This is the webpage for the student geometry seminar in Fall 2016, which houses the list of speakers and their talk titles and abstracts. The seminar met Tuesdays at 3:35 pm in PMA 12.166.

• August 30: organizational meeting

• September 6: Roberta Guadagni

  Lagrangian fibrations and topological mirror symmetry

  For decades, mathematicians have tried to come up with well-posed conjectures and theorems that reflect the idea of mirror symmetry coming from physics. The most intuitive picture involves finding a “special” Lagrangian torus fibration of a space X, then “dualizing” it to get a new space Y also built out of special Lagrangian tori: this gets tricky along the singularities of the fibration. As it turns out though, it is very hard to even find such a fibration, the existence of which has come to be known as “SYZ conjecture.” A more reasonable version of the theory requires the existence of any Lagrangian torus fibration (thus not necessarily special). We'll see why this is more reasonable and I'll draw a few torus fibrations. Not a technical talk.

• September 13: Andrew Lee

  Surface Bundles, Blowups, and Symplectic Fixed Points

  Invariants of symplectic manifolds are often difficult to calculate, and it can be useful to understand how they change under basic operations in the symplectic category. In joint work with Tim Perutz we describe, for a particular geometric situation, how an invariant of symplectomorphisms called fixed-point Floer cohomology changes under a blowup of the underlying symplectic manifold. This is an introductory talk; no knowledge of symplectic geometry is necessary.

• September 20: Yuri Sulyma

  Chern-Weil forms and abstract homotopy theory

  Let G be a Lie group with Lie algebra 𝖌. Classically, Chern-Weil showed that invariant polynomials on 𝖌 give rise to characteristic classes (represented geometrically as differential forms) for principal G-bundles with connection. Much more recently, Freed-Hopkins showed that such Chern-Weil forms are in fact the only natural differential forms associated to principal G-bundles with connection. An interesting feature of their approach is that they use modern homotopy theory to formulate the problem, but the subsequent calculations use only classical ideas. I will rapidly explain the framework they set up, then describe as much of the proof as possible.

• September 27: Tim Magee

  Mirror symmetry for log Calabi-Yaus and combinatorial representation theory

  Roughly speaking, a log Calabi-Yau is a space that comes equipped with a volume form in a natural way. Let X be an affine log CY. We can partially compactify X by adding divisors along which Ω has a pole. The set of these divisors says a lot about X's geometry — for a torus (the simplest example of a log CY) this set is just the cocharacter lattice. We can actually give this set a geometrically motivated multiplication rule too, which I hope not to get into. But this multiplication rule allows us to construct an algebra A, defined purely in terms of the geometry of X, and conjecturally A is the algebra of regular functions on the mirror to X. Viewed as a vector space, A naturally comes with a basis — the divisors we used to define it. So we get a canonical basis for the space of regular functions on X's mirror. Many objects of interest to representation theorists (semi-simple groups, flag varieties, Grassmannians…) are nice partial compactifications of log CY's. This gives us a chance to use the machinery of log CY mirror symmetry to get results in rep theory. Maybe the most obvious application given what I've said so far is finding a canonical basis for irreducible representations of a group. I'll discuss this, as well as how this machinery reproduces cones that make combinatorial rep theorists feel warm and fuzzy inside (like the Gelfand-Tsetlin cone and the Knutson-Tao hive cone). All of the technology involved is a souped-up version of something from the world of toric varieties. I'll discuss things in this context as much as possible in an attempt to keep everything accessible.

• October 4: Arun Debray

  An entrée into Dijkgraaf-Witten theory

  Dijkgraaf-Witten theory is a topological quantum field theory that’s relatively simple to define, exists in all dimensions, and lacks path-integral-related analytic issues. As such, it’s a great introduction to the many facets of TQFT. In this talk, I will discuss what a TQFT is from the mathematician’s perspective, define Dijkgraaf-Witten theory in this paradigm, and use it to shine light on some aspects of TQFT. No knowledge of physics is assumed.

• October 11: Max Stolarski

  Fattening Phenomena in Mean Curvature Flow

  The mean curvature flow is a geometric evolution equation that evolves a hypersurface according to the gradient of the area functional. It has a weak formulation that allows for the consideration of singular solutions and mimics the notion of viscosity solution in PDE. In certain dimensions the existence of unstable stationary cones leads to non-uniqueness phenomena. We’ll survey such non-uniqueness examples for the mean curvature flow in which an initially smooth hypersurface flows to a cone in finite time and then admits non-unique solutions for later times.

• October 18: Richard Hughes

  Introduction to Mirror Symmetry

  Two Calabi-Yau threefolds X and Y are called mirror manifolds if type IIA string theory compactified on X and type IIB string theory compactified on Y yield the same physical theory. We will describe a rough mathematical principle that follows from mirror duality of X and Y, and derive from this a relationship between the Hodge numbers of X and Y. This is an introductory talk, with (almost) entirely mathematical content. Knowledge of physics is emphatically not required; all buzzwords above will be either defined or ignored.

• October 25: Jonathan Lai

  Theta functions with a little mirror symmetry

  Classical theta functions are a special class of holomorphic functions on C^g that are quasi-periodic with respect to a lattice. However, theta functions can also be viewed as sections of a line bundle on an abelian variety and by fixing a bit of data, a canonical basis can be obtained. Classically, the construction of canonical theta functions depended heavily on the group structure of abelian varieties, making generalizations difficult. We will look at how recent use of toric varieties and mirror symmetry have helped provide a generalized notion of theta functions to other varieties, namely K3 surfaces.

• November 1: Richard Wong

  Intro to Modular Representation Theory (and Homotopy Theory)

  In this talk, I will introduce and motivate the main objects of study in modular representation theory, which leads one to consider the stable module category. Perhaps surprisingly, one can do homotopy theory in this setting, as the stable module category has the structure of a stable model category. Therefore, there is a very rich interplay between modular representation theory and stable homotopy theory, which I will try to illustrate. In the spirit of Halloween, one of the ideas I will talk about is the notion of phantom maps.

• November 8: Ethan Leeman

  On Johnson's Description of Atiyah Invariant

  In “Riemann surfaces and Spin Structures,” M. Atiyah defined an invariant for spin structures on a closed oriented surface analytically. D. Johnson then realized this invariant topologically by giving a correspondence between spin structures and quadratic forms on homology. In this talk, we will go through this concrete construction and discuss some consequences in spin bordism.

• November 15: Nicky Reyes

  Motives and noncommutative motives

  I will give a basic introduction to the theory of motives in algebraic geometry and talk about some of the conjectures surrounding them. Then I will talk about their noncommutative analog.

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