# Geometry club

### Schedule of talks for 2012/13:

September 21, 2pm | Chris Palmer | The Maslov Index, Algebraic Eta Invariants and Signature Additivity |

Abstract: This talk examines the role of the Kashiwara-Wall index and complex structures in finding the signature of a manifold. In the first part we interpret the signature defect in Wall's non-additivity theorem as the Kashiwara-Wall index of a particular triple of Lagrangians. Ranicki's real signature is a modified definition of the ordinary signature of a manifold with a correction term given by the algebraic eta invariant of a pair of Lagrangians. Using a relationship between the Kashiwara-Wall index and the algebraic eta invariant we show that the real signature is additive. We conclude by introducing the related problem of finding the signature of a PL manifold via the use of dual cells. If time permits we will also talk about the connection with rational Pontryagin classes. | ||

September 28, 2pm | Will Donovan | Projective varieties and Quiver Grassmannians |

Abstract: I will explain the recent proof by Reineke that every projective variety is a quiver Grassmannian. The result is quite elementary, so I should be able to review all the relevant material as I go along. I hope this will stimulate a lively debate about the virtues and aesthetics of quivers. | ||

October 5, 2pm | Yoshinoro Hashimoto | Moment maps and algebro-geometric stability |

Abstract: When an appropriate class of Lie group acts on a smooth projective variety, we can take quotients in two different categories to construct a space of orbits: GIT (Geometric Invariant Theory) quotient in the category of varieties and symplectic quotient in the category of symplectic manifolds. In fact, these two quotients are equal by the theorem of Kempf and Ness, establishing a beautiful connection between differential and algebraic geometry. The bulk of this talk will be spent on explaining this theorem, starting with basic definitions in Kaehler and symplectic geometry and finally giving an intuitive idea of how the proof is going to work. The final theme of the talk would be the slogan "curvature is an infinite dimensional moment map" and an infinite dimensional analogue of the Kempf-Ness theorem, which is a fundamental philosophical backbone in Kaehler geometry. More specifically, I aim to talk about the Hitchin-Kobayashi correspondence, which states that the existence of Hermitian-Einstein metrics on a vector bundle is equivalent to its Mumford-Takemoto stability. However, as the finite dimensional case is likely to fill up most of the hour, discussion on this will have to be rather brief. | ||

October 19, 2pm | Becca Tramel | Introduction to Toric Varieties |

Abstract: A toric variety is a variety containing the algebraic torus as an open dense subset, such that the natural action of the torus on itself extends to the whole variety. Due to this action, we are able to describe toric varieties with combinatorial objects called fans. Using these fans, many calculations which can be difficult in general are made easier. We will see how fans can be used to describe divisors on toric varieties, as well as how to determine if the variety corresponding to a fan is smooth. If any time remains, I will talk about a research project I worked on with undergraduate students at the University of Connecticut this summer. | ||

November 9, 2pm | Barry Devlin | Toposology |

November 16, 2pm | Jesus Martinez Garcia | Intersection theory and classification of surfaces for the layman |

Abstract: Bezout's theorem tells us that two curves of degrees d and d' intersect at, at most, dd' points on the plane, counted with multiplicity. In this talk I will explain how we compute intersections algebraically, how to interpret it geometrically, how to draw it pictorially and how to enjoy it mathematically. I will explain how to separate curves that intersect using a procedure called blow-up and how intersection theory can be used to classify surfaces. Time providing, I will draft the Minimal Model Programme for surfaces, which is the topic of my next talk. This talk assumes nothing and it should be in the interest of geometers/algebraists and on the taste of topologists, so everyone is welcome. | ||

November 30, 2pm | Jesus Martinez Garcia | Classification of algebraic surfaces and Minimal Model Program |

Abstract: The Minimal Model Program aims to give a simple model of any algebraic variety. In this talk I will recall basic facts of intersection theory of surfaces that appeared in the previous talk. I will introduce the canonical bundle and how we can use it as a canonical intersection tool to contract curves on surfaces. I will use this to describe minimal models of rational surfaces and finally, I will sketch how this is generalised this to higher dimensions. Again, the talk will assume little and recall all needed facts. | ||

January 25, 2pm | Chris Palmer | Introduction to Spectral Sequences I: Foundations |

Abstract: Spectral sequences were originally introduced by Leray in order to compute sheaf cohomology. In this talk I will give the basic definition of convergence of a spectral sequence and give examples including the Serre spectral sequence for a path-space, the spectral sequence for a filtered differential graded module and the spectral sequence for a double complex. There will be a follow-up talk by Dulip where he will look at further geometric and algebraic examples. | ||

February 15, 2pm | Dulip Piyaratne | Introduction to Spectral Sequences II: Examples and Applications |

Abstract: There are many ways to construct spectral sequences. In this talk, I will start by recalling the idea behind the spectral sequence of a double complex; which is also an important example. Then I will use it to prove the well known Snake Lemma. Finally I would like to talk about the Grothendieck spectral sequence. This is very important in mathematics as most of the spectral sequences are instances of it. I hope to discuss an example of the Grothendieck spectral sequence to see what kind of information that we can extract from its convergence. | ||

February 22, 2pm | Supreedee Dangskul | Knots and Seifert Manifolds |

Abstract: A knot is an embedding of an n-sphere into (n+2)-sphere. Given a knot, there exists a compact orientable (n+1)-submanifold bounded by the knot. This submanifold is called a "Seifert manifold". In this talk, I will prove the existence of a Seifert manifold for a knot and then discuss how to improve its connectivity. | ||

March 1, 2pm | Carmen Rovi | TQFTs and the Cobordism Hypothesis |

Abstract: In this talk I will start by reviewing the definition given by Atiyah of a Topological Quantum Field Theory, and I will sketch the ideas that lead to the formulation of the Cobordism hypothesis. | ||

March 8, 2pm | Joseph Karmazyn | Resolutions, Deformations, and Non-Commutative Algebras |

Abstract: When we study algebraic geometry we work with commutative rings, however there are situations where certain non-commutative algebras also hold valuable information. I aim to give some examples of how non-commutative algebras can help us understand resolutions and deformations of singularities. | ||

March 22, 2pm | Patrick Orson | Doubly-Slice Knots |

Abstract: The only classical knot that is the boundary of a disk is the unknot. But if we allow the disk to push into 1 dimension higher, can we bound all knots by disks? The answer is no! A knot that bounds a disk in 1 dimension higher is called a slice knot. I will talk about how this allows us to build an abelian `concordance' group from knots. I will also talk about an extension of this idea called double-sliceness and a nice proof that there is also a doubly-slice concordance group that uses Zeeman's twist-spinning. There will be a lot of pictures! | ||

April 12, 2pm | Noah White | Moduli space of marked curves, monoidal categories and Gaudin subalgebras |

Abstract: Moduli spaces are geometric objects whose points classify geometric or algebraic objects up to some sensible notion of equivalence. I will give a quick introduction to fine moduli spaces through a particularly nice example, the moduli space of stable marked curves of genus zero. This moduli space has strong connections to actions on monoidal categories and the Knizhnik-Zamolodchikov equation which I will attempt to explain. | ||

April 19, 2pm | Dulip Piyaratne | Introduction to GIT and moduli spaces of vector bundles |

Abstract: Geometric Invariant Theory in the modern form is due originally to Mumford. One of the initial motivations was to construct moduli spaces of objects in algebraic geometry. In this talk I will start by recalling some of the important definitions and results in GIT. After discussing the notion of slope stability for vector bundles on algebraic curves, I will then move into construct some moduli spaces of them. | ||

June 03, 2pm | Wafa Alagal | Introduction to k-Very Ample Line Bundles and a Short Proof of "the Göttsche Conjecture" |

Abstract: In this talk, I will start with a quick review of some standard definitions and examples of k-very ample line bundles . Then I will give a proof of "the Göttsche conjecture" by R. Thomas and others. |