Talks

Timothy Duff: Galois/Monodromy Groups in 3D Reconstruction 

Galois groups embody the structure of algebraic equations arising in both enumerative geometry and various scientific applications where such equations must be solved. I will describe a line of work that aims to elucidate the role of Galois groups in applications where data taken from multiple images are used to reconstruct a 3D scene. From this perspective, I will revisit two well-known solutions to camera pose estimation problems, which originate from classical photogrammetry and are still heavily used within modern 3D reconstruction systems. I will then discuss some less-classical problems, for which the insight we gleaned from computing Galois groups led to significant practical improvements over previous solutions. A key ingredient was the use of numerical homotopy continuation methods to (heuristically) compute monodromy permutations. Time-permitting, I will explain how such methods may also be used to automatically recover certain symmetries underlying enumerative problems. 

Laura Escobar: Families of degenerations from mutations of polytopes 

Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to certain varieties which admit a toric degeneration.  In a recent paper with Megumi Harada, we gave a piecewise-linear bijection between Newton-Okounkov bodies of a single variety. This involves a collection of lattices $\{M_i\}_{i\in I}$ connected by piecewise-linear bijections $\{\mu_{ij}\}_{i,j\in I}$. In addition, Kiumars Kaveh and Christopher Manon analyzed valuations into semifields of piecewise linear functions and explored their connections to families of toric degenerations. Inspired by these ideas in joint work in progress with Megumi Harada and Christopher Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties a generalized polytopes can encode a compactification of an affine varieties as well as toric degenerations for the compactification. In this talk, we illustrate these ideas. 

Sean Griffin: Affine Springer fibers and the Delta Conjecture 

In the past few decades, a wealth of connections have been discovered between the combinatorics of Catalan objects (e.g. Dyck paths, parking functions, labeled trees etc) and certain families of algebraic varieties (e.g. Hilbert schemes of points in the plane, compactified Jacobians, and affine Springer fibers). In particular, Hikita proved in 2012 that the Borel-Moore homology of a certain affine Springer fiber gives a geometric model for the Shuffle Theorem concerning parking functions in a n by n grid. The Shuffle Theorem in turn encodes the S_n-character of the ring of diagonal coinvariants by Haiman's work on the n! Theorem. In this talk, I will present a generalization of Hikita's result to the setting of the Delta Conjecture, which conjecturally encodes the S_n-character of the ring of diagonal superspace coinvariants.  This is joint work with Maria Gillespie and Eugene Gorsky.

Eloísa Grifo: Zariski-Nagata Theorems 

A classical theorem of Zariski and Nagata relates symbolic powers, a notion that arises naturally from the theory of primary decomposition, with a classical interpolation question: given a radical ideal in a polynomial ring over a field, its nth symbolic power consists of the polynomials that vanish to order n on the given variety. Over a perfect field, this can be phrased in terms of differential operators. In this talk, we will discuss how to expand Zariski-Nagata to any field and to the mixed characteristic setting. This is joint work with Alessandro De Stefani and Jack Jeffries. 

Emanuele Macri: Hyper-Kähler manifolds and Lagrangian fibrations  

A hyper-Kähler manifold is a complex Kähler manifold that is simply connected, compact, and has a unique holomorphic symplectic form, up to constants. This important class of manifolds has been studied in the past in many contexts, from an arithmetic, algebraic, geometric point of view, and in applications to physics and dynamics.

The theory in dimension two, namely K3 surfaces, is well understood. The aim of the seminar is to give an introduction to the theory of hyper-Kähler manifolds in higher dimension, from a point of view of their classification; in particular, about existence of Lagrangian fibrations. We will present some results in dimension four, obtained in collaboration with Olivier Debarre, Daniel Huybrechts and Claire Voisin. 

Julia Pevtsova: Local rigidity and regularity for categories of modular representations 

Let R be a graded commutative ring, and T be a “nice” tensor triangular R-linear category; for example, T can be a derived category of A-modules where A is a commutative ring or a (often non-commutative) group algebra of a finite group/group scheme. For any homogeneous prime ideal p in R we can consider the fiber category Гp T.  Inspired by the commutative case, we ask - and sometimes answer - questions like “What does it mean for T to be regular? Gorenstein?", “What are the rigid objects in the fiber category?”, and “What is its cohomology?”.  Joint work with D. Benson, S. Iyengar, H. Krause