Workshop Projects

Here is a list of proposed projects for this workshop.

(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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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.

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Gauge transform of linear systems of PDEs

Group leader(s): Anna-Laura Sattelberger

Abstract: The Weyl algebra, denoted $D$, is a non-commutative $\mathbb{C}$-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 $\mathbb{C}$-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 {\tt Dmodules} of Anton Leykin.

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 $\mathbb{C}(x)$-basis ${ 1,s_2,\ldots,s_m } $ of $R/RI$. The $s_i$'s can be chosen to be monomials in the partial derivatives $\partial_1,\ldots,\partial_n$. 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 $F=(f,s_2\bullet f,\ldots,s_m\bullet f)^\top$. Then for any $i=1,\ldots,n$, $\partial_i \bullet F=A_i \cdot F$ for an $m\times m$ matrix $A_i$ with entries in $\mathbb{C}(x)$. 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 $\tilde{A}_i = G^{-1}A_i G - G^{-1}\partial_i \bullet G$. A user-friendly implementation of the gauge transform for connection matrices will be useful for yoga with PDEs both for mathematicians and physicists. In applications in physics, some specific forms of connection matrices are particularly desirable, namely in the case of the dependance on a small parameter $\varepsilon$. For instance, for the so-called $\varepsilon$-factorized form of connection matrices, one can iteratively construct series solutions in $\varepsilon$ via the path-ordered exponential formalism. However, in full generality, there is no algorithmic way to reach specific forms of connection matrices.

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Algebraic Statistics

Group leader(s): Carlos Amendola, Ben Hollering

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