Title: The Topology of 4-Manifolds with Finite Fundamental Group
Speaker: Johnny Nicholson
Abstract: In the 1980s Freedman and Quinn showed that techniques from high-dimensional topology, such as surgery theory and the s-cobordism theorem, can be made to work for topological 4-manifolds. Whilst these techniques can be used to classify simply-connected topological 4-manifolds, similar results in the non-simply connected case are few and far between. This is largely due to the mystery surrounding the second homotopy group \pi_2(M) which has the structure of a \Z[\pi_1(M)] module. In this talk, we will give an overview of two approaches to the classification of 4-manifolds using techniques from high-dimensional topology. The first approach is to start by classifying up to homotopy and then to use classical surgery to classify up to homeomorphism within each homotopy type. Whilst a clean homotopy classification can often be obtained, the second step throws up obstructions which can only
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6/4/2019
Title: Intersections of secant varieties to algebraic curves
Speaker: Mara Ungureanu
Abstract: For a smooth projective curve, the varieties parametrising its secant planes are among the most studied objects in classical enumerative geometry. In order to better understand their geometry, which in turn describes the extrinsic properties of the curve, one is lead to the study of Brill-Noether theory. This allows us to translate such extrinsic geometry problems in terms of objects belonging to the intrinsic geometry of the curve, namely subvarieties of its symmetric product. In this talk we shall introduce some basic notions of Brill-Noether theory, define secant varieties to a curve embedded in projective space and study some unexpected properties of their geometry that arise as non-transversality of intersections inside the symmetric product of the curve.
Keywords: enumerative geometry, algebraic curves, symmetric product, secant planes, Brill-Noether theory
44
6/4/2019
Title: Many reasons to fear the Grothendieck ring of varieties
Speaker: Luigi Lunardon
Abstract: The Grothendieck ring of varieties is an interesting and mysterious ring. In this talk, we introduce it and try to understand something more about its properties. We see how some of its algebraic properties are closely related to many interesting geometric problems. For instance, if the class of the affine line was not a zero-divisor in this ring, it was possible to conclude that the cubic fourfold was rational. Moreover, if this was not enough, we present further geometric reasons to motivate the interest in it. After we have risen your hope that this may actually prove some conjectures, we show that the class of the affine line is a zero-divisor in the Grothendieck ring. Hopefully, even after this shocking revelation, your life will continue as usual; but you will know why this ring is something to be scared of.
Keywords: Grothendieck ring of varieties, class of the affine line, zero d
112
6/4/2019
Title: Introduction to Twistor Theory
Speaker: Benjamin Aslan
Abstract: In this talk, we will learn in which way twistor theory builds a bridge between Riemannian and complex geometry. More precisely, every even-dimensional Riemannian manifold M can be equipped with a twistor space which parametrises certain almost complex structures on M. When M is four-dimensional, the twistor space can itself be equipped with two canoncial almost structures. We will learn how properties about these structures translate into properties of the Riemannian structure of M.
134
6/4/2019
Title: Ricci Flow in Milnor Frames
Speaker: Syafiq Johar
Abstract: In this talk, we are going to talk about the Type I singularity on 4-dimensional manifolds foliated by homogeneous S3 evolving under the Ricci flow. We review the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as by Isenberg, Knopf and Sesum. In the latter, a global frame for the tangent bundle, called the Milnor frame, was used to set up the problem. We shall look at the symmetries of the manifold, derived from Lie groups and its ansatz metrics, and this global tangent bundle frame developed by Milnor and Bianchi. Numerical simulations of the Ricci flow on these manifolds are done, following the work by Garfinkle and Isenberg, providing insight and conjectures for the main problem. Some analytic results will be proven for the manifolds S1×S3 and S4 using maximum principles from parabolic PDE theory and some sufficiency conditions for a neckpinch singularity will be provided. Finally,
32
6/4/2019
Title: The spaces of stability conditions of the Kronecker quiver
Speaker: Caitlin McAuley
Abstract: It is well known that the space of stability conditions of a triangulated category is a complex manifold. In fact, mirror symmetry predicts that this space carries a richer geometric structure: that of a Frobenius manifold. From a quiver, one can construct a sequence of triangulated categories which are indexed by the integers. It is then natural to study the stability manifolds of these categories, and in particular to consider any changes to the manifolds as the integer indexing the triangulated category varies. We will study this construction for the Kronecker quiver, and discuss how the results provide evidence for a Frobenius structure on these stability manifolds
Keywords: stability conditions, quiver representations, triangulated categories, Frobenius manifolds
27
6/4/2019
Title: What's so special about the Lagrangian angle?
Speaker: Christopher Evans
Abstract: The mean curvature of a submanifold is a notoriously irritating quantity to compute. For Lagrangian submanifolds of Calabi-Yau manifolds however, there is a better way. In this talk, we will introduce a function called the Lagrangian angle and see how it determines the mean curvature. We'll also try and generalise our discussion to non-flat manifolds and see some of the problems therein.
20
6/4/2019
Title: Contraction Algebras and the Homological Minimal Model Program
Speaker: Jenny August
Abstract: Contraction algebras are a class of finite dimensional algebras introduced by Donovan and Wemyss as a tool to study minimal models of 3-folds or more generally, flopping contractions. In this talk, I will give an introduction to these algebras, including the key conjecture in this area which states that the derived category of such an algebra completely controls the associated geometry. I will then go on to give evidence towards this conjecture by providing a complete description of the derived equivalence class of these algebras.
Keywords: derived category, flopping contractions, tilting complexes
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6/4/2019