### Schedule of talks for 2014/15:

##### Semester 2

January 23, 2:30pm | Noah White | Schubert calculus and the cactus group |

Abstract: Henriques and Kamnitzer defined an action of the “cactus group” on crystals for semisimple Lie algebras. I will explain, in type A, the connection between this action and a conjecture of Bonnafe and Rouquier on the Kazhdan-Lusztig cells of coxeter groups (in type A, the symmetric group). I will show how this action appears as the monodromy of a certain covering space defined using Schubert calculus or alternatively using the representation theory of the symmetric group and certain generalisations of the Jucys-Murphy elements called the Gaudin Hamiltonians. Notes | ||

January 30, 2:30pm | Jonathan Hickman | Three Perspectives on the Kakeya Problem |

Abstract: Imagine a large collection of long thin tubes (e.g. uncooked spaghetti). If all the tubes point in different directions, then they cannot overlap too much. The Kakeya conjecture essentially asks for a quantitative form of this statement. Many connections have emerged between this conjecture and major questions in Fourier analysis, additive combinatorics, PDE, computer science... and, in light of this, Kakeya has become one of the most important open problems in harmonic analysis. In this simple and very accessible talk I will focus on variants of the Kakeya problem in different algebraic/topological settings, posed in view of modelling certain aspects of the Euclidean case. | ||

February 6, 2:30pm | Rebecca Tramel | Bridgeland stability on the P1 |

Bridgeland stability conditions are a generalization of slope stability of vector bundles on a smooth projective curve to a notion of stability for objects in the derived category of a projective variety. For a such a variety, the space of stability conditions is a manifold. In many examples there are nice connections between the geometry of the variety and the stability manifold. I will give an introduction to Bridgeland stability, using the example of the projective line, and describe this manifold explicitly. If time permits I will give some examples of what happens for surfaces as well. Notes | ||

February 18, 2:00pm | Caterina Campagnolo | Invariants of Surface Bundles |

February 27, 2:30pm | Igor Krylov | Linear System, Birational Map, its Singularities and its Image. |

Every algebraic map from a projective variety is given by some linear system. I will talk about linear systems on Fano varieties which give a birational map to a minimal model and what conditions does it imply on singularities of the system. In particular I will prove non-rationality of smooth quartic threefold. | ||

March 6, 2:30pm | Chunyi Li | Basic Affine Geometric Invariant Theory and Quivers |

In algebraic geometry, there are several ways to describe the quotient space with respect to a Lie group (e.g. C^n quotient by the C*-action, n by m matrices quotient by GL_n). Unfortunately, due to Sjoerd’s slogan: `if objects have non-trivial isomorphisms no (fine) moduli space can exist’, almost all the descriptions are not very `geometric’. I will talk about the geometric invariant theory construction which has some geometric intuition. Some basic examples from the quiver representation will be discussed, as well as an interesting `open’ question. | ||

March 13, 2:30pm | Andrius Stikonas | Applications of Differential Geometry in Physics |

Abstract: In this talk I will show how we can apply language of differential geometry and principal bundles to write equations of electromagnetism in a very compact form. A very similar approach works for geometry of curved manifolds and allows quicker calculation of Riemann tensor in orthonormal frame. Notes | ||

March 20, 1:00pm | Sjoerd Beentjes | Something something moduli |

Abstract: In ideal situations, classifying a certain type of object up to isomorphism results in a nice moduli space parametrising these isomorphism classes. However, one often hears the slogan „if objects have non-trivial isomorphisms no (fine) moduli space can exist”. In this talk I will define the concepts of coarse and fine moduli space, and give some examples related to this slogan. Then, by passing through the idea of doing geometry on the functor of points of a scheme, I will try to motivate the notion of an algebraic (DM) stack as a solution to the non-existence of a fine moduli space in general. Finally, I would like to treat a (very) modest example of a quotient stack in detail. | ||

March 27, 2:30 pm | Chris Campbell | Every Non-Commutative P2 Contains an Elliptic Curve |

Abstract: The revolutionary approach to algebraic geometry that the French School of the last century developed allows many ideas from classical projective geometry to be used in noncommutative algebra. I will discuss some of these ideas and show how they can be used to determine a wider set of rings which these tools allow us to study. I will provide the framework for a proof of the famous classification result of Artin and Schelter on these rings in low dimensions, and hopefully have time to discuss the direction this classification is now taking. There will be some overlap with my previous talk, my aim is to make the title of the talk make sense by the end! | ||

April 30, 2:30pm | Joe Karmazyn | G-Hilbert schemes. |

Abstract: The G-Hilbert scheme was introduced by Ito and Nakamura as a moduli interpretation of the crepant resolution of a SL2 quotient surface singularity. In dimension three crepant resolutions may not exist and if they do exist they may not be unique. However, Bridgeland, King and Reid proved that the G-Hilbert scheme is always a crepant resolution of a SL3 quotient singularity. |

##### Semester 1

September 26, 2:00pm | Noah White | Galois Theory in Enumerative Geometry |

Abstract: I will define the Galois group of an enumerative problem (for example those coming from Schubert calculus). I will detail Harris’ 1979 theorem that in fact, the Galois group is the same as the monodromy group. The main reference for this talk in Harris’ 1979 paper “Galois groups of enumerative problems”. If time permits I will attempt to explain why this theorem is useful in my work. Reference | ||

October 3, 2:30pm | Patrick Orson | A Brief Introduction to K-Theory |

Abstract: Since their introduction by Grothendieck more than 50 years ago, the ideas of K-theory have become core to several areas of algebraic topology, algebraic geometry, analytic topology and even physics. As such, K-theory now means very different things to different people! I will give a /brief/ round-up of the various meanings of the term 'K-theory' before focussing on the classical algebraic K-theory of a ring. I will define the K_0 and K_1 groups of a ring and give an application of their use in constructing obstructions to certain topological phenomena. Notes | ||

October 10, 2:30pm | Adam Boocher | Geometry and the N_p property |

Abstract: The N_p property for a projective variety X is a measuring of the complexity of the defining equations of X. In this talk I'll motivate the definition with some geometric examples and then state a few open conjectures concerning Veronese embeddings of projective space. | ||

October 24, 2:30pm | Igor Krylov | Minimal Model Program |

Abstract: I will give overview of Minimal Model Program for surface and will briefly explain what were the difficulties for a three-dimensional case. I will tell how were they worked around and will present an example of a flip. | ||

October 31, 11:00am | Tom Avery | Synthetic differential geometry |

Abstract: Some concepts in differential geometry are best understood using the language of infinitesimals, despite the fact that infinitesimals do not appear concretely in the underlying mathematical formalism (e.g. vector fields as "families of infinitesimal transformations"). Synthetic differential geometry (SDG) is an alternative approach to differential geometry in which infinitesimals play a central role, not just for intuition, but in the actual mathematical theory. I will explain the difference between synthetic and analytic geometry in general, before delving into the theory of SDG, and describing the synthetic analogues of some classical geometric notions. I'll also talk a bit about models of the theory, which relate back to classical (analytic) differential geometry. | ||

November 7, 2:30pm, 4319a | Carmen Li | Three-dimensional black holes and descendants |

Abstract: We determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole. | ||

November 14, 2:30pm | Hendrik Suess | Frobenius splitting of toric varieties |

Abstract: In positive characteristic some standard tools of algebraic geometry stop working as Kodaira vanishing or resolution of singularities, but on the other hand a new tool occurs in the toolbox: The Frobenius morphism. E.g. if there exists a section of this morphism we get back our Kodaira vanishing theorem. Varieties admitting such a section are called F-split. There is also the (much) stronger version of a diagonally F-split variety, which for example implies that all ample line bundles are very ample and the corresponding embedding is projectively normal. I will talk on the paper http://xxx.tau.ac.il/abs/0802.4302# by Sam Payne, which characterises diagonally split toric varieties. | ||

December 5, 2:30pm | Przemyslaw Pobrotyn | On the Section Conjecture in anabelian geometry |

Abstract: I will briefly discuss some motivations and ideas behind anabelian geometry and its main conjecture, the Section Conjecture. In particular I will define the arithmetic fundamental group and the section property. If time allows I will discuss the proof of the birational section conjecture for certain classes of fields, as stated in Koenigsmann’s 2003 paper ‘On the Section Conjecture in anabelian geometry’, which is he main reference for the talk. |