HCM Graduate Colloquium, Winter Term 2024/25

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Organisers: Michel Alexis, Regula Krapf, Fred Lin, Luise Puhlmann, Christoph Thiele, Radu Toma

This seminar is organised as a BIGS event. The goal is to present topics from all areas of mathematics in an elementary and informal way. The talks should be accessible to a general mathematical audience and are mainly aimed at BIGS students.

Everybody (students, postdocs, faculty, guests) is welcome to attend.
If you would like to give a talk please contact us.

The seminar will take place Wednesdays 15:15 - 16:45 in the Lipschitzsaal. The talks will usually take about one hour and there is the subsequent possibility to ask questions. Coffee, tea and cake will be served beforehand between 15:00 and 15:15 in the Plückerraum.

A predecessor of the HCM Graduate Colloquium is the Basic Notions Seminar which took place until 2017:
Basic Notions Seminar Summer Term 2017

Date	Speaker	Topic
23.10.2024	Ruoyuan Liu (MI)	Role of invariant Gibbs measures in Hamiltonian PDEs
06.11.2025	Jan Bohr (MI)	Geometric optics, scattering and rigidity
27.11.2024	Johannes Linn (MPIM)	Bounding Exponential Sums
18.12.2024	Tba.	Tba.
22.01.2025	Annika Tarnowsky (MPIM)	Computing Differentiable Stack Cohomology

Abstracts

October 23, 2024: Ruoyuan Liu (MI)

Title: Role of invariant Gibbs measures in Hamiltonian PDEs

Abstract. Gibbs measures, rooted in statistical mechanics, provide a probabilistic perspective in the study of Hamiltonian partial differential equations (PDEs). As opposed to individual trajectories, invariant Gibbs measures inform us of the typical behavior of solutions to an Hamiltonian PDEs. In this talk, I will introduce invariant Gibbs measures for Hamiltonian systems, from the finite dimensional setting via ordinary differential equations to the more complicated infinite dimensional PDEs. In particular, I will mention Bourgain's invariant measure argument in constructing a solution for the nonlinear Schrödinger equation.

November 6, 2024: Jan Bohr (MI)

Title: Geometric optics, scattering and rigidity

Abstract. In geometric optics, light propagation is described by paths that (locally) minimise travel time. In a medium with variable refractive index these optical paths can bend, loop around a point, and create other interesting patterns. This leads to a number of interesting questions (can we build an invisible lens? which scattering patterns can be created by a lens?) that can be brought into a precise mathematical form and are subject to current research. In the talk I'll motivate these questions and discuss how Riemannian geometry, complex analysis and twistor spaces enter the picture.

November 27, 2024: Johannes Linn (MPIM)

Title: Bounding Exponential Sums

Abstract. Exponential Sums or Character Sums appear at many places in mathematics and estimating them is often an essential step in solving related problems. In this talk, we want to give an intuition for the meaning of the size of exponential sums and look at some examples of sums, their origin, and techniques to bound them.

January 22, 2025: Annika Tarnowsky (MPIM)

Title: Computing Differentiable Stack Cohomology

Abstract. Higher Differential Geometry is the intersection point between (Higher) Category Theory and Differential Geometry. The field concerns itself with concepts such as that of a Lie groupoid, which is a special type of category where objects and morphisms assemble into manifolds. A Differentiable Stack is a notion that has arisen more recently and is elusive at first glance, but it turns out to be closely related to well-known objects - in particular Lie groupoids - which can be employed for their study. Using this relation, Differentiable Stack Cohomology can be interpreted as a generalisation of Equivariant Cohomology, for which there are models that significantly simplify its computation. This poses the question if a similar method can be applied to Differentiable Stack Cohomology in general. In this talk, I will survey the relevant notions mentioned here and the relations between them as well as the research progress on the given problem including recent advances during my PhD project.