# Geometry club

### Schedule of talks for 2013/14:

##### Summer

April 25, 2:30pm | Becca Tramel | Gromov-Witten Theory, Part 1 |

Abstract: A Gromov-Witten invariant counts the number of curves meeting certain conditions on a symplectic manifold. For example, we can ask how many rational curves of degree d pass through 3d-1 points in the projective plane. Kontsevich gave a recursive formula, which reduces solving this question for any d to knowing that there is a unique line through 2 points in the plane. In this talk, I will introduce the moduli space of stable n-pointed maps, and explain how it can be used to give this recursion. Notes | ||

May 9, 2:30pm | Igor Krylov | Gromov-Witten Theory, Part 2 |

Abstract: I will define Gromov-Witten invariants for projective space and I will discuss how to get relations needed for computing Gromov-Witten invariants. Then I will define quantum product and show a connection between recursive relations of Gromov-Witten invariants and associativity of quantum product. Notes | ||

June 13, 2:30pm | Joe Karmazyn | The McKay correspondence, Part 1 |

Notes | ||

June 20, 2:30pm | Evgeny Shinder | The McKay correspondence, Part 2 |

Notes | ||

June 27, 2:30pm | Martin Kalck | The McKay correspondence, Part 3 |

Abstract: TBA | ||

July 11, 2:30pm | Patrick Orson | Double L-theory and localisation |

Abstract: I will give an introduction to the chain complex methods of symmetric algebraic L-theory; a way of algebraically modelling the symmetric properties of topological manifolds, such as Poincare duality, and manifold cobordism. I will describe how I have applied these techniques to the study of the 'doubly-slice' problem in high-dimensional knot theory, by defining an algebraic `double-cobordism' relation. This new algebra admits a localisation exact sequence which I will describe and compare to the classical Witt group and L-group localisation exact sequences of Milnor-Husemoller and Vogel-Ranicki. Notes |

##### Semester 2

January 24, 2:30pm | Patrick Orson and Noah White | Intersection Homology, Part 1 |

Abstract: This will be the first talk in a short series. Intersection homology is an attempt to define a homology theory for singular spaces that has some of the nice properties of singular homology on compact manifolds. In this first talk we will give motivation for and a definition of intersection homology along with many examples. Notes | ||

January 31, 2:30pm | Noah White | Intersection Homology, Part 2 |

Abstract: I will continue last week's introduction to intersection homology by formulating the construction in the language of sheaves. This lets us use the machinery of the derived category to prove properties of intersection homology as well as easily extending to the non-simplicial case. Notes | ||

February 7, 2:30pm | Patrick Orson | Intersection Homology, Part 3 |

Abstract: In the third intersection homology talk I will address Sullivan's original motivating question for intersection homology: "to find a class of spaces with singularities for which the signature of manifolds extends as a cobordism invariant." I will give a flavour of the significance of this deep question and explain some work by Goresky-Macpherson and Siegel in its solution. This will take us into the area of surgery theory! I hope to get far enough to talk about the tight correspondence between this question and Ranicki's algebraic L-theory. Notes | ||

February 14, 2:00pm | Carmen Rovi | Non-multiplicativity of the signature of fibre bundles |

Abstract: The signature $\sigma(M)$ of a closed oriented $n$-dimensional manifold $M$ is defined to be the signature of the intersection form if $n \equiv 0 \pmod 4$, and $0$ otherwise. Consider a fibre bundle $F \to E \to B$. What is the relationship between the signature of the total space $E$ and the signature of base and fibre? I shall discuss the main theorems related to this 60 year old question and my current work on the obstruction to multiplicativity mod $8$. Time permitting I will explain the relation between this problem and Banagl's twisted signature topological field theories. | ||

February 21, 2:30pm | Igor Krylov | Index of canonical singularities |

Abstract: I will give an introduction to singularities of algebraic varieties. I will define an index of a singularity and state Shokurov conjecture. Then I will discuss some special cases, in particular boundedness of index of canonical singularities. Notes | ||

March 7, 2:30pm | Tom Avery | Model Categories |

Abstract: Model categories are an important tool in homotopy theory, and provide a way of generalising it to more abstract contexts. I will describe some of the motivation behind the notion of a model category and state some of the most important results. I will also explain how model structures can be constructed via the small object argument and describe some of the most important examples. The talk should be accessible to people with very little knowledge of homotopy theory (e.g. me), and will only require basic category theory. Notes | ||

March 14, 2:00pm | Joe Karmazyn | Cyclic Quotient Surface Singularities |

Abstract: I'll talk about a very nice geometric example: cyclic quotient surface singularities and their resolutions. These can be described in a completely combinatorial manner, so the talk will require (virtually) no algebraic geometry knowledge and should hopefully show everyone some easy to construct examples of singularities and resolutions. Aside from being a example of resolution of singularities these objects occur in many different guises and areas, arising as the simplest example in many larger stories. If I have time - and anyone wants to know - I'll try to briefly discuss some of these too. Notes | ||

April 11, 2:30pm | Adam Boocher | What's a syzygy? |

Abstract: If X is a projective variety in P^n, then the Hilbert series usually does an excellent job of describing the geometry of this embedding. However, in some cases, the Hilbert function isn't fine enough to detect more subtle geometry. In this talk I'll discuss a refinement of the Hilbert series - the set of graded Betti numbers. I'll roughly follow the first bit of Eisenbud's "Geometry of Syzygies" and explain why people care about these numbers. Notes |

##### Semester 1

September 19, 11am (Thurs, 5325) | Noah White | Invariants of knots from representation theory |

Abstract: I will give a short introduction to tensor categories and how one can "draw" these categories. I will show how one can use this idea to obtain an invariants of knots (and 3-manifolds) from such a category called Reshetikhin-Turaev invariants. Most interesting examples of appropriate tensor categories come from representation theory (often via mathematical physics). I'll draw lots of pictures and won't assume too much apart from what a category is. Notes (thank you Becca for taking notes). | ||

September 27, 2pm | Patrick Orson | Invariants of knots from analysis |

Roughly speaking, the `eta-invariant' is an invariant of a smooth structure on an even dimensional manifold \( W \) with boundary \( M \). It's defined using the tools of index theory and analysis and its definition can be twisted by picking a unitary representation of the fundamental group of \( W \). If \( M \) is a knot exterior then correct choice of representation can give an eta-invariant that is an invariant of the knot or the cobordism class of the knot. I will talk about how to choose the correct representation and some resulting invariants. Notes | ||

October 3, 4pm (Thurs, 5325) | Joe Karmazyn | Quiver GIT and Tilting |

Abstract: Given an algebraic variety, \( X \), with a tilting bundle, \( T \), you can produce an algebra \( A=\mathrm{End}(T) \) - which is commonly non-commutative. You may always chose to view such an \( A \) as the path algebra of a quiver with relations. Alternatively given an arbitrary quiver and relations you can use quiver GIT quotients to produce various algebraic varieties from the data of the quiver and relations. I will give lots of examples from the case of rational surfaces, and discuss how these two constructions interact. Notes | ||

October 11, 3pm | Becca Tramel | Stability and the Derived Category |

Abstract: My goal is to give a brief introduction to Bridgeland stability. I will start with the example of slope stability on curves, and look at examples. Then, I will explain why for varieties of higher dimension we need to look in the derived category if we hope to define something similar. I will define the heart of a bounded t-structure, and talk about tilting in the derived category. Notes | ||

October 25, 2:30pm | Chris Campbell | Elliptic Curves in Noncommutative Projective Geometry |

Abstract: Noncommutative projective geometry uses the tools developed in the last century of algebraic geometry in the context of noncommutative rings. Classifying rings that are 'geometric' in some appropriate sense is an open problem but in low dimensions a lot of work has been done and I will discuss some of the main results in this area, giving some results in which elliptic curves play a surprising role. | ||

November 1, 2:30pm | Adrien Brochier | Knot theory and quantization |

Abstract: "Everybody" knows that quantum groups can be used to produce knot invariants. I will argue that the natural story goes the other way around, and that a purely knot theoretic statement is the secret reason of the very much existence of quantum groups. | ||

November 8, 2:30pm | Dulip Piyaratne | Stability conditions on three folds |

Abstract: I will start by quickly recalling the definition and some important properties of Bridgeland stability conditions. Construction of such stability conditions on higher dimensional varieties is a challenging problem. I will discuss what kind of hearts of t-structures that we need in threefold case. Then I will explain how the conjectural construction introduced by Bayer, Macri and Today give rise to positive answers in some projective three folds. | ||

November 15, 2:30pm | Francois Petit | Moduli space of perfect objects |

Abstract: In this talk I will present the constructions and some of the properties of the derived moduli stack of perfects objects by Toen and Vaquie. | ||

November 22, 3:00pm | PG open day | Short talks |

This week there are two short talks from geometry/topology/algebra: Speaker: Joe Karmazyn Title: Algebras defined by superpotentials. Abstract: Often we can write down algebras using generators and relations, but is there a good way to produce relevant examples rather than just random junk? I'll talk about a class of non-commutative algebras with relations defined by superpotentials which have many nice properties, and try to convince you that these produce lots of interesting examples which can help us understand problems in geometry. Speaker: Carmen Rovi Title: The non-multiplicativity of the signature of fibre bundles Abstract: The signature \( \sigma(M) \) of a closed oriented \( n \)-dimensional manifold \( M \) is defined to be the signature of the intersection form if \( n \equiv 0 \pmod 4 \) and \( 0 \) otherwise. Consider a fibre bundle \( F \to E \to B \). I shall be talking about the following 60 year old question: What is the relationship between the signature of the total space \( E \) and the signature of base and fibre? Slides | ||

November 29, 2:30pm | Guillaume Pouchin | Introduction to Hall algebras |

In this talk we will give a gentle introduction to Hall algebras, with explicit computations in the case of quivers, followed by an overview of the theory. | ||

December 6, 12:00pm | Ciaran Meachan | Symmetries of K3 surfaces |

Abstract: The (derived) symmetry group is a mysterious and intriguing object. A precise description of this group was conjectured by Bridgeland almost ten years ago and this talk will focus on trying to give a flavour of the statement and, in particular, why it is both interesting and difficult. If there is time, I will discuss connections with my own work. Notes |