Fall 2022

Title: Combinatorics of log Voronoi cells for linear concentration models

Abstract: The logarithmic Voronoi cell of a point on a statistical model is the set of all positive-definite matrices which are "closer" to that point than to any other point on the model. The structure of these logarithmic Voronoi cells is well understood for some special statistical models. One topic that is less well understood is how these cells vary as the point on the model varies. In this talk, we will discuss the variance of the combinatorial structure of the logarithmic Voronoi cells in terms of wall-crossings, where the wall-crossings are defined by certain algebraic hypersurfaces. This is joint work in progress with Yulia Alexandr, Margaret Regan, and Madeleine Weinstein.

Prerequisites: linear algebra, definitions of covariance matrices and algebraic varieties.

Title: Combinatorics of Root Multiplicities of Real Polynomials

Abstract: We relate a variant of the poset of compositions to a cellular decomposition of the space of univariate monic polynomials or binary forms of a fixed degree. We show how using this decomposition we can define a combinatorial complex calculating the homology of spaces of polynomials with given root multiplicities.

Prerequisites: For most of the talk I just use basic linear algebra and real analysis. If one takes it for granted that what I call homology is an important invariant, then no prior knowledge of homology is needed. In topological terms I will use homeomorphism and ball. If one knows what a CW-complex is, very good, if not then one has to believe me that my decomposition of the space of polynomials is a nice one.

Title: Two-variable singularities and symplectic topology

Abstract: Start with a two-variable complex polynomial f with an isolated critical point at the origin. We explain how to associate to it a smooth surface with boundary: the Milnor fiber of f, M_f, given by a smoothing of f near the origin. This comes equipped with a distinguished collection of S^1s on M_f. We will show how to encode all of this information combinatorially. We will then discuss consequences for the symplectic topology of M_f. No prior knowledge of singularity theory or symplectic topology will be assumed.

Prerequisites: a first course in geometry (concept of a smooth surface); group presentations (generators / relations); some familiarity with Riemann surfaces would be useful (branch cuts / branch locus; the idea that a complex polynomial cuts out a shape)

Title: Sampling and learning from random polynomials: two stories

Abstract: This talk is motivated by probabilistic models of random monomial ideals that mirror and extend those from random graphs and simplicial complexes literatures. Our results provide precise probabilistic statements about various algebraic invariants of (coordinate rings of) monomial ideals: the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers.

We will tackle the following related questions: What is a systematic way, in a probabilistic-model sense, to generate binomial ideals randomly? What can be (machine) learned from such data sets? How do we 'test out the waters' to see if a problem is 'learnable'? How do we generate, share, and make available large training data sets for machine learning in computational algebra?

These topics are based on joint work with various collaborators and students and form a two-step process in learning on algebraic structures.

Prerequisites: A commutative algebraist's interest in randomness has many facets, of which this talk highlights two: 1) how to use basic statistics and machine learning for improving computations with polynomials and 2) how to generate samples of ideals in a `controlled' way. Since the topics cover several areas, I will explain all definitions intuitively. However, it would be helpful for students to be familiar with the following basic concepts: ideals, monomial ideals, Groebner bases (Buchberger's algorithm); probability of an event, expected value of a random variable; simple linear regression, and an idea of what machine learning might be doing (no details required whatsoever).

Title: Boundary h*-polynomials of rational polytopes

Abstract: We will talk about the geometric interpretation of h*-polynomials of rational polytopes and about the history of symmetric decompositions of h*-polynomials. Then we will look at decompositions of h*-polynomials from the perspective of boundaries of polytopes, and see how this perspective reveals properties of reflexive and Gorenstein polytopes. This is joint work with Matthias Beck.

Prerequisites: Maybe some familiarity with generating functions, but nothing really :)

Title: Exotic Surfaces in 4-Space

Abstract: The goal of this talk is some recent results about the difference between smooth and non-smooth topology in four dimensions. We will start by motivating the distinction by thinking about curves in the plane and in 3-space. We will then discuss how to describe smooth surfaces in 4-space, and some longstanding open problems and recent results. The last part of the talk will hint at the techniques used to prove these results.

This is almost entirely work of other people, though small parts are joint with Ozsváth-Thurston or Sarkar.

Prerequisites: At least the first 2/3 of the talk should be accessible to anyone familiar with continuous and differentiable functions in several variables.

Title: Zero forcing and graph complements

Abstract: Zero forcing on a graph is a graph coloring iterative process where, starting with an initial set of blue vertices, we try to “force” other non-blue vertices blue according to a color change rule. The zero forcing number of a graph was first defined in 2008 by an AIM (American Institute of Mathematics) working group as a method of bounding the maximum nullity of a graph. Motivated in part by considering the zero forcing number of the complement of trees, we consider the zero forcing number for the complement of more general graphs under some conditions. We will explore and discuss the zero forcing number, its connection with the maximum nullity of a graph, and some current work I am exploring with zero forcing on graphs.

Prerequisites: Some knowledge of linear would be helpful. In particular, matrices, eigenvalues, nullity of a matrix.

Title: Self-Similar Tilings of the Plane

Abstract: A history of tilings of the plane over the past 100 million years leads to the concept of a self-similar tiling and a construction of self-similar tilings developed jointly with Michael Barnsley of the Australian National University.

Prerequisites: Nearly the whole talk will be accessible to any undergraduate math student - many pictures.

Title: Comparing the topological and the smooth 4-genus of satellite knots.

Abstract: The study of 4-dimensional objects is special: a manifold can admit infinitely many non-equivalent smooth structures, and manifolds can be homeomorphic but not diffeomorphic. This difference between topological and smooth structures, can be addressed in terms of the study of knots as boundaries of surfaces embedded in 4D space. In this talk I will focus on some knot operators known as satellites and will show that satellites can bound very different surfaces in the smooth and topological category. This is joint work with Allison Miller and Peter Feller.

Prerequisites: The plan is for the talk to be self-contained, that is, I will define the main objects before stating any theorems. The main topics will be knot theory, surfaces, embeddings.

Title: A new combinatorial Hirzebruch-Riemann-Roch

Abstract: Given a complex algebraic variety with an action of (C^*)^n that has finitely many fixed points, we can write down a graph which computes two important rings -- K-theory and cohomology, which are related by the Hirzebruch-Riemann-Roch formula. In this talk I will do a complete worked example of this, and show in some of the nicest, but most well-studied cases (related to projective spaces, Grassmannians, flag varieties, toric varieties, etc. in Lie types A and B), that there's a modified Hirzebruch-Riemann-Roch formula which is adapted to the underlying combinatorics.

Prerequisites: Rings, Group actions, and Projective Spaces.

Title: Braids, Knots, and the Link Between

Abstract: It has been known for about 100 years that every knot or link can be represented as a closed braid. Over the years new proofs that use more modern techniques have been developed. We will give a brief introduction to the braids groups and links. Then we will give a proof of Alexander’s Theorem due to Vogel using Seifert circles.

Prerequisites: It would be nice but not necessary if students knew the definition of a group.