The Utrecht Geometry Center seminar is the biweekly pure mathematics colloquium from Utrecht University. In its current incarnation, talks are mostly given by our own members, explaining their work to the whole department in an approachable manner.

**When:**Tuesdays 17.00-18.00**(note the change in time!)**. We encourage attendees to stay around afterwards for informal discussion.**Where:**Online over Zoom.**How:**The biweekly invitations to the Zoom meetings are sent to the mailing lists for staff and master students. If you have not received an announcement, please do write to one of the organisers (see below).**Videos:**Recordings of previous talks can be accessed on our Youtube channel. Links to each individual talk can be found below together with their titles and abstracts.

#### Upcoming talks:

January 26, 2021

February 9, 2021

February 23, 2021

#### Previous talks:

January 12, 2021

**Title:** *Brownian motion in Riemannian manifolds*

**Speaker:** Rik Versendaal

When studying stochastic processes, and diffusions in particular, one of the most important processes is Brownian motion. Brownian motion is, in a way, the natural analogue of the normal distribution for processes. Furthermore, it is intimately related to the heat equation, since the Laplacian describes the infinitesimal evolution of Brownian motion. We will look at how to define and construct Brownian motion in a Riemannian manifold, the so-called Riemannian Brownian motion. There are various ways to do this, both geometric and probabilistic in nature.

First of all, as mentioned above, we can consider the process generated by the Laplacian of the Riemannian manifold, i.e., the Laplace-Beltrami operator. Second, we can use an invariance principle. In Euclidean space, this states that the paths of suitably scaled random walks converge to Brownian motion. One can define an analogue of random walks in manifolds, so-called geodesic random walks, and use these to obtain Riemannian Brownian motion in the limit.

Finally, Riemannian Brownian motion can also be obtained in a geometric way from a Euclidean Brownian motion. The idea is that we can transfer curves in Euclidean space to a manifold by suitably rolling the manifold along the curve. By Malliavin's transfer principle, it turns out that this also makes sense for stochastic processes. In particular, if we roll the manifold along a Euclidean Brownian motion, we will obtain a Brownian motion in the manifold.

If time permits, we will look into some results regarding large deviations for Riemannian Brownian motion. These large deviations are concerned with quantifying exponentially small probabilities of atypical trajectories of Brownian motion with vanishing variance. In particular, the action of a trajectory determines the exponential rate of decay of the probability. This can be shown to hold even in time-evolving Riemannian manifolds, i.e., manifolds where the metric depends on time.

December 22, 2020

**Title:** *From Poisson Geometry to (almost) geometric structures*

**Speaker:** Marius Crainic

I will report on an approach to general geometric structures (with an eye on integrability) based on groupoids endowed with multiplicative structures; Poisson geometry (with its symplectic groupoids, Hamiltonian theories and Morita equivalences) will provide us with some guiding principles. This allows one to discuss general "almost structures" and an integrability theorem based on Nash-Moser techniques (and this also opens up the way for a general "smooth Cartan-Kahler theorem").

This report is based on collaborations/discussions with Francesco Cataffi (almost structures), Ioan Marcut (Nash-Moser techniques), Maria Amelia Salzar (Pfaffian groupoids).

December 8, 2020

**Title:** *Abelian varieties over finite fields*

**Speaker:** Valentijn Karemaker

Abelian varieties are algebro-geometric objects with a rich arithmetic structure. Over finite fields, the rational points on these varieties can be understood through arithmetic invariants like the zeta function. I will introduce these varieties and describe some ways in which they have come up in my work.

November 24, 2020

**Title:** *Solving polynomial equations in many variables in primes*

**Speaker:** Shuntaro Yamagishi

Solving polynomial equations in primes is a fundamental problem in number theory. For example, the twin prime conjecture can be phrased as the statement that the equation x-y-2=0 has infinitely many solutions in primes. In this talk, I will talk about some results related to solving F = 0 in primes, where F is a more general higher degree polynomial. It will be a light introduction to the topic, and knowing what prime numbers and polynomials (in more than one variable) are should be enough to follow most of the talk.

November 10, 2020

**Title:** *Rational points on Fano varieties*

**Speaker:** Marta Pieropan

Fano varieties form one of the fundamental classes of building blocks in the birational classification of algebraic varieties. In this talk I will discuss how their special geometric properties can be used to study their arithmetic. I will focus on a few conjectures about their rational points over number fields (potential density, Manin’s conjecture) and how they can be investigated by determining the asymptotic behavior of certain counting functions of rational points.

All the main objects involved will be introduced and illustrated by examples. A summary of the literature on the topic will be discussed, including my own contribution.

October 27, 2020

**Title:** *Counting quaternion algebras, with applications to spectral geometry*

**Speaker:** Lola Thompson

We will introduce some classical techniques from analytic number theory and show how they can be used to count quaternion algebras over number fields subject to various constraints. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts can be used to produce quantitative results in spectral geometry. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.