Graduate Geometry and Topology Current Literature Seminar

Travis Mandel: How to Construct a Mirror Family

SYZ mirror symmetry is a geometric approach to understanding mirror symmetry, which essentially is a relationship between the symplectic structure of one manifold to the complex structure its "mirror." It purports that the two manifolds should admit dual special Lagrangian torus fibrations over some base manifold. I'll explain this relationship using the example of the complement of an anti-canonical divisor on a surface. In particular, I'll explain how Gross, Hacking, and Keel, motivated by ideas of Auroux, give an explicit construction of a mirror family with a canonical basis of global sections.

Yuecheng Zhu: From SYZ conjecture to Gross-Siebert program

The first part will be about a geometric description of Mirror pairs conjectured by Strominger, Yau and Zaslow. The second part will be about a paralleled story in algebraic geometry called Gross-Siebert program. In the end, I hope I will explain why we care about Gross-Siebert program.

Tom Mainiero: Spectral Networks: a 3/2 hour tour

With their powers combined, G, M, and N once again save us from the mysterious powers of extended supersymmetry. We will draw the appropriate squiggly arrows to describe the machinery of spectral networks (using a pinch of inspiration from physics), their connection to wall crossing phenomena, and the nonabelianization map; then, perhaps, we will live to see another day.

Mio Alter: Equivariant and differential cohomology

I'll introduce S¹-equivariant versions of cohomology and (topological) K-theory with examples and describe how these come up in differential cohomology. I will assume familiarity with singular and de Rham cohomology and not much else.

Sam Ballas: The Ending Lamination Conjecture

Let M be a hyperbolic manifold, then two natural questions are how many hyperbolic structures does M admit and what data do we need to determine such a structure. In dimension 2, Teichmuller theory tells us that the lengths of finitely many curves suffices to determine the structure. In dimension 3 the answer is more complicated and involves asymptotic data coming from the ends of M. We will explain how hyperbolic manifolds give rise to ending data and outline the proof of the ending lamination conjecture, which asserts that this data uniquely determines the hyperbolic structure on M.

The notes in this section were taken by an audience member and are provided as-is.

Aaron Royer: The Homotopy Type of the Cobordism Category

Sam Taylor: The Virtual Haken Conjecture (Notes)

In this talk we explore the web of ideas culminating in Ian Agol’s recent proof of the virtual Haken conjecture. The conjecture, made by Friedhelm Waldhausen in the 1960s, is among the most important questions in 3 manifold theory. The proof, however, leaves the world of 3 manifolds and, instead, focuses on understanding the cubical geometry of certain hyperbolic groups. We will explain the conjecture and its importance as well as introduce the background necessary to understand the role that recent advances in geometric group theory play in its resolution.

Laura Starkson: An Isomorphism of two Floer homology theories for 3-manifolds (Notes)

There are a number of complicated invariants of low-dimensional manifolds that have been defined over the last couple decades, all inspired by ideas in gauge theory and Floer theory, but defined using very different auxiliary geometric information. While there was suspicion that many of these theories were capturing the same information, it was not clear how to prove any equivalence of the theories. Recently two different proofs have been written that three of these theories: monopole Floer homology, embedded contact homology, and Heegaard Floer homology, are all isomorphic. In this talk I will introduce two of these theories: embedded contact homology and Heegaard Floer homology, and discuss some of the ideas that go into one proof (by Colin, Ghiggini, and Honda) that they are isomorphic.

Lee Cohn: The Cobordism Hypothesis (Notes)

We will begin with motivations/examples of oo-categories. Next, we will define the category, Bord^fr_n, of framed cobordisms and symmetric monoidal functors out of this category (topological field theories aka TFTs). We will then discuss the main theorem classifying fully extended topological field theories (the Cobordism Hypothesis). The talk will finish with applications to representation theory.

Pavel Safronov: The Fundamental Lemma (Notes)

The Langlands program is a series of conjectures relating number theory (representations of Galois groups) and representation theory (modular and automorphic forms). One famous special case is the Taniyama-Shimura conjecture, which was proved by Wiles, Taylor and others in the course of the proof of Fermat's last theorem. In my talk I will discuss an important result called the Fundamental Lemma. I will touch upon its recent proof by Ngo, which, surprisingly, involves ideas from geometry. Namely, Ngo was able to deduce the Fundamental Lemma from the properties of the Hitchin fibration.

Hendrik Orem: Koszul Duality and Representation Theory (Notes)

Koszul duality describes a relationship between (derived) categories of modules over two rings which are suitably dual to one another. Recent work has shown that this kind of duality appears in the representation theory of semismiple Lie algebras and related objects. These Koszul dualities provide additional structure to categories of representations, and illuminate deep connections between otherwise distant objects. I will use the example of Koszul self-duality for the principal block of category O to build up to recent progress in this area.

Nick Zufelt: Bordered Heegaard Floer Homology (Notes)

Bordered Floer Homology is a variant of Heegaard Floer homology that is defined for 3-manifolds with boundary. Probably the most exciting consequence is a pairing theorem which allows one to decompose a 3-manifold along a surface and compute its Heegaard Floer invariant from that of its pieces. In this talk, we'll introduce the bordered invariant and work toward a conceptual understanding of how to use it to compute the original invariant.

Davi Maximo: Min-max theory and the Willmore conjecture

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R³ is at least 2π². This conjecture was recently confirmed by Fernando Codá Marques (IMPA) and André Neves, using the min-max theory of minimal surfaces. In this talk, I will go through the steps of their proof, focusing mostly on the ideas involved. If time permits, I will also mention a related application of the min-max theory also found by the authors, this time jointly with Ian Agol, on their solution of the Freedman-He-Wang conjecture about the energy of links.

Yuan Yao: Birational Geometry of Hilbert Schemes and Bridgeland Stability

Birational geometry of moduli spaces (e.g., Hilbert schemes or moduli of curves) has long been a key problem in algebraic geometry and Bridgeland stability is a relatively new method to attack it. In this talk I will explain the rough idea of how to apply the highly abstract algebraic structure "stability"–defined using derived category and etc.–to study the concrete geometric objects. The first half is aimed to provide the background and preliminaries for general audience, while the real theorems will be presented during the second half. Main reference: arXiv:1203.0316

Rahul Shah: The Gaiotto-Moore-Neitzke construction of Hyperkahler metrics

Physically, one expects to find a Hyperkahler metric on the target space of (an approximation of) a 4-dimensional N=2 super symmetric field theory. It has been known for some time how to explicitly write down a metric away from a certain 'bad' locus. GMN correct the metric to extend it and in doing so, produce explicit formulae for the corrected metric. We will gently introduce Kahler and Hyperkahler metrics and discuss the main ideas in producing the corrections to the metric. We will then give a detailed account of this construction in a particular example.

Aaron Fenyes: Waves in shallow water: 135 years at the beach

The study of water waves has its roots in the study of partial differential equations, but it makes contact with geometry in several ways. One of the most surprising involves two closely related equations that can be used to model waves in shallow water: the Korteweg-de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation. I will sketch the 19th-century origins of these equations in the geometry of fluid flow, outline the 20th-century discovery of solution families parameterized by the infinite-dimensional Grassmannian, and describe how people have recently begun exploring these solution families using equipment from tropical geometry and combinatorics.