Computations and Applications of Algebraic Geometry and Commutative Algebra

December 9-13, 2024

Organizers: John Cobb, Anand Deopurkar, Sione Ma’u, and Hal Schenck.

Overview

This is a special session at the AMS-AustMS-NZMS joint meeting in Auckland, New Zealand, taking place December 9-13, 2024. The goal of this meeting is to bring together experts in algebraic geometry and commutative algebra, with a emphasis on computations and applications.

Talks

There are thirteen 25-minute talks by:

• Christin Bibby (Louisiana State University)
  A Serre spectral sequence for the moduli space of tropical curves

  Abstract: The map $M_{g,n} \to M_g$ on moduli spaces of genus $g$ algebraic curves, given by forgetting marked points, is a fibration whose fiber is a configuration space of a surface. One can then "in principle" compute the cohomology of $M_{g,n}$ using the Serre spectral sequence. We present a tropical analogue of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs. We use this to obtain new calculations in the top weight cohomology of the moduli spaces $M_{2,n}$ and $M_{3,n}$.
• Michael Brown (Auburn University)
  Computing sheaf cohomology over noncommutative projective schemes

  Abstract: Given a commutative graded algebra $A = \bigoplus_{i \ge 0} A_i$ over a field $k$ such that $A_0 = k$, one has an associated projective scheme $X = \operatorname{Proj}(A)$. When $A$ is not commutative, no such scheme $X$ exists, but one may nevertheless define a category associated to $A$ that has many of the same homological properties as the category of coherent sheaves on a projective scheme. In particular, there is a noncommutative analogue of sheaf cohomology for objects in this category. Our goal is to develop a method for computing noncommutative sheaf cohomology in the case where $A$ is Koszul and Gorenstein. As an application, we prove a noncommutative generalization of the Horrocks splitting criterion, which is a necessary and sufficient condition for a vector bundle on projective space to split as a sum of line bundles.
• Anand Deopurkar (Australian National University)
  How twisty is that orbit?

  Abstract: Fix a polynomial $P$. How complicated is the set of all polynomials obtained from $P$ by changes of coordinates? This question, and many others, generalise as follows: given a representation an algebraic group, how complicated is the orbit of a given element? I will describe some answers, featuring toric varieties, Newton polyhedra, and stacks.
• Jörg Frauendiener (University of Otago)
  A computational approach to Riemann surfaces with applications in Physics

  Abstract: Integrable systems have a wide range of applications in fluid mechanics, general relativity and other areas of science. Many such systems have solutions which can be described in terms of Theta functions defined on Riemann surfaces. In this talk, we discuss our computational approach to evaluate these functions given data to define a Riemann surface. We show how to obtain a homology basis, the independent holomorphic differentials and from those the corresponding Riemann matrix. We demonstrate some applications, such as solutions to some integrable equations, the visualisation of stationary, axisymmetric relativistic systems. Finally, we discuss an application to the Schottky problem.
• Elizabeth Gross (University of Hawai`i at Mānoa)
  Computational algebraic geometry for evolutionary biology

  Abstract: A main goal of phylogenomics is to understand the evolutionary history of a set of species. These histories are represented by directed graphs where the leaves represent living species and the interior nodes represent extinct species. While it is common to assume the evolutionary history is a tree, when events such as hybridization are present, networks are more realistic. However, allowing for networks, rather than simply trees, complicates the process of inference. One recent approach to phylogenetic network inference is rooted in computational algebraic geometry. In this talk, we discuss the role computational algebraic geometry and symbolic computation has played in the statistical problems related to network inference with a focus on problems related to identifiability and model selection.
• Changho Han (Korea University)
  Extending the Torelli map to alternative compactifications of the moduli space of curves

  Abstract: It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension $g$), extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus $g$ with axis-like singularities? As a joint work with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with axis-like singularities and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.
• Martin Helmer (North Carolina State University)
  Effective Whitney Stratification and Applications

  Abstract: In this talk I will describe two algorithms to compute Whitney stratifications of real and complex algebraic varieties. I will begin with an overview of how the structure of the conormal variety is related to this problem and how we can exploit this to give an algorithm to compute a Whitney stratification. The main computational step in this algorithm involves finding the associated primes of a polynomial ideal. I will then explore how this approach can be made more efficient by using techniques for equidimensional decomposition of varieties rather than computing the full set of associated primes. This modified algorithm will yield a quite significant speedup but may fail to produce a minimal Whitney stratification. Time permitting, I will additionally present an algorithm to coarsen any Whitney stratification of a complex variety to a minimal Whitney stratification. Finally I will illustrate applications of the methods to the study of Feynman integrals in mathematical physics. This talk contains content of three separate joint works, one with Vidit Nanda (Oxford), one with Rafael Mohr (Sorbonne Université), and one with Felix Tellander (Oxford) and Georgios Papathanasiou (City University, London).
• Arvind Kumar (New Mexico State University)
  Generalized Hamming weights and symbolic powers of Stanley-Reisner ideals of matroids

  Abstract: It is well-known that the first generalized Hamming weight of a code, more commonly called the minimum distance of the code, corresponds to the initial degree of the Stanley-Reisner ideal of the matroid of the dual code. Our starting point in this paper is a generalization of this fact -- namely, the $r$-th generalized Hamming weight of a code is the smallest degree of a squarefree monomial in the $r$-th symbolic power of the Stanley-Reisner ideal of the matroid of the dual code (in the appropriate range for $r$). It turns out that the squarefree monomials in successive symbolic powers of the Stanley-Reisner ideal of a matroid suffice to describe all symbolic powers of the Stanley-Reisner ideal. This implies that generalized Hamming weights -- which can be defined in a natural way for matroids -- are fundamentally tied to the structure of symbolic powers of Stanley-Reisner ideals of matroids. We illustrate this by studying initial degree statistics of symbolic powers of the Stanley-Reisner ideal of a matroid in terms of generalized Hamming weights and working out many examples that are meaningful from a coding-theoretic perspective. Our results also apply to projective varieties known as matroid configurations introduced by Geramita-Harbourne-Migliore-Nagel.
• Diane Maclagan (University of Warwick)
  Toric Bertini theorems in arbitrary characteristic

  Abstract: The classical Bertini theorem on irreducibility when intersecting by hyperplanes is a standard part of the algebraic geometry toolkit. This was generalised recently, in characteristic zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for subvarieties of an algebraic torus, with hyperplanes replaced by subtori. I will discuss joint work with Gandini, Hering, Mohammadi, Rajchgot, Wheeler, and Yu in which we give a different proof of this theorem that removes the characteristic assumption. The proof surprisingly hinges on better understanding algebraically closed fields containing the field of rational functions in n variables, which involve polyhedral constructions. An application is a tropical Bertini theorem.
• Scott Mullane (University of Melbourne)
  The Kodaira classification of the moduli space of pointed hyperelliptic curves

  Abstract: The moduli space of pointed hyperelliptic curves is a seemingly simple object with perhaps unexpectedly interesting geometry. I will report on joint work with Ignacio Barros completing the classification of both the Kodaira dimension and the structure of the effective cone of these moduli spaces.
• Greg Smith (Queen’s University)
  Cohomology of toric vector bundles

  Abstract: A toric vector bundle is a vector bundle on a toric variety $X$ equipped with a torus action that is compatible with the canonical action on $X$. Klyachko proves that toric vector bundles are classified by finite-dimensional vector spaces with a suitable family of filtrations. Building on this equivalence of categories, we construct a complex of modules over the Cox ring of $X$ which simultaneously encodes the cohomology of a toric vector bundle and many of its twists by line bundles. Beyond improved computational efficiency, this approach leads to new insights into virtual resolutions. This talk is based on joint work with Michael Perlman.
• Frank Sottile (Texas A&M University)
  The Critical Point Degree of a Bloch Variety

  Abstract: Given an operator on a ${\mathbb Z}^d$-periodic graph, its Bloch variety encodes its spectrum with respect to the unitary characters of ${\mathbb Z}^d$. Finer questions about the spectrum involve understanding the critical points of the projection to ${\mathbb R}$. Previous work with Faust gave a bound for the number of complex critical points in terms of the volume of the Newton polytope of the dispersion polynomial. This talk will present background and then describe refined bounds on the number of critical points that are combinatorial in nature and involve an analysis of asymptotic behavior of the Bloch variety. This is joint work with Faust and Robinson.
• Prashanth Sridhar (Auburn University)
  Noncommutative geometry over dg-algebras

  Abstract: Pioneering work of Artin-Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of such a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss extensions of some classical and modern results in the theory of nc projective geometry to nc spaces associated to dg-algebras. This is joint work with Michael K. Brown.

Schedule