebras for quivers with potential
  15th January 2019, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.02 -- Show/hide abstract
  Abstract: For a quiver with potential, Kontsevich and Soibelman defined an algebra called the cohomological Hall algebra. After work of Davison and Meinhardt, the CoHA is actually a localized bialgebra and in many cases it satisfies a PBW theorem. We study a version of the Hall algebra defined using the K_0 group of certain categories of singularities associated to the quiver with potential, and explain that similar definitions as the ones used for CoHA put a localized bialgebra structure on this algebra. We also prove a PBW theorem for this algebra using semi-orthogonal decompositions inspired by the work of Spenko and Van den Bergh.
• EDGE: Yongnam Lee (KAIST) On function fields and deformations of hypersurfaces
  4th December 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: In this talk, we consider dominant rational maps from very general hypersurfaces and smooth deformations of hypersurfaces under some suitable conditions. The first part is based on a joint work with Gian Pietro Pirola, and the second part is a joint work with Fabrizio Catanese.
• EDGE: Francesco Sala (Kavli IPMU) 2D Cohomological Hall algebra of a curve
  27th November 2018, 3:30pm to 4:30pm Bayes Centre, 5th floor -- Show/hide abstract
  Abstract: Given a curve X, one can associate with it the following abelian categories of homological dimension two: the category of Higgs sheaves on X, the category of vector bundles on X with flat connection, the category of finite-dimensional representations of the fundamental group of X. The corresponding moduli stacks of objects are stacks of coherent sheaves over different "forms" of the curve: the Dolbeaut, de Rham, and Betti form of X. In the present talk, I will introduce convolution algebras associated with such stacks and provide some characterization result and some conjectures relating them. This is based on a joint paper with Olivier Schiffmann and a work in progress with Mauro Porta.
• EDGE: Jeff Hicks (UC Berkeley) Dimers and Tropical Lagrangians
  20th November 2018, 3:30pm to 4:30pm Bayes Centre Room 5.02 -- Show/hide abstract
  Abstract: Mirror symmetry is a conjectured duality between the symplectic geometry of a space X and complex geometry on a mirror space Y. These spaces are conjectured to share dual torus fibrations over a common affine base Q. A proposed mechanism for mirror symmetry is that the symplectic geometry of X and complex geometry of Y can be mutually compared to tropical geometry on the base Q. Starting with the combinatorial data of a dimer on a torus, we construct a Lagrangian in X whose valuation projection is a tropical hypersurface in Q. We will then explore a mutation and wall-crossing story for these tropical Lagrangians. The construction of these tropical Lagrangians will also tell us why they are homologically mirror to sheaves supported on complex hypersurfaces with matching tropical valuation.
• Special EDGE: Ivan Smith (Cambridge) Lagrangian pinwheels
  15th November 2018, 3:30pm to 4:30pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: Pinwheels are certain simple cell complexes; Lagrangian pinwheels arise as vanishing cycles of cyclic quotient singularities. We give a symplectic analogue of a theorem of Hacking and Prokhorov classifying the Wahl-type degenerations of the projective plane in terms of Markov triples, and contrast this with a finiteness theorem for pinwheels which admit Lagrangian embeddings in surfaces of general type, partially answering a question of Kronheimer. This talk reports on joint work with Jonny Evans.
• EDGE: Mauricio Corrêa (Oxford) Moduli spaces of reflexive sheaves and classification of distributions on P^3
  13th November 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is a integrable phenomena. This parte of the talk is a work joint with M. Jardim (Unicamp) and O. Calvo-Andrade (Cimat). We also study foliations by curves via the investigation of their singular schemes and conormal sheaves and we provide a classification of foliations of degree at most 3 with conormal sheaves locally free. This parte of the talk is a work joint with M. Jardim (Unicamp) and S. Marchesi (Unicamp).
• EDGE: Aaron Bertram (Utah) Stability Conditions on Projective Space
  13th November 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: I describe a one-parameter family of stability conditions on projective space of any dimension that seems to converge to Gieseker stability. In dimensions two and three this is known due to previous work of many people, but it is open in higher dimension. One feature of this family is that the ``walls'' are easily computed, as I will show with several examples. This is joint work with my students, Matteo Altavilla, Dapeng Mu and Marin Petkovic.
• Special EDGE: Alexander Kuznetsov (Moscow) Residual categories
  12th November 2018, 3:00pm to 4:00pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: From the homological projective duality perspective it is important to construct Lefschetz semiorthogonal decompositions which are as close as possible to being rectangular. I will explain an approach to this problem based on the notion of a residual category. Among other things it shows that the "out-of-rectangular" part of the derived category has an interesting structure. I will show some examples of how this part looks for homogeneous varieties and discuss the connection with quantum cohomology of these varieties. This is a joint work with Maxim Smirnov.
• ICMS workshop: Constructions and obstructions in birational geometry (NO EDGE)
  6th November 2018, 3:30pm to 4:30pm Bayes Centre, 5th floor
• EDGE: Gregory G. Smith (Queen's University) Sums of squares on real projective varieties
  30th October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: How does classical algebraic geometry help one recognize sums of squares? After sampling some of the unexpected connections between real and complex geometry, we will look at new bounds on the number of terms in a sum-of-squares expression for a quadratic form on a real projective variety. This talk is based on joint work with G. Blekherman, R. Sinn, and M. Velasco.
• EDGE Special Seminar: Victor Przyjalkowski (HSE Moscow) Weighted complete intersections
  25th October 2018, 3:00pm to 4:00pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: We observe a classification and the main properties of one of the main class of examples of higher dimensional Fano varieties --- smooth complete intersections in weighted projective spaces. We discuss their main properties and boundness results. We also discuss extremal examples from Hodge theory point of view and their relations with derived categories structures and their semiorthogonal decompositions. If time permits, we discuss mirror symmetry for the complete intersections and invariants of their Landau--Ginzburg models related to ones of the complete intersections.
• EDGE: Angela Ortega (HU Berlin) Generic injectivity of the Prym map for double ramified coverings
  23rd October 2018, 3:00pm to 4:00pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: Given a finite morphism of smooth curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. In this talk we will consider the Prym map between the moduli space of double coverings over a genus g curve ramified at r points, and the moduli space of polarised abelian varieties of dimension (g-1+r)/2 with polarisation of type D. We will show the generic injectivity of the Prym map in the cases (a) g=2, r=6 and (b) g=5, r=2. In the first case the proof is constructive and can be extended to the range r > max{6, 2(g+2)/3}. This is a joint work with Juan Carlos Naranjo.
• EDGE Special Seminar: Ziquan Zhuang (Princeton) Superrigidity and K-stability
  18th October 2018, 2:10pm to 3:10pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: Superrigidity and K-stability are properties of Fano varieties that have many interesting geometric implications. For instance, birational superrigidity implies non-rationality and K-stability is related to the existence of Kähler-Einstein metrics. Nonetheless, both properties are hard to verify in general. In this talk, I will first explain how to relate birational superrigidity to K-stability using alpha invariants; I will then outline a method of proving birational superrigidity that works quite well with most families of index one Fano complete intersections and thereby also verify their K-stability. This is partly based on joint work with Charlie Stibitz and Yuchen Liu.
• EDGE: Hiro Lee Tanaka (Harvard) Morse theory on a point, and associative algebras
  16th October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: I'll talk about the first step in a project to enrich Morse theory and Lagrangian Floer theory over coefficient rings that are deeper than the integers (e.g., over ring spectra). Such a thing is useful because invariants contained in spectra are way more sensitive than invariants contained in chain complexes. Also, even if you don't care about Morse theory or symplectic geometry, this "first step" is fun in its own right: We construct a stack of broken lines (which one can think of as the appropriate moduli space of Morse trajectories on a point), and then we prove that a factorizable sheaf on this stack is the same thing as a possibly non-unital associative algebra. I'll try to explain what any of this has to do with Floer theory in the exact case if time allows. This is joint work with Jacob Lurie.
• EDGE: Laura Pertusi (Milano) Geometry of fourfolds with an admissible K3 subcategory
  16th October 2018, 2:00pm to 3:00pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: The derived category of a cubic fourfold admits a semiorthogonal decomposition whose non trivial component is a subcategory of K3 type by a result of Kuznetsov. This allowed to prove many properties on the geometry of the hyperkaehler manifolds associated to the cubic fourfold. More recently, Kuznetsov and Perry found a semiorthogonal decomposition with a K3 type component in the case of an other class of fourfolds, known as Gushel-Mukai fourfolds. The aim of this talk is to discuss a generalization of some results on lattice theory, proved for cubic fourfolds by Addington, Thomas and Huybrechts, in the setting of Gushel-Mukai fourfolds. In particular, we discuss the conditions under which their associated hyperkahler fourfold is birational to a moduli space of stable sheaves (resp. to the Hilbert square) on a K3 surface.
• MAXIMALS Doubleheader
  9th October 2018, 2:00pm to 4:30pm
• EDGE: Helge Ruddat (Mainz) Smoothing toroidal crossing varieties
  2nd October 2018, 3:30pm to 4:30pm Bayes Centre Seminar Room 5.10 -- Show/hide abstract
  Abstract: I explain the proof of a new result on smoothing toroidal and normal crossing varieties by constructing certain kinds of log structures that enjoy Hodge to de Rham degeneration with a suitable definition of differential forms. We generalize work by Friedman and work by Gross and Siebert. This is a joint project with Simon Felten and Matej Filip.
• EDGE: Clark Barwick (Edinburgh) Inverse Galois Problems
  25th September 2018, 3:30pm to 4:30pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: There is a big class of problems that assert, under various circumstances, that certain kinds of Galois-theoretic data can be ‘realised’ at the level of fields, rings, or schemes. These problems go under various names – anabelian conjectures, Abhyankar conjectures, section conjectures, inverse Galois problems, etc. – but the idea is always the same. We survey these problems, and we discuss those that have actually been solved.
• EDGE: Nick Sheridan (Edinburgh) Cubic fourfolds, K3 surfaces and mirror symmetry
  18th September 2018, 3:05pm to 3:55pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett) or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface’ to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric’, i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric’: i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context. (joint work with Ivan Smith).
• EDGE V. Balaji (Chennai Mathematical Institute) On semi-simplicity of tensor products in positive characteristics
  5th July 2018, 2:05pm to 2:55pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: We work over an algebraically closed field k of characteristic p > 0. In 1994, Serre showed that if semi-simple representations V_i of a group Γ are such that ∑(dim V_i − 1) < p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where Γ is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of Γ in G, (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the V_i are semi-simple representations of a group scheme G. In my talk I present the recent work of mine (2017) with Deligne and Parameswaran where we consider the case when G is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes G of reductive groups G. As an application, we obtain an analogue of classical Luna’s étale slice theorem in positive characteristics.
• EDGE Filippo Viviani (Rome) - On the first steps of the minimal model program for the moduli space of stable pointed curves
  28th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: I will report on a joint work with G. Codogni and L. Tasin, in which we investigate the possible first steps of the minimal model program for the moduli space of stable pointed curves. We prove that these first steps have a modular interpretation and we relate them to the first steps of the so called Hassett-Keel program, which studies certain log canonical models of the moduli space of stable pointed curves and their modular interpretations.
• EDGE Miles Reid (Warwick) - The Tate-Oort Group and Godeaux Surfaces in Mixed Characteristic
  15th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: We construct Godeaux surfaces and Calabi-Yau 3-folds whose Pic^tau contains 5-torsion, in mixed characteristic at 5. The same ideas give Campedelli surfaces and Calabi-Yau 3-folds with 7-torsion, and (in progress) Godeaux surfaces with 3-torsion. The aim is to put varieties in characteristic p with action of Z/p, al_p and mu_p into a single family, together with the characteristic zero case with a Z/p action. Joint work with Kim Soonyoung.
• GLEN seminar: Maxence Mayrand (Oxford)
  15th June 2018, 11:00am to 12:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Stratified hyperkähler spaces http://www.maths.ed.ac.uk/~jmartens/GLEN.html
• GLEN seminar: Hendrik Suess (Manchester)
  15th June 2018, 9:45am to 10:45am JCMB 5323 -- Show/hide abstract
  Abstract: On irregular Sasaki-Einstein manifolds in dimension 5. http://www.maths.ed.ac.uk/~jmartens/GLEN.html
• GLEN seminar: Andrew Dancer (Oxford)
  14th June 2018, 3:30pm to 4:30pm JCMB 5323 -- Show/hide abstract
  Abstract: Integrability in Riemannian Geometry http://www.maths.ed.ac.uk/~jmartens/GLEN.html
• GLEN seminar: Leonardo Mihalcea (Virginia Tech)
  14th June 2018, 2:00pm to 3:00pm JCMB 5323 -- Show/hide abstract
  Abstract: Chern classes of Schubert cells, Hecke algebras, and stable envelopes.
  
  http://www.maths.ed.ac.uk/~jmartens/GLEN.html
• EDGE Nicolò Sibilla (Kent) - Log schemes, root stacks and parabolic bundles
  7th June 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Log schemes are an enlargement of the category of schemes that was introduced by Deligne, Faltings, Illlusie and Kato, and has applications to moduli theory and deformation problems. Log schemes play a central role in the Gross-Siebert program in mirror symmetry. In this talk I will introduce log schemes and then explain recent work joint with D. Carchedi, S. Scherotzke, and M. Talpo on various aspects of their geometry. I will discuss a comparison result between two different ways of associating to a log scheme its etale homotopy type, respectively via root stacks and the Kato-Nakayama space. Our main result is a new categorified excision result for parabolic sheaves, which relies on the technology of root stacks.
• EDGE: Daniel Halpern-Leistner (Cornell) - Wall crossing in moduli problems large and small
  31st May 2018, 3:05pm to 3:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Geometric invariant theory is an essential tool for constructing moduli spaces in algebraic geometry. Recently a theory has emerged in my work and the work of others which treats the results and structures of geometric invariant theory in a broader context. The theory of Theta-stability applies directly to moduli problems without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme. I will give an overview of the picture that has emerged, including a discussion of Harder-Narasimhan theory and relatively simple criteria for the existence of good moduli spaces. Then I will discuss applications to wall crossing formulas of K-theoretic Donaldson invariants of algebraic surfaces.
• EDGE: Xiaolei Zhao (Northeastern/UCSB) - Twisted cubics on cubic fourfolds and stability conditions
  31st May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components. Along the way, we will derive several properties of cubic fourfolds as consequences. This is based on a joint work with Chunyi Li and Laura Pertusi.
• EDGE: Anna Barbieri (Sheffield) - A Riemann-Hilbert problem for uncoupled BPS structures
  24th May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Abstract: BPS structures locally describe the space of Bridgeland stability conditions of a CY3 category together with a generalised Donaldson-Thomas theory. On the other hand, Riemann-Hilbert problems are inverse problems in the theory of differential equations. After defining the notion of BPS structures I will introduce and motivate a Riemann-Hilbert problem naturally attached to BPS structures.
• EDGE: Fabio Bernasconi (Imperial College London) - Pathologies for Fano varieties and singularities in positive characteristic
  17th May 2018, 3:05pm to 3:55pm JCMB, Seminar room, 5323 -- Show/hide abstract
  Abstract: Abstract: In this talk, after motivating the study of singularities in the context of the Minimal Model Program, we will explain the connection between vanishing theorems on Fano-type varieties and cohomological properties of certain classes of singularities. In particular, we will explain how to construct del Pezzo surfaces in low characteristic violating Kodaira vanishing and how to deduce the existence of klt not Cohen-Macaulay threefold singularities. Time permitting, we will discuss some possible obstructions to the existence of flips.
• EDGE: Davide Cesare Veniani (Mainz) - Recent advances about lines on quartic surfaces
  17th May 2018, 2:05pm to 2:55pm JCMB, Seminar room, 5323 -- Show/hide abstract
  Abstract: Abstract: The number of lines on a smooth complex surface in projective space depends very much on the degree of the surface. Planes and conics contain infinitely many lines and cubics always have exactly 27. As for degree 4, a general quartic surface has no lines, but Schur's quartic contains as many as 64. This is indeed the maximal number, but a correct proof of this fact was only given quite recently. Can a quartic surface carry exactly 63 lines? How many can there be on a quartic which is not smooth, or which is defined over a field of positive characteristic? In the last few years many of these questions have been answered, thanks to the contribution of several mathematicians. I will survey the main results and ideas, culminating in the list of the explicit equations of the ten smooth complex quartics with most lines.
• EDGE: Francesca Carocci (Imperial College London) - Reduced vs Cuspidal GW invariants for the quintic 3-fold
  10th May 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Abstract: Moduli spaces of stable maps of genus g>0 are highly singular and with many irreducible components which affect the enumerative meaning of the invariants arising from them . In this talk we will try to give a flavour of how bad these spaces can be, already in the simplest example in genus 1. We will then hint at two possible approaches to deal with the so called "degenerate contributions" , namely: Li-Zinger reduced invariants and Viscardi-Smyth cuspidal invariants. We will then explain in which sense these two approaches coincide for the quintic 3-fold.
• EDGE: Alexander Kasprzyk (Nottingham) - Bounding the anticanonical degree of toric Gorenstein Fano varieties
  26th April 2018, 3:05pm to 3:55pm JCMB, room 4325B -- Show/hide abstract
  Abstract: Abstract: In recent work with Balletti and Nill we proved a sharp bound on the volume of a reflexive polytope -- hence on the anticanonical degree of a toric Gorenstein Fano variety. In this talk I will review some of the known classifications of Fano polytopes and remind the audience of their connections to geometry. I will then explain how we proved a sharp bound in the case of reflexive polytopes. Finally, time permitting, I will point out an implication for the study of Fano manifolds via Mirror Symmetry.
• EDGE: Thomas Baier (IST Lisbon) - Mabuchi Geodesic Rays on Hamiltonian K-Spaces and Mixed Polarizations
  26th April 2018, 2:05pm to 2:55pm JCMB, room 4325B -- Show/hide abstract
  Abstract: The aim of this talk (based on joint work in progress with José Mourão and João Pimentel Nunes) is to link aspects of the metric geometry of manifolds with (non-abelian) Hamiltonian symmetry, and their geometric quantization: We provide a description of Mabuchi geodesic rays of K-invariant Kähler metrics, parametrized by convex functions on the moment map image (not unlike the well-known toric case). In the second part, we discuss K-invariant mixed polarizations (in the sense of geometric quantization), and how they arise in this setting.
• EDGE: Chris Elliott (IHES) - The Multiplicative Hitchin System in Supersymmetric Gauge Theory
  19th April 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Abstract: Multiplicative Higgs bundles are an analogue of ordinary Higgs bundles where the Higgs field takes values in a Lie group instead of its Lie algebra.  In this talk I'll discuss two contexts where multiplicative Higgs bundles appear in supersymmetric gauge theory.  I'll explain how hyperkähler structures on these moduli spaces arise physically and mathematically and relate to the theory of Poisson Lie groups, and finally I'll introduce a speculative multiplicative analogue of the geometric Langlands conjecture.  This is based on joint work in progress with Vasily Pestun.
• EDGE: Swarnava Mukhopadhyay (Bonn) - Hyperplane arrangements and invariants of tensor products.
  12th April 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract:

  We study the map from compactly supported cohomology to the usual cohomology of a complement of a hyperplane arrangement and give an explicit formula for a generalized version of this map. Our motivation comes from work of Schechtman and Varchenko who connected invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. Our results allow to equip certain spaces of invariants with interesting mixed Hodge structures via the Knizhnik-Zamolodchikov connection. This is a joint work with P. Belkale and P. Brosnan. 

• EDGE: Giulio Codogni (Ecole Polytechnique Fédérale de Lausanne) - Positivity of the Chow-Mumford line bundle for families of K-stable klt Fano varieties
  5th April 2018, 2:05pm to 2:55pm JCMB, seminar room 5323 -- Show/hide abstract
  Abstract: The Chow-Mumford (CM) line bundle is a functorial line bundle defined on the base of any family of polarized varieties, in particular on the base of families of klt Fano varieties. It is conjectured that it yields a polarization on the conjectured moduli space of K-semi-stable klt Fano varieties. This boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with  K-semi-stable/K-polystable fibers. In this talk, I will present a proof of the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming stability only for very general fibers. These results work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. I will also  present an application to the classification of Fano varieties. This is a joint work with Zs. Patakfalvi.
• EDGE: Nils Carqueville (Vienna) - Topological quantum field theories: state sums and defects
  29th March 2018, 2:05pm to 2:55pm JCMB, Seminar room, 5323 -- Show/hide abstract
  Abstract: A general framework will be discussed which unifies group orbifolds and state sum
  models, in the context of topological quantum field theory (TQFT) in arbitrary
  dimension. After a review of the 2-dimensional case, I will outline general aspects
  of the construction and and discuss examples in 3 dimensions, including Turaev-Viro
  theory as and orbifold, as well as surface defects in quantised Chern-Simons theory.
  (Based on joint work with I. Runkel and G. Schaumann.)
• EDGE: Andrea Fanelli (Dusseldorf) - Del Pezzo fibrations in positive characteristic
  15th March 2018, 2:05pm to 2:55pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract:
  

  In this talk, I will discuss some pathologies for the generic fibre of del Pezzo fibrations in characteristic p>0, motivated by the recent developments of the MMP in positive characteristic. The main application of the joint work with Stefan Schröer concerns 3-dimensional Mori fibre  spaces.

• EDGE: Naoki Koseki (Tokyo) - Perverse coherent sheaves on blow-ups at codimension two loci
  9th March 2018, 2:05pm to 2:55pm JCMB, Room 4312 -- Show/hide abstract
  Abstract:  Let X be the blow up of a smooth projective variety Y along codimension two 
  smooth closed subvariety. We will discuss how the moduli space of sheaves on X 
  and that of Y can be related. The key method is the notion of perverse coherent sheaves 
  introduced by T.Bridgeland. Our main result is a higher dimensional generalization of the 
  result of H.Nakajima and K.Yoshioka. 
  As an application of our main result, we will also discuss about the birational geometry of 
  Hilbert scheme of two points. 
• EDGE: Takero Fukuoka (Tokyo/Warwick) - Relative linear extensions of sextic del Pezzo fibrations
  1st March 2018, 2:05pm to 2:55pm JCMB, Seminar Room 5325 -- Show/hide abstract
  Abstract: Abstract: An extremal contraction from a non-singular projective 3-fold onto a smooth curve is so-called a del Pezzo fibration. It is classically known that every del Pezzo surface $S$ is a (weighted) complete intersection of a certain Fano variety. In order to study del Pezzo fibrations, it is important to relativize such descriptions for those. The main result of this talk shows that the sextic del Pezzo fibrations are relative linear sections of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$-fibrations and $\mathbb{P}^2 \times \mathbb{P}^2$- fibrations, which are constructed as Mori fiber spaces with smooth total space. As an application, we will classify the singular fibers of sextic del Pezzo fibrations.
• EDGE: Sjoerd Beentjes (University of Edinburgh) - The crepant resolution conjecture for Donaldson-Thomas invariants
  15th February 2018, 2:05pm to 2:55pm Seminar Room 5323 JCMB -- Show/hide abstract
  Abstract: Abstract: Donaldson-Thomas (DT) invariants are integers virtually enumerating curves on a Calabi-Yau threefold. These invariants are typically organised in a generating series. The crepant resolution conjecture predicts a transformation rule for the DT generating series of two related Calabi-Yau threefolds, in the setting of the McKay correspondence. In this talk, I will state this conjecture, explain why it is really a conjecture about expansions of rational functions, and sketch a proof using wall-crossing arguments and the motivic Hall algebra. This is joint work with John Calabrese and Jørgen Rennemo.
• EDGE: Julio Andrade (University of Exeter) - Two Problems Involving the Divisor Functions
  8th February 2018, 2:05pm to 2:55pm EDGE: JCMB, Seminar room, 5323 -- Show/hide abstract
  Abstract: Abstract: In this talk, I will discuss two problems involving the divisor functions. The first problem is about the auto-correlation of the values of the divisors functions and I will describe how we can completely solve the problem in the function field setting using a combination of analytic and algebraic techniques. The second problem is about the maxima pairwise of the divisors functions and I will present some recent results that improve some old results of Erdos and Hall.
• EDGE: Andrea Ricolfi (Bonn) - The DT/PT correspondence for smooth curves
  30th January 2018, 2:05pm to 2:55pm JCMB, Seminar room 5323 -- Show/hide abstract
  Abstract: Abstract: Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are enumerative invariants attached to Calabi-Yau 3-folds. They are related by a wall-crossing type formula called the “DT/PT correspondence”. After a general introduction, we will discuss a variation of this correspondence, relating the “local” DT and PT invariants, encoding the contribution of a single smoooth curve to the full (classical) invariants.
• Joint EDGE-MAXIMALS: Peter Samuelson (University of Edinburgh) - The Hall algebra of the Fukaya category of a surface
  16th January 2018, 3:05pm to 3:55pm JCMB, Seminar Room 5323 -- Show/hide abstract
  Abstract: Abstract: The construction of the Fukaya category of a symplectic manifold is inspired by string theory: roughly, objects are Lagrangians, morphisms are intersection points, and composition of morphisms comes from "holomorphic disks." For surfaces, a combinatorial construction of the (partially wrapped) Fukaya category was recently given by Haiden, Katzarkov, and Kontsevich. We will discuss this category and some recent results involving its (derived) Hall algebra (joint with B. Cooper).
• EDGE: Tran Bach (Edinburgh) - On k-normality and Regularity of Normal Toric Varieties
  7th December 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: We will give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.
• EDGE: Alex Lemmens (University of Leuven) - Syzygies of toric varieties
  30th November 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: Syzygies are a basic homological invariant for modules over a ring. One gathers the dimensions of the syzygy spaces in a table of integers called the graded Betti table. There are deep connections between the geometry of algebraic varieties (for instance curves) and their syzygies. In this talk we study the link between the graded Betti table of a projective toric variety and the combinatorics of the defining polytope.
• EDGE: Stefan Schreieder (University of Munich) - The rationality problem for quadric bundles
  23rd November 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: We study the rationality problem for quadric bundles X over rational bases S. By a theorem of Lang, such bundles are rational if r>2^n-2, where r denotes the fibre dimension and n=dim(S) denotes the dimension of the base. We show that this result is sharp. In fact, for any r at most 2^n-2, we show that many smooth r-fold quadric bundles over rational n-folds are not even stably rational. Our result is based on a generalization of the specialization method of Voisin and Colliot-Thélène—Pirutka.
• EDGE: Soheyla Feyzbakhsh (University of Edinburgh) - Applications of Bridgeland stability conditions to classical algebraic geometry
  16th November 2017, 2:05pm to 2:55pm Seminar room JMCB 5323 -- Show/hide abstract
  Abstract: I will explain how wall-crossing with respect to Bridgeland stability conditions provides a new upper bound for the number of global sections of sheaves on K3 surfaces. This, in particular, extends and completes a program proposed by Mukai to reconstruct a K3 surface from a curve on that. Furthermore, the upper bound characterizes special vector bundles on curves on K3 surfaces, which have the maximum number of global sections for the minimum degree. Therefore, it gives an explicit expression for Clifford indices of curves on K3 surfaces.
• EDGE: Navid Nabijou (Imperial College London) - Relative Quasimaps and a Lefschetz-type Formula
  9th November 2017, 2:05pm to 2:55pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: The theory of stable quasimaps provides an alternative system of curve counts to the usual Gromov-Witten invariants. In joint work with Luca Battistella, we define moduli spaces of relative stable quasimaps to a pair (X,Y), where Y is a hyperplane section in X. Intuitively these spaces parametrise quasimaps in X with specified contact orders to Y, and can be used to define relative quasimap invariants. By investigating these moduli spaces we obtain a Lefschetz-type formula, expressing certain quasimap invariants of Y in terms of the invariants of X. Since the I-function from mirror symmetry is equal to a generating function for quasimap invariants, this result can be viewed as a “quantum Lefschetz theorem for I-functions.” It agrees with an earlier formula obtained by Ciocan-Fontanine and Kim.
• EDGE: Andrea Petracci (Nottingham) - Deformations of toric singularities and Mirror Symmetry
  2nd November 2017, 2:05pm to 2:55pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: Abstract: Deformations of affine toric varieties have been studied by Klaus Altmann and Anvar Mavlyutov: they also give constructions of such deformations starting from Minkowski decompositions of polyhedra. On the other hand, deformations of projective toric varieties are not well understood. In this talk, I will present an approach, based on mirror symmetry, to construct deformations of Gorenstein affine toric 3-folds in the context of the Gross-Siebert program. This approach can be globalised and gives a combinatorial recipe to construct smoothings of Gorenstein toric Fano 3-folds. This talk is partly based on work in progress in collaboration with Alessio Corti, Paul Hacking, and Thomas Prince.
• EDGE: Ben Davison (Glasgow) - The integrality conjecture for coherent sheaves on a Calabi-Yau 3-fold.
  26th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: BPS numbers are certain invariants that "count" coherent sheaves on Calabi-Yau 3-folds. Because of subtleties in the definition, especially in the presence of strictly semistable sheaves, it is not a priori clear that the numbers are in fact integers. I will present a recent proof with Sven Meinhardt of this integrality conjecture. The conjecture follows from a stronger conjecture, namely that a certain constructible function on the coarse moduli space of semistable sheaves defined by Joyce and Song is integer valued. This conjecture in turn is implied by the stronger conjecture that their function is in fact the pointwise Euler characteristic of a perverse sheaf. We prove all of these conjectures by defining this perverse sheaf.
• EDGE: Nicola Pagani (Liverpool) - The indeterminacy of the Abel-Jacobi maps
  19th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: The Abel-Jacobi morphisms are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space of smooth pointed curves. When the source and target moduli spaces are compactified, these morphisms can be reinterpreted as rational maps, and it is natural to ask for their locus of indeterminacy. We explicitly characterize the indeterminacy locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification \bar{M}_{g,n} by means of stable curves). In particular, we deduce that for evey Abel-Jacobi map there exists a compactification such that the map extends to a well-defined morphism on \bar{M}_{g,n}. This offers an approach to define and then compute the classes of several different extensions of the "Jacobian double ramification cycles" (= the pullbacks of the zero section via the Abel-Jacobi maps). This is a joint work with Jesse Kass.
• EDGE: Adam Boocher (University of Utah) - Inequalities on Betti Numbers
  12th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: If X is a projective variety then the Betti numbers of its coordinate ring are a sequence of numbers that encode the symmetry and complexity of the defining equations. They are a strict refinement of the Hilbert function and can detect subtle geometric information. In this talk I'll discuss two inequalities concerning the Betti numbers. One, the Buchsbaum-Eisenbud, Horrocks rank conjecture says roughly that the Koszul complex is the ``smallest'' possible resolution. The second, a conjecture of Conca for Koszul rings, is motivated by the Taylor resolution for monomial ideals. I'll discuss recent work on these topics and how the algebra structure of Koszul homology has given us hints about to the structure of these algebras.
• EDGE: Mara Ungureanu (HU Berlin) - From billiards in polygons to de Jonquières divisors
  5th October 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: The aim of this talk is to give a broad overview of the circle of ideas surrounding the de Jonquières divisors on algebraic curves. I will introduce the notion of translation surfaces and their parameter spaces by means of billiards on rational polygons. These parameter spaces lend themselves to a natural algebro-geometric interpretation as strata of holomorphic differentials on the moduli space of curves and as such can be related to certain cohomological field theories. Finally, the de Jonquières divisors will emerge as generalisations of the previous structures and I will discuss related recent progress and open questions.
• EDGE: Gwyn Bellamy (Glasgow) - Counting symplectic resolutions of quotient singularities
  28th September 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: In this talk I will explain how one can use the representation theory of symplectic reflection algebras to count the number of symplectic resolutions admitted by a symplectic quotient singularity. In particular, I will explain the role played by Calogero-Moser families in this story. This is based partially on joint work with T. Schedler and U. Thiel, and builds on the general theory developed by Namikawa.
• EDGE: Lorenzo Foscolo (Heriot-Watt) - Complete non-compact G2 manifolds from asymptotically conical Calabi-Yau 3-folds
  21st September 2017, 2:05pm to 2:55pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: G2 manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as analogues of Calabi-Yau 3-folds. In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds are currently known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds, the metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in algebraic geometry about isolated singularities and their resolutions can be used to produce examples of complete G2 manifolds and interesting submanifolds inside them.
• EDGE: Elizabeth Gasparim (Universidad Católica del Norte) - A Landau--Ginzburg model without projective mirrors
  8th August 2017, 2:05pm to 2:55pm JCMB 5323 -- Show/hide abstract
  Abstract: Using methods of Lie theory we construct a family of Symplectic Lefschetz fibrations. I will explain the construction, and discuss mirror symmetry in a simple case, the adjoint orbit of sl(2). This is joint work with Ballico, Barmeier, Grama, and San Martin.
• EDGE: Benjamin Schmidt (UT Austin) - The Genus of Space Curves
  29th June 2017, 5:30pm to 6:30pm JCMB 5326 -- Show/hide abstract
  Abstract: A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. In this talk I will give an introduction to the topic, introduce stability conditions in the derived category, and explain how the two can be related. This is based on joint work in progress with Emanuele Macri.
• EDGE: Francis Bischoff (Toronto University) - Morita equivalence and the generalized Kahler potential
  15th June 2017, 2:10pm to 3:00pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: In this talk I will present a new approach to Generalized Kahler geometry in which a GK structure of symplectic type can be described in terms of a holomorphic symplectic Morita equivalence along with a brane bisection. I will then explain how this new approach can be applied to the problem of describing a GK structure in terms of holomorphic data and a single real-valued function (the generalized Kahler potential). This is joint work with Marco Gualtieri and Maxim Zabzine.
• EDGE: De-Qi Zhang (NUSingapore) - Polarized endomorphisms of projective varieties
  30th May 2017, 10:10am to 11:00am JCMB 5323 -- Show/hide abstract
  Abstract: An endomorphism f on a normal projective variety X is polarized if the f-pullback of an ample divisor H on X is linearly equivalent to the multiple qH for some natural number q larger than 1. Examples of such f include self-maps of the projective spaces (or more generally Fano varieties of Picard number 1) and multiplication map of complex tori. We show that we can run the f-equivariant minimal model program (MMP) on smooth or mildly singular X, and conclude that the building blocks of polarized endomorphisms are those on Fano varieties or complex tori and their quotients. This is a joint work with S. Meng.
• EDGE: Jason Lo (CSU) - Behaviour of slope stable sheaves under a Fourier-Mukai transform
  25th May 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: Over the years, Fourier-Mukai transforms have been used in constructing birational maps or isomorphisms between moduli spaces, explaining symmetries in counting invariants, and constructing Bridgeland stability conditions on Abelian threefolds. All these problems involve the question of how stable objects behave under a Fourier-Mukai transform. In this talk, I will consider this question for slope-stable torsion-free sheaves on a product elliptic threefold. If time permits, I will also discuss this question for stable 1-dimensional sheaves.
• EDGE: Xiaolei Zhao (Northeastern) - Canonical points on K3 surfaces and hyper-Kähler varieties.
  4th May 2017, 2:10pm to 3:00pm JCMB 5323 -- Show/hide abstract
  Abstract: The Chow groups of algebraic cycles on algebraic varieties have many mysterious properties. For K3 surfaces, on the one hand, the Chow group of 0-cycles is known to be huge. On the other hand, the 0-cycles arising from intersections of divisors and the second Chern class of the tangent bundle all lie in a one-dimensional subgroup. A conjecture of Beauville and Voisin gives a generalization of this property to hyper-Kähler varieties. In my talk, I will recall these beautiful stories, and explain a conjectural connection between the K3 surface case and the hyper-Kähler case. If time permits, I will also explain how to extend this connection to Fano varieties of lines on some cubic fourfolds. This is based on a joint work with Junliang Shen and Qizheng Yin.
• EDGE: Richard Szabo (Heriot-Watt) - Orientifolds and Real bundle gerbes
  20th April 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: We describe a generalisation of the notions of bundle gerbe and bundle gerbe modules appropriate to spaces with involution, which realise a twisted version of Atiyah's Real K-theory. This construction is applied to the topological classification of D-branes in orientifolds of string theory with H-flux.
• EDGE: Emilio Franco (Porto) - Moduli spaces of Lambda-modules on abelian varieties
  6th April 2017, 2:40pm to 3:30pm Seminar room JCMB 5323 -- Show/hide abstract
  Abstract: Let \Lambda be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of \Lambda-modules for particular choices of \Lambda. Simpson studied the moduli problem for the classification of \Lambda-modules over Kahler varieties, proving the existence of a moduli space Lambda-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of \Lambda-modules of rank 1. We also proof that polystable \Lambda module decompose as a direct sum of rank 1 \Lambda-modules. This allow us to describe the module spaces for arbitrary rank and trivial Chern classes.
• EDGE/EMPG: Dennis The (University of Tromsø) - Exceptionally simple PDE
  5th April 2017, 12:10pm to 1:00pm JCMB 5323 -- Show/hide abstract
  Abstract: In back-to-back articles in 1893, Cartan and Engel gave the first realisations of $​G_2$, the smallest of the exceptional complex simple Lie groups, as the symmetries of a geometric object. I will show how to generalize this story in a remarkably uniform manner to obtain analogous explicit geometric realisations for any complex simple Lie group (…except for $SL_2$).
• EDGE: Alexey Bondal (Steklov/Higher School of Economics) - Flobers: Flops and Schobers
  4th April 2017, 12:10pm to 1:00pm JCMB 4312 -- Show/hide abstract
  Abstract: I shall explain how the ideology of schobers, the categorification of the perverse sheaves on stratified topological spaces, should amplify our understanding of some parts of the Minimal Model Program in Birational Geometry.
• EDGE: Philip Boalch (Paris Sud) - Nonperturbative hyperkahler manifolds and H3 surfaces
  30th March 2017, 2:40pm to 3:30pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: In the 80s and 90s many hyperkahler manifolds were constructed out of additive/Lie algebraic objects such as coadjoint orbits, cotangent bundles and quiver representations. I'll explain how some of these have multiplicative or ``more transcendental'' analogues in wild nonabelian Hodge theory (i.e. moduli of Higgs bundles or connections or solutions of Hitchin's equations, on noncompact curves). In particular 12 deformation classes of complete hyperkahler four-manifolds occur, including multiplicative analogues of the hyperkahler manifolds of Eguchi-Hanson and Atiyah-Hitchin. We call them H3 surfaces in honour of Higgs, Hitchin and Hodge. For the most part we will work algebraically and describe the construction of the underlying holomorphic symplectic manifolds as finite dimensional multiplicative symplectic quotients. This construction was recently completed in joint work with D. Yamakawa, extending the author's construction in the untwisted case 2002-2014.
• EDGE: Mihai Fulger (Princeton) - Seshadri constants for curves
  23rd March 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: The Seshadri constants of nef divisors are important local measures of positivity of the divisor. We develop a natural dual theory for movable curves and show how it mirrors the case of divisors. For example they give a criterion of "ampleness", their vanishing locus is an analogue of the nonample locus for divisors, and in a sense they measure jet separation at smooth points. We also present examples.
• EDGE: Jenia Tevelev - The Craighero-Gattazzo surface is simply-connected
  16th March 2017, 4:10pm to 5:00pm JCMB 5323 -- Show/hide abstract
  Abstract: We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface, we use an algebraic reduction mod p technique and deformation theory. Joint work with Julie Rana and Giancarlo Urzua.
• EDGE: Giovanni Mongardi (Milan) - The last involutions
  16th March 2017, 2:10pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: In this joint work with C. Camere and G. and M. Kapustka, we give two construction of the last two families of nonsymplectic involutions on deformations of K3^[2] using twisted sheaves on K3 surfaces and special lagrangians, proving that these families are also unirational.
• EDGE: Marcos Jardim - Moduli spaces of instanton sheaves on projective space
  9th March 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: Instanton bundles were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970s as the holomorphic counterparts, via twistor theory, to anti-self-dual connections (a.k.a. instantons) on the sphere S^4. We will revise some recent results regarding some of the basic geometrical features of their moduli spaces, and on its possible degenerations. We will describe the singular loci of instanton sheaves, and how these lead to new irreducible components of the moduli space of stable sheaves on the projective space.
• EDGE: Yanki Lekili (KCL) - Fukaya categories of plumbings and multiplicative preprojective algebras
  2nd March 2017, 2:10pm to 3:00pm JCMB Seminar room (5th floor) -- Show/hide abstract
  Abstract: Given an arbitrary graph, I will construct an open symplectic 4-manifold obtained by plumbing cotangent bundles of 2-spheres. I will then describe a rigorous and explicit computation of the wrapped Fukaya category of this symplectic 4-manifold using techniques of Legendrian surgery. It turns out that the endomorphism algebra of a generating set of objects of this Fukaya category can be identified with the (derived) multiplicative preprojective algebra associated to the original graph. In the case, the underlying graph is Dynkin, we have a (derived) Koszul duality between wrapped and compact Fukaya categories. This leads to computations of Hochschild cohomology of these categories in this case. In the case the graph is of extended Dynkin type, a version of homological mirror symmetry can be confirmed as a result of our computations. This talk is an update on a previous joint work with Tolga Etgu.
• EDGE: Boris Kruglikov(Tromsø) - The gap phenomenon and submaximally symmetric structures
  16th February 2017, 2:10pm to 3:00pm JCMB 5215 -- Show/hide abstract
  Abstract: In 2014 together with Dennis The we resolved the gap problem in complex or split-real parabolic geometry, i.e. we computed the amount of submaximal symmetry for every geometry in the class. Results of this type have been known for selected geometries since Ricci, Tresse, Fubini, Cartan, Egorov, Kobayashi, Sinyukov, Yano and some others via specific techniques. However it was in our paper that we first presented a universal solution for a large class of geometries, including conformal structures, systems of second order ODE, almost Grassmanian and Lagrangian structures, generic parabolic distributions, exceptional geometries etc. In later development we covered CR-structures, c-projective structures and some other real (non-split) specification. I will review the results and overview further developments and problems. References: [1] Boris Kruglikov, Dennis The, The gap phenomenon in parabolic geometries, Journal für die reine und angewandte Mathematik (Crelle's Journal) DOI 10.1515/crelle-2014-0072 (2014). [2] Boris Kruglikov, Vladimir Matveev, Dennis The, Submaximally symmetric c-projective structures, International Journal of Mathematics 27, No. 3, 1650022 - 34 pp, (2016). [3] Boris Kruglikov, Submaximally symmetric CR-structures, Journal of Geometric Analysis DOI: 10.1007/s12220-015-9663-x (2015). [4] Boris Kruglikov, Henrik Winther, Lenka Zalabova, Submaximally symmetric quaternionic structures, arXiv: 1607.02025 (2016).
• EDGE: Ruadhai Dervan (Cambridge) - K-stability for Kähler manifolds
  9th February 2017, 2:10pm to 3:00pm JCMB Seminar room 5323 -- Show/hide abstract
  Abstract: K-stability is an important algebro-geometric concept introduced by Tian and Donaldson, which is related to the existence of "canonical" Kähler metrics on smooth projective varieties. We formulate a notion of K-stability for Kähler manifolds, and prove that Kähler manifolds admitting canonical Kähler metrics are K-stable. We also show how Kähler techniques can prove stronger results even in the projective case. Some of this is joint work with Julius Ross.
• EDGE: George Dimitrov (ICTP) - Norms of non-commutative projective spaces via Bridgeland stability conditions
  2nd February 2017, 2:10pm to 3:00pm JCMB Seminar Room 5323 -- Show/hide abstract
  Abstract: Tom Bridgeland assigned to any triangulated category a complex manifold: the space of stability conditions on it. In a joint work with Ludmil Katzarkov we prove that the Bridgeland stability spaces on wild Kronecker quivers are CxH and these calculations suggest a new notion of a norm. To a triangulated category T which has property of a phase gap, we attach a number ||T||_epsilon \in [0,(1-epsilon)\pi] depending on a parameter epsilon \in (0,1). In this talk, I will tell more about this.
• EDGE: Alex Perry (Columbia University)- Derived categories of Gushel-Mukai varieties
  26th January 2017, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: I will discuss the derived categories of Fano varieties of Picard number 1, degree 10, and coindex 3. In particular, I will describe an interesting semiorthogonal component of the derived category of such a variety, and discuss its behavior for some birationally special families of fourfolds. This is joint work with Alexander Kuznetsov.
• EDGE: Caucher Birkar(Cambridge)- Birational geometry of Fano varieties
  19th January 2017, 2:10pm to 3:00pm JCMB 1501
• EDGE: Jinhyung Park(KIAS) - Newton-Okounkov bodies and asymptotic invariants of divisors
  8th December 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: A Newton-Okounkov body is a convex body in Euclidean space associated to a divisor on an algebraic variety with respect to an admissible flag. After briefly recalling basics of Newton-Okounkov bodies of ample or big divisors, I introduce two natural ways to associate Newton-Okounkov bodies to pseudoeffective divisors. We then study various asymptotic invariants of pseudoeffective divisors using these convex bodies. This is joint work with Sung Rak Choi, Yoonsuk Hyun, and Joonyeong Won.
• EDGE: Theo Raedschelders (Vrije Universiteit Brussel) - Derived categories of noncommutative quadrics and Hilbert schemes of points
  1st December 2016, 2:10pm to 3:00pm JCMB1501 -- Show/hide abstract
  Abstract: Recent work of Orlov's suggests that for a smooth projective rational surface S, it should be possible to embed Perf-S into Perf-M_S, where M_S is also smooth projective and represents some moduli problem on S. Moreover, noncommutative deformations of S should embed into commutative deformations of M_S. I will discuss Orlov's work and consider the specific example of a smooth quadric surface in more detail. This is joint work with Pieter Belmans.
• EDGE: Maxym Fedorchuk (Boston) - Invariant theory of Artinian Gorenstein algebras
  24th November 2016, 2:10pm to 3:00pm JCMB1501 -- Show/hide abstract
  Abstract: I will discuss the interplay between hypersurface singularities, their Milnor algebras, and classical invariant theory of homogeneous forms. In particular, I will prove that a contravariant that associates to a smooth homogeneous form the Macaulay inverse system of its Milnor algebra preserves GIT stability. I will discuss some applications of this result, for example to the direct sum decomposability of polynomials, and many related open problems.
• EDGE: Michael Wemyss (Glasgow) - On the Classification of 3-fold Flops
  17th November 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstract
  Abstract: I will outline a still largely conjectural framework in which smooth flops of irreducible curves in 3-folds are classified by certain elements of the free algebra in two variables. The correspondence is explicit, and by manipulating the noncommutative side, we are able to use this framework to produce many new examples of 3-fold flops, the first since the early 1980s. In particular, we show that there are more type D flops than simply the standard Laufer-type examples, and we also give the first examples in type E. Our flops come equipped with their Gopakumar--Vafa invariants, and I will discuss some of the consequences of our constructions to curve counting invariants. This is all joint work with Gavin Brown.
• EDGE: Piotr Pragacz (Polisch Academy of Sciences) - Duality on Grassmann bundles and applications
  10th November 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: The duality of Schubert calculus allows one to present any class on a Grassmannian as an integer combination of Schubert classes. We state and prove a duality theorem on a Grassmann bundle using its Gysin map and skew Schur functions. We give new Gysin formulas for flag bundles and an alternative derivation of the Kempf-Laksov formula. This is a joint work with Lionel Darondeau.
• EDGE: Dimitra Kosta (Edinburgh) - Maximum Likelihood Estimation for models corresponding to toric del Pezzo surfaces
  3rd November 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstract
  Abstract: I will present the correspondence between some statistical models and toric varieties and show how one can obtain a closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric del Pezzo surfaces with Du Val singular points.
• EDGE: Seung-Jo Jung (KIAS, Seoul) - G-constellations and quotient singularities
  27th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: Let G be a finite group in GL_n(C). A G-equivariant sheaf F on C^n is called a G-constellation if H^0(F) is isomorphic to the regular representation of G as a G-representation. In this talk, we discuss moduli interpretations of many interesting birational models of C^n/G using G-constellations.
• EDGE: Diletta Martinelli (Edinburgh) - On the number of minimal models of a smooth threefold of general type
  20th October 2016, 2:10pm to 3:10pm JCMB 1501 -- Show/hide abstract
  Abstract: Finding minimal models is the first step in the birational classification of smooth projective varieties. After it is established that a minimal model exists some natural questions arise such as: is it the minimal model unique? If not, how many are they? After recalling all the necessary notions of the Minimal Model Program, I will explain that varieties of general type admit a finite number of minimal models. I will talk about a recent joint project with Stefan Schreieder and Luca Tasin where we prove that in the case of threefolds this number is bounded by a constant depending only on the Betti numbers. I will also show that in some cases it is possible to compute this constant explicitly.
• EDGE: Jesus Martinez-Garcia (Max Planck) - Moduli space of cubic surfaces and their anticanonical divisors
  13th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate this compactifications to the (conjectural) moduli space of log K-semistable pairs showing that any log K-stable pair is an element of our moduli and that there is a canonically defined CM line bundle that polarizes our moduli. This is joint work with Patricio Gallardo (University of Georgia) and Cristian Spotti (Aarhus University).
• EDGE: Julian Holstein (Lancaster) - The derived period map
  6th October 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: In this talk I develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. (I will not assume previous knowledge of derived algebraic geometry.)
• EDGE: Roberto Fringuelli (Edinburgh) - The Picard group of the universal moduli space of vector bundles on stable curves.
  29th September 2016, 2:10pm to 3:00pm JCMB 1501 -- Show/hide abstract
  Abstract: In this talk, we present the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We show some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of a universal family of an open substack of the rigidification. In the remaining time, we discuss some consequences for the associated moduli varieties.
• EDGE: Igor Pak (UCLA) - What is a formula?
  19th September 2016, 2:10pm to 3:10pm JCMB 6201 -- Show/hide abstract
  Abstract: Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will give a mini-survey aiming to answer these questions. At the end, I will present some recent results counting certain permutation classes, and finish with open problems.
• EDGE: Benjamin Bakker (Humboldt-Universitaet Berlin) - The birational geometry of complex ball quotients
  21st June 2016, 1:45pm to 2:45pm JCMB 5327 -- Show/hide abstract
  Abstract: Quotients of the complex ball by discrete groups of holomorphic isometries naturally arise in many moduli problems---for instance, those of low genus curves, del Pezzo surfaces, certain K3 surfaces, and cubic threefolds. On the other hand, the complex ball is the only bounded symmetric domain to admit nonarithmetic lattices, so birationally classifying such quotients is of particular interest. In joint work with J. Tsimerman, we show that in dimension $n\geq4$ every smooth complex ball quotient is of general type, and further that the canonical bundle $K_{\overline{X}}$ is ample on the toroidal compactification $\overline{X}$ for $n\geq 6$. The proof uses a hybrid technique employing both the hyperbolic geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. We will also discuss applications to bounding the number of cusps and the Green--Griffiths conjecture.
• EDGE: Asher Auel (Yale) - Brill-Noether theory for cubic fourfolds
  2nd June 2016, 2:00pm to 3:00pm JCMB 4325A -- Show/hide abstract
  Abstract: Certain cubic fourfolds have K3 surfaces associated to them via Hodge theory. I will discuss how the Brill-Noether properties of special divisors on curves in those associated K3 surfaces can be reflected in the geometry of the cubic fourfolds. For example, cubic fourfolds containing two disjoint planes have an associated K3 surface of degree 14 that is Brill-Noether special in the sense of Lazarsfeld and Mukai. One application is to a description of the boundary of the locus of cubic fourfolds that have pfaffian presentations.
• EDGE: Dario Beraldo (Oxford) - On the notion of temperedness in geometric Langlands
  5th May 2016, 2:00pm to 4:00pm JCMB 6206 -- Show/hide abstract
  Abstract: Using the rich algebraic structure enjoyed by Hochschild cochains, I will define a new notion of "sheaf of categories", called $ShvCat^{HC}$, on an arbitrary prestack. This theory resembles the theory of D-modules, in the same way as Gaitsgory's notion of sheaf of categories resembles the theory of quasi-coherent sheaves. For a nice stack \mathcal{Y}, the \infty-category ShvCat^{HC}(\mathcal{Y}) is equivalent to the \infty-category of modules for an explicit monoidal DG category, denoted QCoh^{HC}(\mathcal{Y}). This category is closely related to ind-coherent sheaves on the formal completion of the diagonal of \mathcal{Y}. As an application, I will show that the two DG categories appearing in the geometric Langlands correspondence, Dmod(Bun_G) and IndCoh_N(LocSys_{\check{G}}), are both equipped with a QCoh^{HC}(LocSys_{\check G})-action. The compatibility of geometric Langlands with geometric Satake predicts that these actions be intertwined by the conjectural Langlands equivalence. The action of QCoh^{HC}(\mathcal Y) on a DG category \mathcal{C} allows to define tempered objects of \mathcal{C}. In particular, we obtain a new definition of tempered D-modules on Bun_G, thereby proving a conjecture of Arinkin and Gaitsgory.
• EDGE: Alexey Bondal (Steklov/IPMU/Higher School of Economics) - Noncommutative moduli of genus 0 n-punctured curves
  21st April 2016, 2:00pm to 3:00pm JCMB 4325B -- Show/hide abstract
  Abstract: Generalizations of the moduli space of stable n-punctured curves of genus zero will be discussed. One such generalized version is assigned to any finite dimensional algebra. In some cases, they are identified with moduli of quiver representations, thus giving a new geometric point of view on the later.
• EDGE: Emile Bouaziz (Edinburgh) - The Twisted Chiral de Rham Complex
  31st March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: The Chiral de Rham Complex of a smooth complex variety is an infinite dimensional refinement of the usual de Rham complex whose cohomology has modular properties. In joint work in progress with Ian Grojnowski we are studying an analogous object in the presence of a potential on the variety, refining instead the twisted de Rham complex, familiar from vanishing cycles theory.
• EDGE: Kai Behrend (UBC) - The spectrum of the inertia operator on the motivic Hall algebra
  24th March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: Following an idea of Bridgeland, we study the operator on the K-group of algebraic stacks, which takes a stack to its inertia stack. We prove that the inertia operator is diagonalizable when restricted to nice enough stacks, including those with algebra stabilizers. We use these results to prove a structure theorem for the motivic Hall algebra of a projective variety, and give a more conceptual definition of virtually indecomposable stack function. This is joint work with Pooya Ronagh.
• EDGE: Bruce Bartlett (Oxford) - Three-dimensional bordism representations
  17th March 2016, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstract
  Abstract: The collection of all oriented compact 1-, 2- and 3-manifolds can be assembled into a bicategory, which has a fairly simple "generators and relations" presentation obtained using Morse theory. Via the graphical calculus of "internal string diagrams", a representation of this bordism bicategory corresponds to an algebraic structure known as a modular category. I will give an overview of these results and speculate on the relationship to factorization homology.
• EDGE: Roland Abuaf (IHES) - Compact hyperkähler categories
  10th March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: Compact Calabi-Yau categories play a preeminent role in non-commutative geometry and in the mathematical background of string theory. Indeed, many manifestations of Kontsevich's Homological Mirror Symmetry conjectures are best understood when interpreted in the framework of 3-Calabi-Yau categories. In this talk, I want to introduce and discuss the basic properties of a new class of compact Calabi-Yau categories : the hyperkaehler categories. They are categorical analogues of compact hyperkaehler manifolds. The theory of non-commutative resolutions of singularities allows us to construct a large number of deformation classes of such categories in each dimension. For instance, I can construct at least 243 deformation classes of such categories in dimension 4 (compare with the only 2 deformation classes of hyperkaehler spaces of dimension 4 which are known in commutative geometry). If time permits, I would like to discuss a specific modular example (of dimension 4), for which Hochschild co-homology reveals some very intriguing features.
• EDGE: Igor Krylov - Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group
  3rd March 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: Cremona group of rank b is the group of birational transformations of the projective n-space. One was to study Cremona group is to study its finite subgroups. This problem can be translated to the geometric language: instead of subgroups of Cremona group isomorphic to a group G we can study rational G-Mori fiber spaces. This idea works particularly well for simple subgroups of Cremona group. I prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration X_n of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K^2-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that X_n are superrigid, in particular not rational, for n>2.
• EDGE: Kaie Kubjas (Aalto) - Semialgebraic geometry of nonnegative and psd rank
  25th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: One of many definitions gives the rank of an $m \times n$ matrix $M$ as the smallest natural number $r$ such that $M$ can be factorized as $AB$, where $A$ and $B$ are $m \times r$ and $r \times n$ matrices respectively. In many applications, we are interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank and are closely related to mixture models in statistics. Another rank I will consider in my talk is the positive semidefinite (psd) rank. Both nonnegative and psd rank have geometric characterizations using nested polytopes. I will explain how to use these characterizations to derive a semialgebraic description of the set of matrices of nonnegative/psd rank at most $r$ in some small cases, and to study boundaries of this set. The talk is based on joint work with Rob H. Eggermont, Emil Horobet, Elina Robeva, Richard Z. Robinson, and Bernd Sturmfels
• EDGE: Mohammad Akhtar (IHES) - Mutations and the Classification of Fano Varieties
  18th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: The classification of Fano varieties is an important long-standing problem in algebraic geometry. Mirror symmetry predicts that this problem should be equivalent to classifying a suitable class of Laurent polynomials up to an appropriate notion of equivalence. Recent work of Coates, Corti, Golyshev et al. suggests that the correct equivalence relation to impose is algebraic mutation of Laurent polynomials. This talk will introduce algebraic mutations and discuss the notion of combinatorial mutations, which are transformations of lattice polytopes induced by algebraic mutations of Laurent polynomials supported on them. Our focus will be on the case of surfaces, where the theory is particularly rich. Particular attention will be given to the role played by combinatorial mutations in the classification of Fano orbifold surfaces. This is joint work with Tom Coates, Alessio Corti and Alexander Kasprzyk.
• EDGE: Yuki Hirano (Tokyo Metropolitan University/Edinburgh) - Derived Knoerrer periodicity and Orlov's theorem for gauged Landau-Ginzburgh models
  11th February 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: We prove Kn"orrer periodicity type equivalence between derived factorization categories of gauged LG models, which is an analogy of a theorem proved by Shipman and Isik independently. Combining the Kn"orrer periodicity type equivalence and the theory of variations of GIT quotients, we obtain a LG version of a Orlov's theorem which describes semi-orthogonal decompositions between categories of graded matrix factorizations and derived categories of hypersurfaces in projective spaces.
• EDGE: Matilde Marcolli (Caltech) - Motives in Quantum Field Theory
  1st February 2016, 1:00pm to 2:00pm JCMB 5327 -- Show/hide abstract
  Abstract: I will give an overview of the algebro-geometric approach to Feynman integral in perturbative quantum field theory and the occurrence of motives and periods in parametric Feynman integrals in momentum space, focusing on joint work with Paolo Aluffi.
• EDGE: Roberto Svaldi (Cambridge) - A geometric characterization of toric varieties.
  28th January 2016, 2:10pm to 3:00pm JCMB 5327 -- Show/hide abstract
  Abstract: Given a pair (X, D), where X is a proper variety and D a divisor with mild singularities, it is natural to ask how to bound the number of components of D. In general such bound does not exist. But when -(K_X+D) is positive, i.e. ample (or nef), then a conjecture of Shokurov says this bound should coincide with the sum of the dimension of X and its Picard number. We prove the conjecture and show that if the bound is achieved, or the number of components is close enough to said sum, then X is a toric variety and D is close to being the toric invariant divisor. This is joint work with M. Brown, J. McKernan, R. Zong.
• Big data/EDGE seminar: Marta Casanellas Rius (Barcelona) - The link between pure mathematics and phylogenetics
  22nd January 2016, 2:10pm to 3:00pm JCMB 4325A -- Show/hide abstract
  Abstract: Many of the usual statistical evolutionary models can be viewed as algebraic varieties and a deep understanding of these varieties may solve open problems in phylogenetics. We show how different mathematical areas such as linear and commutative algebra, algebraic geometry, group representation theory, or numerical methods show up when one studies these varieties. Moreover, we prove that an in-depth geometric study leads to improvements on phylogenetic reconstruction methods. We illustrate these improvements by showing results on simulated data and by comparing them to widely used methods in phylogenetics. In order to follow this talk it is not required a previous knowledge on algebraic varieties or phylogenetics.
• EDGE: Lotte Hollands (Heriott-Watt) - Spectral networks and the T3 theory
  3rd December 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: A spectral network is a collection of trajectories on a (punctured) Riemann surface. Given a spectral network we can define a notion of "abelianization", which relates flat SL(K) connections on the Riemann surface to flat C^* connections on a covering. For any spectral network abelianization gives a construction of a local Darboux coordinate system on the moduli space of flat SL(K) connections. In this seminar we will take a look at a particularly rich example for rank K=3, with interesting applications to WKB analysis and to quantum physics. This is based on work in progress with Andy Neitzke.
• EDGE seminar: James Pascaleff (UIUC) - Equivariant structures on Lagrangian submanifolds
  19th November 2015, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: In this talk, based on joint work with Y. Lekili and N. Sheridan, I will describe some equivariant structures in symplectic geometry and their relationship to mirror symmetry. This will be illustrated with some low dimensional examples.
• EDGE seminar: Wouter Castryck (Gent) - Geometric invariants encoded in the Newton polygon
  19th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Let f be a sufficiently generic bivariate Laurent polynomial with Newton polygon D, and let C be the plane curve defined by f. It is known by work of Khovanskii that the geometric genus of C equals the number of points in the interior of D having integer coordinates. Building on recent work of Kawaguchi, we give similar combinatorial interpretations for various other geometric invariants, such as the gonality, the Clifford index and the scrollar invariants. We will also report on work in progress, where the hope is to find combinatorial interpretations for the graded Betti numbers of the canonical model, a problem which is motivated by Green's canonical syzygy conjecture. The talk will contain joint work with Filip Cools and Alexander Lemmens.
• EDGE seminar: Renzo Cavalieri (CSU) - Open invariants and Crepant Transformations
  12th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited it as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Corti, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and make some verifications. Our approach is well tuned with Iritani's approach to the CRC via integral structures, and it seems to suggest that open invariants should play a prominent role in mirror symmetry.
• EDGE seminar: Oscar García-Prada (ICMAT) - Higgs bundles and representations of surface groups
  12th November 2015, 1:10pm to 2:00pm JCMB 6311 -- Show/hide abstract
  Abstract: In this talk I will show how Higgs bundles over a compact Riemann surface can be used to study the moduli space of representations of the fundamental group of the surface in a non-compact semisimple Lie group. Special attention will be given to the case in which the symmetric space defined by the Lie group is of Hermitian type.
• EDGE seminar: Nikita Kalinin (Geneva) - Tropical geometry in sandpiles, singularity theory and Legendrian geometry
  5th November 2015, 3:10pm to 4:00pm JCMB 6311 -- Show/hide abstract
  Abstract: We encounter tropical curves and their different aspects in three different situations. 1) A planar tropical curve may appear as the scaling limit of the result of the relaxation of a maximal stable state of a sandpile configuration on a large polygon, perturbed in several points. 2) A singular point on a planar curve over a valuation field ``influences’’ a part of the Newton polygon of the curve. This allows us to prove some Nagata’s type estimates for curves in toric varieties, just by area counting. 3) We consider algebraic curves in $\mathbb C P^3$ tangent to the distribution, given by the form $ydx-xdy+wdz-zdw=0$. If the tropical limit of a family of such curves contains a part, locally looking like a tropical line, then we can observe some nice divisibility property. This property is a corollary of a much more general lemma in tropical integration theory (joint work in progress with G. Mikhalkin). This talk will contain no proofs and a lot of pictures.
• EDGE seminar: Agnieszka Bodzenta (Edinburgh) - Perverse schobers and flops
  5th November 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: I will consider X and Z related by flops f: X \to Y, g: Z \to Y of relative dimensions one and the fiber product W of X and Z over Y. I will show that an appropriate quotient of the derived category of W admits a semi-orthogonal decomposition into the derived category of X and derived category of the null category for g. I will prove that derived categories of null categories of f and g form a spherical pair in the quotient of D(W) and that the associated spherical twist is the flop-flop equivalence of D(X). This is joint work with A. Bondal.
• Hodge seminar: Sarah Zerbes (UCL) - Euler systems and the conjecture of Birch and Swinnerton-Dyer
  2nd November 2015, 1:10pm to 2:00pm JCMB 5327 -- Show/hide abstract
  Abstract: The Birch—Swinnerton-Dyer conjecture is one of the most mysterious open problems in number theory, predicting a relation between arithmetic objects, such as the points on an elliptic curve, and certain complex-analytic functions. A powerful approach to the conjecture is via a tool called an ‘Euler system’. I will explain the idea behing this approach, and some recent new results in this direction.
• EDGE: Alexander Pukhlikov (Liverpool) - Birationally rigid Fano-Mori fibre spaces
  29th October 2015, 3:10pm to 4:00pm JCMB 6206 -- Show/hide abstract
  Abstract: Starting from the pioneer work of Iskovskikh and Manin on three-dimensional quartics, birational rigidity has been gradually understood as one of the key phenomena in higher-dimensional birational geometry. I will discuss the recent progress in birational rigidity of fibre spaces over a non-trivial base in the general context of higher-dimensional birational geometry of rationally connected varieties.
• EDGE: Jørgen Rennemo (Oxford) - Homological projective duality for Sym^2 P^n
  22nd October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: In 2011, Hosono and Takagi constructed an interesting example of two derived equivalent, non-birational Calabi-Yau 3-folds. This example be explained by phrasing it in terms of Kuznetsov's theory of homological projective duality. With this as motivation, we compute the homological projective dual of Sym^2 P^n, and by taking n = 4 we recover Hosono and Takagi's example. I will explain this result and its proof, which is based on the honestly-less-complicated-than-it-sounds technique of "gauged LG models and variation of GIT stability".
• EDGE: Joseph Karmazyn (Bath) - Noncommutative Knörrer periodicity
  15th October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: The singularity category of a type $A_n$ Kleinian surface singularity can be understood by using Kn\"{o}rrer periodicity to show it is equivalent to the singularity category of the finite dimensional algebra $k[x]/(x^n)$. However, Knörrer periodicity only applies to hypersurface singularities so cannot be used to study more general cyclic quotient surface singularities​. I will discuss joint work with Martin Kalck that extends the Knörrer periodicity phenomenon outside of the Kleinian case by producing finite dimensional algebras with singularity categor​ies equivalent to general cyclic quotient surface singularities. Intriguingly the algebras produced are noncommutative in general. The proof of this result proceeds via studying different subcategories of the derived category of a type $A$ configuration of rational curves in a surface, and it provides a geometric interpretation of these equivalences that is new even in the Kleinian case.
• EDGE: Dmitri Orlov (Steklov) - Derived noncommutative schemes, their geometric realizations, and quiver algebras
  8th October 2015, 3:10pm to 4:00pm JCMB 6311
• EDGE: Victor Przyjalkowski (Steklov) - On Hodge numbers for Landau-Ginzburg models
  8th October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Abstract: We discuss various definitions of Hodge numbers for Landau-Ginzburg models and their relation to Hodge numbers of Fano varieties.
• EDGE: Igor Krylov (Edinburgh) - Rationally connected non Fano type varieties
  1st October 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Abstract: The class of varieties of Fano type is a generalization of Fano varieties which is very well behaved under the MMP. It is known that all varieties of Fano type are rationally connected. The converse is true in a sense in dimension 2. I will give counterexamples in dimension 3 and higher using the technique of singularities of linear systems which is typically used for proving birational rigidity.
• EDGE: Matthew Woolf (Edinburgh) - Rational curves on general hypersurfaces in positive characteristic
  24th September 2015, 2:10pm to 3:00pm JCMB 6311 -- Show/hide abstract
  Abstract: Abstract: In characteristic 0, it is a classical fact that smooth hypersurfaces of large degree cannot be rationally connected, i.e., have a rational curve passing through a general pair of points. In positive characteristic, though, there are examples of smooth hypersurfaces of a fixed dimension and characteristic and arbitrarily large degree which are rationally connected. In this talk, we will see that nevertheless, the general hypersurface outside of the Fano range is not rationally connected. We will also discuss some applications of this result to other questions about rational curves on hypersurfaces in positive characteristic