## Juni 2014 – Juli 2018

Vortrag im Rahmen des Seminars “Algebra und Geometrie”

I will discuss the concept of affine representation for topological dynamical systems. This leads naturally to the study of dynamics induced onto the space of probability measures. Some qualitative relationships between a dynamical system and its lifting to the probability space are shown. I will also give a differentiable approach to dynamics on the space of probability measures. This structure implies that notions from differentiable dynamics may be carried over to the representation of a system that has no differentiable structure itself.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Graphene is a two-dimensional material made up of a single atomic layer of carbon atoms arranged in honeycomb pattern. Many of its remarkable electronic properties, e.g. quasi-particles (wave-packets) that propagate as massless relativistic particles and topologically protected edge states, are closely related to the spectral properties of the underlying single-electron Hamiltonian: -Laplacian + V(x), where V(x) is a potential with the symmetries of a hexagonal tiling of the plane. Taking inspiration from graphene, there has been a great deal of activity in the fundamental and applied physics communities related to the properties of waves (photonic, acoustic, elastic,…) in media whose material properties have honeycomb symmetry. In this talk l will review progress on the mathematical theory.

Sherlock Holmes and doctor Watson take a ride on an air-balloon. After a sudden gust of wind takes them in an unknown direction, they spot a man on the ground and inquire about their location. After a short moment of consideration, the man answers: “You are on the air balloon”. “This man is a mathematician!” — concludes Sherlock, while wind carries the balloon further. “But how do you know?” — wonders Dr. Watson. “For three reasons, dear Doctor. First, he thought before answering, second, his answer is absolutely correct and, finally, his answer is absolutely useless.”

While we, with J. Portegies, were developing the theory of tropical probability spaces, the so fitting description of mathematical work given by Sherlock was rather frustrating, because the value of any theory is in its applications outside of itself.

Now that we have developed fairly sophisticated tools and are learning how to use them, the first fruits, though small and green, start appearing.

I will introduce the toolbox of tropical probability spaces and will show how it can be used to deduce a non-Shannon inequality for entropies of four random variables and their joints.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Martensitic transformation is a shear-dominant, lattice distortive and diffusionless solid-solid transformation occurring by nucleation and growth. Shape memory alloy exhibits a martensite microstructure, which is a complex pattern of martensitic domains. In this study, the character of the interfaces between the martensite domains, dynamics of the formation of the microstructure and the emergence of power-law in the domain size distribution are investigated by various recent microscopy techniques in shape memory alloys. The experimental results are analyzed in the framework of the nonlinear elasticity theory of the microstructure which was founded by Ball and James, to bridge the theory and experiment and to elucidate underlying problems to be solved.

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2018/geometry2018/index.html

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2018/geometry2018/index.html

Automated invariant generation is a fundamental challenge in program analysis and verification, with work going back several decades. In this talk I will present a select overview and survey of previous work on this problem, and I will describe a new algorithm to compute all polynomial invariants for the class of so-called affine programs (programs that allow affine assignments and non-deterministic branching). Our main technical contribution is a mathematical result of independent interest: an algorithm to compute the Zariski closure of a finitely generated matrix semigroup.

This is joint work with Ehud Hrushovski, Joel Ouaknine, and Amaury Pouly.

More information you can find on the conference homepage:

https://www.mis.mpg.de/calendar/conferences/2018/geometry2018/index.html

At the conference FPSAC 2017 in London, Luc Lapointe gave a plenary talk about his work, together with many other collaborators, about superspace. After attending the talk, Mike Zabrocki and I were very interested in this work and we start reading about it and having questions (and conjectures!). In this talk, I would like to present what it is known about superspace, what I am interested in and other open questions.

Colloquium of the Max Planck Institute

Linear recurrence sequences (LRS), such as the Fibonacci numbers, permeate vast areas of mathematics and computer science. In this talk, we consider three natural decision problems for LRS over the integers, namely the Skolem Problem (does a given LRS have a zero?), the Positivity Problem (are all terms of a given LRS positive?), and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). Such questions have applications in a wide array of scientific areas, ranging from theoretical biology and software verification to quantum computing and statistical physics. Perhaps surprisingly, the study of decision problems for linear recurrence sequences (and more generally linear dynamical systems) involves techniques from a variety of mathematical fields, including analytic and algebraic number theory, Diophantine geometry, and algebraic geometry. I will survey some of the known results as well as recent advances and open problems.

This is joint work with James Worrell

About 30 years ago Nigel Hitchin introduced a new class of so called completely integrable systems. These geometric objects had their origin in a set of differential equations but surprisingly turn out to have various alternative descriptions that allow for arithmetic and topological applications. Thus a wide range of methods have been applied to get a better understanding of the underlying spaces.

In this talk I will try to explain a little bit about the history of these spaces and of some of the methods used to study them. After giving some more recent insights into their geometry I would like to explain what the P=W conjecture of de Cataldo, Hausel and Migliorini is about, which gives an unexpected link between two distinct algebraic structures of the manifolds appearing in Hitchin’s construction.