# TBA

## Speaker:

## Institution:

## Time:

## Host:

## Location:

TBA

# TBA

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

# Story of Holomorphic Dynamics

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

Holomorphic Dynamics (in a narrow sense) is the theory of the iteration of rational maps on the Riemann sphere. It was founded in the classical work by Fatou and Julia around 1918. After about 60 years of stagnation, it was revived in the 1980s, bringing together deep ideas from Conformal and Hyperbolic Geometry, Teichmüller Theory, the Theory of Kleinian Groups, Hyperbolic Dynamics and Ergodic Theory, and Renormalization Theory from physics, illustrated with beautiful computer-generated pictures of fractal sets (such as various Julia sets and the Mandelbrot set). We will highlight some landmarks of this story.

# TBA

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

# Holonomy - a success concept of modern differential geometry

## Speaker:

## Institution:

## Time:

## Host:

## Location:

Holonomy is a prime example of mathematical intuition and creativity - it generalises our school knowledge about the sum of angles in a triangle and led to `Berger’s holonomy theorem’ from 1954 which turned out to be a most successful research programme for differential geometry for over 50 years. We are going to tell the story of this development, how holonomy relates to curvature and advanced symmetry concepts, including a small detour to theoretical physics and what Calabi-Yau manifolds have to do with it. We conclude by a small outlook to recent results.

# Einstein Metrics, 4-Manifolds, and Gravitational Instantons

## Speaker:

## Institution:

## Time:

## Host:

## Location:

A Riemannian metric is said to be **Einstein** if it has constant Ricci curvature. Certain peculiar features of 4-dimensional geometry make dimension four into a “Goldilocks zone” for Einstein metrics, with just the right amount of local ﬂexibility managing to coexist with strong global rigidity results. This talk will ﬁrst describe some aspects of the interplay between Einstein metrics and smooth topology on compact symplectic 4-manifolds without boundary. We will see how ideas from Kähler and conformal geometry allow us to construct Einstein metrics on many such manifolds, while a complimentary tool-box shows that these existence results are optimal in certain specific contexts. The talk will then conclude with a brief discussion of analogous results concerning complete Ricci-ﬂat 4-manifolds.

# Isospectral connections, frame flow ergodicity, and polynomial maps between spheres

## Speaker:

## Institution:

## Time:

## Location:

Classifying real polynomial maps between spheres is a challenging problem in real algebraic geometry. Remarkably, this question has found recent applications in two seemingly unrelated fields:

- in spectral theory, it allowed to solve Kac's celebrated isospectral problem (Can one hear the shape of a drum?) for the connection Laplacian.

- in dynamical systems, it allowed to prove ergodicity for a certain class of partially hyperbolic flows (extensions of the geodesic flow on negatively-curved manifolds).

I will explain these problems and how they all connect together. No prerequisite required -- the talk is intended for a broad audience.

*Joint work with Mihajlo Cekić.*

# On KPZ universality and stochastic flows

## Speaker:

## Institution:

## Time:

## Location:

We will start by introducing the phenomenon of the KPZ (Kardar-Parisi-Zhang) universality. KPZ problem was a very active research area in the last 20 years. The area of KPZ is essentially interdisciplinary. It is related to such fields as probability theory, statistical mechanics, mathematical physics, PDE, SPDE, random dynamics, random matrices, and random geometry, to name a few.

In most general form the problem can be formulated in the following way. Consider random geometry on the two-dimensional plane. The main aim is to understand the asymptotic statistical properties of the length of the geodesic connecting two points, which are far away from each other, in the limit as distance between the endpoints tends to zero. One also wants to study the geometry of random geodesics, in particular how much they deviate from a straight line. It turn out that the limiting statistics for both the length and the deviation is universal, that is it does not depend on the details of the random geometry. Moreover, many limiting probability distributions can be found explicitly.

In the second part of the talk we will proceed with discussion of the geometrical approach to the problem of the KPZ universality which provides an even broader point of view on the problem of universal statistical behavior.

No previous knowledge of the subject will be assumed.

# Dynamics on homogeneous spaces: a quantitative viewpoint

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Host:

## Location:

Rigidity phenomena in homogeneous spaces have been extensively studied over the past few decades with several striking results and applications. We will give an overview of activities pertaining to the quantitative aspect of the analysis in this context with an emphasis on recent developments.