December 6, 2023December 6, 2023 by alkhwarizmi

Paul Breiding

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$\begin{quote} \begin{center} \textbf{Using Homotopy Continuation} \end{center} \medskip My lecture is on polynomial homotopy continuation (PHC) and it consists of two parts. In the first part I will explain the theory that one needs to understand in order to effectively use PHC. In the second part I will show example applications. \end{quote}$

December 6, 2023December 10, 2023 by alkhwarizmi

Bernd Sturmfels

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$\begin{quote} \begin{center} \textbf{The Quadratic Formula Revisited} \end{center} \medskip High school students learn how to express the solution of a quadratic equation in one unknown in terms of its three coefficients. Why does this formula matter? We offer an answer in terms of discriminants and data. This lecture invites the audience to a journey towards non-linear algebra. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Yue Ren

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$\begin{quote} \begin{center} \textbf{Tropical Geometry of neural networks - Introduction, overview, and outlook} \end{center} \medskip In this talk, we provide a basic introduction to tropical geometry with special emphasis on its applications in machine learning. We will cover mathematical topics such as classical and tropical Grassmannians, the duality between tropical hypersurfaces and regular subdivisions. We will touch upon machine learning questions such as quantifying the expressivity of neural networks, and dealing with hierarchical data such as in phylogenetics. Part of the talk will be dedicated to an outlook on potential future applications and the mathematical challenges that they entail. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Marianne Akian

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$\begin{quote} \begin{center} \textbf{Repeated zero-sum games: from operators to algorithms and tropical geometry} \end{center} \medskip Repeated zero-sum 2-player deterministic or stochastic games with a finite state space can be studied with the help of the Shapley dynamic programming equation. This equation allows one to compute not only the value function but also optimal strategies of the two players. When considering mean-payoff criteria, under an ergodicity condition, the dynamic programming equation reduces to a fixed point problem of a monotone operator acting on a projective space. It is thus related to nonlinear Perron-Frobenius theory. This approach also applies to entropy games -- a multiplicative version of mean-payoff stochastic games. This talk will present an overview of existence and uniqueness results for such non-linear fixed point problems. It will include applications to the convergence and complexity analysis of algorithms, showing also how these techniques can be used to study and solve problems from tropical geometry. This talk is based on joint works with Antoine Bereau, Stéphane Gaubert, Julien Grand Clément, Jérémie Guillaud, Alexander Guterman, Antoine Hochart, Ulysse Naepels, Yang Qi, Omar Saadi, Basile Terver. \end{quote}$