We propose a new framework of Markov α-potential games to study Markov games. 

We show that any Markov game with finite-state and finite-action is a Markov α-potential game, and establish the existence of an associated α-potential function. Any optimizer of an α-potential function is shown to be an α-stationary Nash equilibrium. We study two important classes of practically significant Markov games, Markov congestion games and the perturbed Markov team games, via the framework of Markov α-potential games, with explicit characterization of an upper bound for αand its relation to game parameters. 

Additionally, we provide a semi-infinite linear programming based formulation to obtain an upper bound for α for any Markov game. 

Furthermore, we study two equilibrium approximation algorithms, namely the projected gradient- ascent algorithm and the sequential maximum improvement algorithm, along with their Nash regret analysis.

This talk is part of the Erlangen AI Hub.

A 2024 collection of articles in the Bulletin of the AMS asked "Will machines change mathematics?", suggesting that  "Pure mathematicians are used to enjoying a great degree of research autonomy and intellectual freedom, a fragile and precious heritage that might be swept aside by a mindless use of machines." and challenging readers to  "decide upon our subject’s future direction.”

This was a response to the mathematical capabilities of emerging technologies, alone or in combination. These techniques include  software such as LEAN for  providing formal proofs; use of LLMs to produce credible, if derivative, research papers with expert human guidance; specialist algorithms such as AlphaGeometry; and sophisticated use of machine learning to search for examples.  Their development (at huge cost in compute power and energy) has been accompanied by an unfamiliar and exuberant level of hype from well-funded start-ups claiming to “solve mathematics” and the like. And it raises questions beyond the technical concerning governance, funding and the nature of the mathematical profession.

To try and understand what’s going on we look historical examples of changes in mathematical practice - as an example we consider key developments in the early days of computational group theory.

The speaker is keen to hear of colleagues using LLMs, LEAN or similar things in research, even if they can’t come to the talk.

Mathematicians often like to think of maths as objective. Science communicator Hana Ayoob joins us to discuss how the fact that humans do maths means that the ways maths is developed, used, and communicated are not neutral.

TBA

Speaker Prof Dr Maria Lukacova will talk about 'Numerical analysis of oscillatory solutions of compressible flows'

Oscillatory solutions of compressible flows arise in many practical situations.  An iconic example is the Kelvin-Helmholtz problem, where standard numerical methods yield oscillatory solutions. In such a situation,  standard tools of numerical analysis for partial differential equations are not applicable. 

We will show that structure-preserving numerical methods converge in general to generalised solutions, the so-called dissipative solutions. 
The latter describes the limits of oscillatory sequences. We will concentrate on the inviscid flows, the Euler equations of gas dynamics, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations.

We discuss a concept of K-convergence that turns a weak convergence of numerical solutions into the strong convergence of
their empirical means to a dissipative solution. The latter satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress.  We will also discuss suitable selection criteria to recover well-posedness of the Euler equations of gas dynamics. Theoretical results will be illustrated by a series of numerical simulations.