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Mahrud Sayrafi edited this page Nov 1, 2024 · 18 revisions

Here is a list of proposed projects for this workshop.

Game Theory

Group leader(s): Irem Portakal

Abstract: The existence of Nash equilibria was proven by Nash in 1950 using fixed-point theorems from topology, and he did so in just one page. This breakthrough sparked great interest in the game theory community, and it has remained a hot topic ever since. A similar shift is now occurring with the use of algebraic geometry in game theory. Computing different types of equilibria is a nonlinear algebra problem, using both classical functions from Macaulay2 and newer packages like GraphicalModels. Our goal is to create a GameTheory package that combines all of these tools in one.

References:

Algebraic Analysis

Group leader(s): Anna-Laura Sattelberger, Mahrud Sayrafi

Abstract: The Weyl algebra, denoted $D$, is a non-commutative $ℂ$-algebra that encodes linear differential operators with polynomial coefficients in the variables $x_1,\ldots,x_n$. To an operator $P\in D$, one associates the partial differential equation $P\bullet f(x_1,\ldots,x_n)=0$. Systems of such PDEs correspond to left ideals in the Weyl algebra. Given a $D$-ideal $I$, one has a big freedom in choosing generators of $I$—for instance, by varying the monomial order on $D$, one obtains different Gröbner bases of $I$.

In applications, one often enlarges the coefficient ring to rational functions in the $x$-variables. The resulting ring of differential operators is denoted $R$. The quotient $R/I$ is a vector space over the field of rational functions. Its dimension is the "holonomic rank" of the $D$-ideal.

If $I$ is holonomic, it follows from the theorem of Cauchy–Kovalevskaya–Kashiwara, that outside the singular locus of $I$, the $ℂ$-vector space of holomorphic solutions to $I$ has finite dimension, given by the holonomic rank of $I$. In Macaulay2, the Weyl algebra is implemented in the package Dmodules.

Now let $I$ be a $D$-ideal of holonomic rank $m$. In physics, it is common to write D-ideals in connection form. To do so, one chooses a $ℂ(x)$-basis $\{1,s_2,\dots,s_m\}$ of $R/RI$. The $s_i$'s can be chosen to be monomials in the partial derivatives $\partial_1,\dots,\partial_n$. For instance, the standard monomials of $I$, considered as an ideal in $R$, give rise to a basis of $R/RI$. For a solution $f$ of $I$, denote $F=(f,s_2\bullet f,\dots,s_m\bullet f)^\top$. Then for any $i=1,\dots,n$, $\partial_i \bullet F=A_i \cdot F$ for an $m\times m$ matrix $A_i$ with entries in $ℂ(x)$. The matrices $A_i$ are called "connection matrices" and are obtained by reducing the $\partial_is_j$'s modulo $I$. Changing the considered basis of $R/RI$ results in a gauge transform of the connection matrices, i.e., for $\tilde{F}=GF$ for an invertible $m\times m$ matrix $G$, the corresponding connection matrices will be $\tilde{A}_i = G^{-1}A_i G - G^{-1}\partial_i \bullet G$.

Project goals:

  1. A Gröbner basis implementation over the rational Weyl algebra.
  2. A user-friendly implementation of the gauge transform for connection matrices.

This will be very useful for yoga with PDEs both for mathematicians and physicists.

Algebraic Statistics

Group leader(s): Carlos Amendola, Ben Hollering

Abstract: Graphical models are fundamental objects in statistics which use a graph to represent conditional independence structures and causal relationships between random variables. When the random variables involved are Gaussian or discrete, the parameterization of the statistical model is actually a rational map and thus can be studied with tools from algebraic geometry. This project will be focused around extending the functionality in GraphicalModels.m2 to new families of statistical models as well as unifying some of the various packages used for computation concerning graphical models.

(Positive) Tropical Geometry

Group leader(s): Yue Ren

Abstract: This project revolves around tropical geometry in OSCAR and M2. Participants are free to propose topics and pursue their own project.

The general plan is to recall different notions of positive tropicalizations, and algorithms ways to compute them. When possible, we will implement them in OSCAR and M2.

The goal is to create a comprehensive toolset for future experiments on positive tropicalizations.

Literature:

  • D. Bendle, J. Boehm, Y. Ren, B. Schroeter: Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians https://arxiv.org/abs/2003.13752
  • K. Rose, M.L. Telek: Computing positive tropical varieties and lower bounds on the number of positive roots https://arxiv.org/abs/2408.15719
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