Spring 2023

Title: Galois groups in Enumerative Geometry and Applications

Abstract:  In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.  A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

Prerequisites: groups and permutations, fields and field extensions

Title: Looking for the Center (trace ideals over commutative rings)

Abstract: Do you know which matrices commute with all the others? Do you know a beautiful structural algebraic reason behind the answer? The well-known trace map on matrices can be generalized to a map on any module over a commutative ring.  The image of such a map is called a trace ideal. I will speak on some recent developments in the theory of trace ideals with applications to our understanding about the relationship between modules, rings and ideals. Ultimately, this will help us understand the commutative centers of endomorphism rings with applications to rigid modules and classifications of commutative rings. 

Prerequisites: groups, vector spaces, rings, ideals

Program:

1:30-2:00 Federico Ardila (SFSU)

2:05-2:35 Yulia Alexandr (UC Berkeley)

2:40-3:10 Patrick O'Melveny (SFSU)

3:10-3:40 Coffee/Tea

3:40-4:10 Svala Sverrisdottir (UC Berkeley)

4:15-4:45 Ayush Bhardwaj (SFSU)

4:50-5:20 Bernd Sturmfels (UC Berkeley and MPI Leipzig)

Titles and Abstracts:

Speaker: Federico Ardila

Title: Combinatorial Intersection Theory: A Few Examples

Abstract: Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X). When X is the toric variety of a simplicial fan, Brion gave a presentation of A(X) in terms of generators and relations, and Fulton and Sturmfels gave a "fan displacement rule” to intersect classes in A(X), which holds more generally in tropical intersection theory. In these settings, intersection theoretic questions translate to algebraic combinatorial computations in one point of view, or to polyhedral combinatorial questions in the other. Both of these paths lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectural inequalities. 

This talk will survey a few problems on matroids and root systems that arise in combinatorial intersection theory. It will feature joint work with Montse Cordero, Graham Denham, Chris Eur, June Huh, Carly Klivans, and Raúl Penaguião. The talk will not assume previous knowledge of the words in the abstract.

Speaker: Yulia Alexandr

Title: Moment varieties for mixtures of products

Abstract: I will present recent joint work with Joe Kileel and Bernd Sturmfels on the moment varieties of conditionally independent mixture distributions on $\mathbb{R}^n$. These are the secant varieties of toric varieties that express independence in terms of univariate moments. I will introduce these varieties and their images under certain coordinate projections, using familiar examples. I will then present what we know about their dimensions and defining polynomials. I will also report on some computational results, featuring both symbolic and numerical methods.

Speaker: Patrick O'Melveny

Title: Log-Concave Sequences, Mixed Volumes, and the Normal Complex of a Fan

Abstract: Unimodal and log-concave sequences arise again and again in surprising places across algebra, geometry, and combinatorics. Notably, the resolution of the conjecture of the log-concavity of characteristic polynomials of matroids has garnered some attention recently. A classic generator of log-concave sequences comes from the area of convex geometry, the mixed volume function and the Alexandrov–Fenchel inequalities. We present joint work with Lauren Nowak and Dusty Ross on the generalization of mixed volumes to non-convex objects known as normal complexes and conditions under which the Alexandrov–Fenchel inequalities still hold.

Speaker: Svala Sverrisdottir 

Title: The variety of four dimensional Lie algebras

Abstract: The projective variety of Lie algebra structures on a 4-dimensional vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we answer a 1987 question by Kirillov and Neretin.

Speaker: Ayush Bharadwaj

Title: Complex Critical Points of Deep Linear Neural Networks

Abstract: Artificial neural networks are the workhorse behind highly successful deep learning techniques used in domains like natural language processing, computer vision and drug discovery. Training a neural network involves minimizing a highly non-convex loss (cost) function which makes it challenging to identify all minima. Even specifying a good upper bound on the number of minima can be hard. Mehta, Chen, Tang, and Hauenstein show that, for the special case of deep linear neural networks, the complex critical points of the loss function are precisely the solutions to a polynomial system. We extend their work by identifying an improved bound on the number of complex critical points of the loss function for 1-hidden-layer linear networks trained on a single data point. We also show that for any number of hidden layers, the complex critical points with zero coordinates arise in certain patterns which we completely classify for the 1-hidden layer case.

Speaker:  Bernd Sturmfels

Title:    Subspaces fixed by a nilpotent matrix

Abstract:  I will discuss recent work with Marvin Hahn, Gabi Nebe and Mima Stanojkovski on a problem in linear algebra. It concerns the linear subspaces that are fixed by a given nilpotent n x n matrix. We classify them for small n, using computer algebra. Mutiah, Weekes and Yacobi conjectured that their radical ideals in the Grassmannian are generated by linear forms known as shuffle equations. We prove this conjecture for n at most 7, and we disprove it for n = 8. It remains open for nilpotent matrices arising from the affine Grassmannian.

Title: Identifiability and Indistinguishability of Linear Compartmental Models

Abstract: An important question that arises when modeling is if the unknown parameters of a model can be determined from real (and sometimes noisy) data, the so-called parameter estimation problem.  A key first step is to ask which parameters can be determined given perfect data, i.e. noise-free and of any time duration required.  This is called the structural identifiability problem.  If all of the parameters can be determined from data, we say the model is identifiable.  However, if there is some subset of parameters that can take on an infinite number of values yet yield the same data, we say the model is unidentifiable. If a model is unidentifiable assuming perfect data, then it is almost certainly unidentifiable with real, noisy data, thus knowing this information a priori helps with experimental design.  We examine this question for an important class of models called linear compartmental models used in many areas, such as pharmacokinetics, physiology, cell biology, toxicology, and ecology.  We also examine a somewhat related question called indistinguishability, which examines if two distinct models yield the same dynamics.  For both of these questions, we will consider the underlying graph corresponding to our model and use tools from graph theory and computational algebra to describe and analyze our models.  This is joint work with Cashous Bortner, Elizabeth Gross, Anne Shiu, and Seth Sullivant.

Prerequisites: Linear algebra, some basic notions from graph theory

Title: Stirling numbers for complex reflection groups

Abstract: The ordinary Stirling numbers count set partitions and permutations of 1, 2, ..., n by number of subsets and number of cycles, respectively.  We show how to generalize these concepts to a complex reflection group.  The ordinary Stirling numbers are recovered in type A.  We show that often these Stirling numbers can be expressed in terms of elementary and homogeneous symmetric functions.  All terminology concerning Stirling numbers, symmetric functions, and complex reflection groups will be defined.  This is joint work with Joshua Swanson.

Prerequisites: disjoint cycle decomposition of a permutation, ranked partially ordered set (poset), group theory, Hilbert series

Title: An Ehrhart Theory for Tautological Intersection Numbers

Abstract: The tautological intersection theory of the moduli space of stable pointed curves is a jewel of modern algebraic geometry. In particular, tautological intersection numbers exhibit a tremendous amount of structure. In this talk, I will discuss how tautological intersection numbers can be organized into polynomial families. Furthermore, these polynomials turn out to be Ehrhart polynomials of partial polytopal complexes. This connection with Ehrhart polynomials provides a novel enumerative interpretation of tautological intersection numbers.

Prerequisites: Some familiarity with Ehrhart theory and moduli of algebraic curves would be nice, but not absolutely necessary. I will introduce everything from scratch.

Title: Intersection Bodies of Polytopes 

Abstract: By computing the volume of slices of a polytope P, we can construct its intersection body IP, a fascinating (sometimes nonconvex) object coming from convex geometry.  In joint work with Katalin Berlow, Marie-Charlotte Brandenburg, and Chiara Meroni, we study the intersection body of a polytope and prove it is a semialgebraic set.  We further investigate its algebraic boundary and find an upper bound on the degree of its irreducible components.  This talk will include at least a few nice pictures!

Prerequisites: Linear Algebra.

Title: Shuffle-Compatibility: From Linear to Cyclic

Abstract: Since the early work of Richard Stanley, it has been observed that several permutation statistics—numerical parameters that encode properties of permutations—have a remarkable property related to shuffles. This notion of a shuffle-compatible permutation statistic was first explicitly studied by Gessel and Zhuang in 2018, who developed a unifying framework for shuffle-compatibility in which quasisymmetric functions play an important role. Since then, shuffle-compatibility has become an active topic of research. The first half of my talk will give an overview of the theory of shuffle-compatibility from my joint work with Ira Gessel; the second half will focus on more recent work—joint with Jinting Liang and Bruce Sagan—on shuffle-compatibility of cyclic permutation statistics, in which the role played by quasisymmetric functions is replaced by the cyclic quasisymmetric functions introduced by Adin, Gessel, Reiner, and Roichman.

Prerequisites: Basic familiarity with permutations, formal power series, vector spaces, and rings.

Title: Compactifying a Moduli Space: what are the missing objects?

Abstract: Imagine you have a hobby of collecting insects and classifying them according genus and species. You have a very large fly collection, organized by wing shape and body size, but the collection has a few holes in it. Based on the characteristics of flies near the hole, and the symmetry of nature, you strongly suspect that there is an as-yet-undiscovered species of fly out there that would fit perfectly into the missing spot in your collection. You don your exploring hat and set out to find that fly!

A moduli space is a mathematician's version of an insect collection: it is an organized collection of some kind of mathematical object. The moduli space of orbifold curves is a popular space in modern algebraic geometry that also has applications to a branch theoretical physics called Gromov-Witten theory. However, this moduli space has some holes in it, i.e., we expect there should be more orbifold curves out there than what people typically study. I will present a project joint with Martin Olsson to find the correct definition of the missing orbifold curves (for those who know, we define a generalization of Hassett's weighted stable curves to orbifold curves and show the resulting stack is proper). Besides the visceral satisfaction of filling in a hole in our moduli space, this project may facilitate computations in Gromov-Witten theory. 

The first portion of the talk will be spent explaining the concept of a moduli space and what it means to compactify such a space. This will require no background.

Prerequisites: In the later part of the talk I'll assume finite abelian groups (e.g. Z_4 x Z_7). Some other technical words that will arise are closed, bounded, and separated (from topology) and section and fiber of a morphism X --> Y, and algebraic curve. I will explain all those words in the talk with pictures, but if you want technical definitions you should look them up ahead of time.

Title: Kirchhoff polynomials and configuration hypersurfaces

Abstract: A finite graph determines a Kirchhoff polynomial, which is a

squarefree, homogeneous polynomial in a set of variables indexed by

the edges.  The Kirchhoff polynomial appears in an integrand in the

study of particle interactions in high-energy physics, and this provides

some incentive to study the motives and periods arising from the

projective hypersurface cut out by such a polynomial.

From the geometric perspective, work of Bloch, Esnault and Kreimer

(2006) suggested that the most natural object of study is a polynomial

determined by a linear matroid realization, for which the Kirchhoff

polynomial is a special case.

I will describe some ongoing joint work with Delphine Pol, Mathias

Schulze, and Uli Walther on the interplay between geometry and matroid

combinatorics for this family of objects.

Prerequisites: some basic commutative algebra.

Title: Navigating Mathematical Spaces: my journey and community

Abstract: In this talk I will present different stories from my mathematical journey.  I will highlight adversities, successes, and most importantly my mathematical community.  I present formative mathematical experiences and how these experiences have informed my mathematical trajectory and my outlook for my work. My hope is that my journey will allow you to also dream and imagine how younger mathematicians can shape their mathematical path.  I will also give a glimpse into the mathematical objects that I care about and some reflections on important mathematical concepts in my research area. 

Title: Signed permutohedra

Abstract: Postnikov has shown that generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, have remarkable formulas for their volumes and lattice point counts. Additionally, generalized permutohedra can always be decomposed into certain easy-to-understand polytopes. I will discuss a generalization of these results to signed generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, e_i + e_j, or e_i. Joint work with Chris Eur, Alex Fink, and Hunter Spink. 

Prerequisites: Familiarity with convex sets and rings. 

Title: Geometry of Algebraic Data

Abstract: Algebraic data is commonly encountered in science, but analyzing it meaningfully can be challenging. Understanding the geometrical shape of the data can reveal important information about clustering and voids, which are central to certain analyses. Algebraic Topology and Algebraic geometry are two branches of mathematics that offer powerful tools for analyzing algebraic data.In this talk, we will provide an introduction to analyzing algebraic data using Algebraic geometry and present recent results that demonstrate the power of these approaches. 

Prerequisites: algebraic varieties, projective spaces, distance function, tangency and normality. But I will not prove things so I will avoid technicalities and try to intuitively explain.