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.

**UGC talks:**They will take place on Tuesdays 16.00-17.00, in person at HFG611 and online over Zoom.**MI talks:**They will take place on Tuesdays 16.00-17.00, in person at the HFG Library and online over Zoom.**How to attend online:**The biweekly invitations to the Zoom meetings are sent to the UGC and master students mailing lists. Invitations for the MI-talks are sent to all staff. 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 18, 2022

**Title:** *On the cohomology of moduli spaces of stable pointed curves: genus 4 and small n, and the vanishing of H^7 and H^9*

**Speaker:** Carel Faber

After an introduction to moduli spaces of curves, I will discuss joint work with Jonas Bergström and Sam Payne. For n at most 3, we determine the relevant terms of the point count of the moduli space M_{4,n} of smooth n-pointed curves of genus 4 over finite fields (S_n-equivariantly). Using several earlier results, we then determine the cohomology of the moduli space overline{M}_{4,n} of stable n-pointed curves of genus 4 and the full count for M_{4,n} and we deduce the mentioned vanishing for all g and n

#### Previous talks:

November 16, 2021

**Title:** *Where symplectic becomes complex*

**Speaker:** Gil Cavalcanti

Generalised complex structures are a simultaneous generalisation of complex and symplectic manifolds, but differently from those two cases, generalised complex structures are not homogeneous and have nontrivial local invariants. In fact, in a connected manifold the structure can be symplectic at some points and complex at other points, that is, its type can change. This type changing behaviour is a rich source of puzzles. I will review the basics of generalised complex geometry, including its applications, then explain the main results that allow us to deal with these type change points and possible consequences of these. This talk is based on joint work with Bailey and Leer-Duran.

November 2, 2021

**Title:** *Lie groups, Whittaker functions and Fourier inversion (MI-talk) *

**Speaker:** Erik van den Ban

We will explain certain recent results in harmonic analysis on the class of real reductive Lie groups for the simplest example: the special linear group SL(2,R). In that setting our results are expressible in terms of a spectral decomposition associated with the classical Whittaker ODE.

The talk will not presuppose knowledge of the theory of Lie groups.

October 19, 2021

**Title:** *From K-theory to calculus*

**Speaker:** Niall Taggart

Topological K-theory is an extension of algebraic K-theory to the world of geometry. Building from vector spaces (say over the real or complex numbers) Atiyah constructed an invariant of topological spaces which behaves similarly to how algebraic K-theory behaves as an invariant of rings. Depending on one's choice of the base field we obtain different invariants which are intricately related.

Also starting with real or complex vector spaces as the foundations, one can describe a categorification of differential calculus which studies functors from your category of vector spaces to the category of topological spaces. This calculus has much in common with differential calculus including the ability to examine the rate-of-change of a functor and a version of Taylor’s Theorem which provides a filtration of a topological space by “polynomial” parts.

In this talk, I will aim to provide some intuition behind these two seemingly unrelated constructions and discuss how the intricate relationship between real and complex K-theory offers deep insight into the calculus. If time permits I will discuss some other categorifications of calculus and how they relate to each other.

October 5, 2021

**Title:** *Integrality of instanton numbers*

**Speaker:** Frits Beukers

In a famous 1995 paper on mirror symmetry Candelas, de la Ossa, Greene and Parkes discovered a remarkable relation between certain numbers (named instanton numbers) arising from a classical family of Calabi-Yau 3-folds and counting the number of rational curves of given degree on a general quintic threefold in projective space. This discovery was driven by arguments from physics, but mathematically it was not known that these numbers are integers for a long time. In this talk I describe some work together with Masha Vlasenko that gives some insight in this integrality.

September 21, 2021

**Title:** *The characters behind the large sieve*

**Speaker:** Lasse Grimmelt

Multiplicative and Additive characters of Z/qZ play an important role in number theory. We will take a look how they interact with the so called large sieve inequality. We show how their basic orthogonality properties are combined with some simple analysis to derive surprising results. These results are foundation to much of our progress in prime number research in the last 70 years. While motivated by this applications, the talk will focus on the role of characters.

June 8, 2021

**Title:** *Kodaira dimension of moduli spaces*

**Speaker:** Martin Moeller (U. Frankfurt). ** Note: The talk will start at 17:30. **

We give a brief overview about the motivation for studying the Kodaira dimension of moduli spaces and summarize results for classical moduli spaces. We then turn attention to moduli spaces of flat surfaces and give recent results about their Kodaira dimension.

June 8, 2021

June 1, 2021

**Title:** *Canonical decomposition of rational maps*

**Speaker:** Mikhail Hlushchanka

There are various classical and more recent decomposition results in mapping class group theory, geometric group theory, and complex dynamics (which include celebrated results by Bill Thurston). The goal of this talk is to introduce a novel powerful decomposition of rational maps based on the topological structure of their Julia sets. Namely, we will discuss the following result: every postcritically-finite rational map with non-empty Fatou set can be canonically decomposed into crochet maps (these have very "thinly connected" Julia sets”) and Sierpinski carpet maps (these have very "heavily connected" Julia sets). If time permits, I will discuss applications of this result in various aspects of geometric group theory. Based on a joint work with Dima Dudko and Dierk Schleicher.

May 18, 2021

**Title:** *Why did Greek geometers construct?*

**Speaker:** Viktor Blasjo

Why did Greek mathematicians spend hundreds of years trying to make an angle the third of another, or a cube twice the volume of another, in dozens of different ways? What sin could be so grave that they imposed on themselves such a Sisyphean task? In my view, constructions were a foundational program to ensure consistency, validate diagrammatic reasoning, and protect against hidden assumptions. Pushing this view to its logical conclusion leads to accepting that geometry is based in physical reality, not in abstract thought - a view that is in much better agreement with ancient sources than many commentators, both ancient and modern, have cared to admit. Furthermore, this perspective suggests new interpretations and reconstructions of operationalist aspects of solutions to the classical problems that are missing in surviving sources. Notably that: Archytas's cube duplication was originally a single-motion machine; Diocles's cissoid was originally traced by a linkage device; Greek conic section theory was based on the conic compass, and in a few cases string constructions.

April 20, 2021

**Title:** *Endomorphisms of algebraic groups from the viewpoint of dynamical systems*

**Speaker:** Gunther Cornelissen

Consider the following two problems:

(a) count the number of irreducible polynomials over a finite field, asymptotically in thedegree of the polynomial (i.e., a polynomial analogue of the prime number theorem) and

(b) compute the number of invertible nxn matrices over a finite field.

These apparently different looking problems can both be interpreted in tems of dynamics of an endomorphism of an algebraic group (where the endomorphism is “Frobenius” and the algebraic group is the additive group or GL(n), and we count orbits, or fixed points). The talk is about a vast generalisation of these results to arbitrary endomorphisms of arbitrary algebraic groups with finitely many fixed points. The results will be explained by example and by picture.

Typical technical keywords are: adelic distortion of linear recurrent sequences, non-hyperbolicity in dynamical systems, Steinberg’s formula for reductive groups, rationality of Artin-Mazur zeta functions, analogue of the Riemann Hypothesis for orbit counts (ongoing joint work with Jakub Byszewski and Marc Houben).

April 6, 2021

**Title:** *Partial Combinatory Algebras - Variations on a Topos-theoretic Theme*

**Speaker:** Jaap van Oosten

One of the cornerstones of Logic is the theory of computable functions. In this talk, we show that a common axiomatics underlies both the set of computable functions and certain spaces of continuous functions. The models of these axioms - partial combinatory algebras - serve as building blocks of certain elementary toposes. In the talk, I shall endeavour to familiarize you with the basic ideas. Towards the end, I will point out some recent work by Jetze Zoethout.

March 23, 2021

**Title:** *Proof of a Magnificent Conjecture*

**Speaker:** Martijn Kool

Solid partitions are piles of boxes in the corner of a 4-dimensional room. Their enumeration is a mystery since MacMahon proposed an incorrect formula around 1916. Motivated by super-Yang-Mills theory on (complex) 4-dimensional affine space, Nekrasov recently assigned a measure to solid partitions and proposed a conjectural formula for their weighted enumeration.

We give a geometric definition of this measure using the Hilbert scheme of points on 4-dimensional affine space. Although this Hilbert scheme is very singular and has “higher obstruction spaces”, we can use recent work of Oh-Thomas to localise our invariants and prove Nekrasov’s conjecture. Joint work with J. V. Rennemo.

March 9, 2021

**Title:** *From exotic spheres to equivariant homotopy*

**Speaker:** Mingcong Zeng

In 1956, Milnor discovered that there are several smooth structures on the 7-spheres that are not diffeomorphic to the standard one. In 1963, Kervaire and Milnor provided a method of counting the number of smooth structures on spheres of dimension greater than 4. In this talk, I will start with their results, and talk about how this geometric problem was transformed into a problem in homotopy theory, and discuss how equivariant homotopy comes into play. Finally, I will talk about some recent results with my collaborators on equivariant homotopy and some confusing problems I am thinking about.

February 23, 2021

**Title:** *Generating sets for symplectic capacities*

**Speaker:** Fabian Ziltener

This talk is about joint work with my former Ph.D.-student Dušan Joksimović.

Symplectic geometry originated from classical mechanics, where the canonical symplectic form on phase space appears in Hamilton's equation. It is related to dynamical systems and algebraic geometry, among other fields.

Roughly speaking, a (symplectic) capacity is a real-valued function on the class of all symplectic manifolds, satisfying some natural conditions. The set of all capacities may intuitively be viewed as the dual of the class of all symplectic manifolds. Helmut Hofer et al. posed the following problem: *Find a minimal set of capacities that generates all capacities.*

The main result presented in this talk is that every such generating set has cardinality bigger than the continuum. This diminishes the hope of finding manageable generating sets of symplectic capacities.

February 9, 2021

**Title:** *Periodic versions of algebraic K-theory*

**Speaker:** Lennart Meier

Algebraic K-theory is a fundamental invariant of rings. Originally conceived in algebraic geometry by Grothendieck, today it plays a significant role also in topology and number theory. It was an insight of Thomason that one can define periodic versions of algebraic K-theory. Recent work of Bhatt-Clausen-Mathew and Land, Mathew, Tamme and myself shows that this periodic algebraic K-theory has very nice properties. If time permits, I will also discuss our work on higher periodic versions.

January 26, 2021

**Title:** *Random Vector Functional Link Neural Networks as Universal Approximators ( Note: the talk will start at 5:30pm)*

**Speaker:** Palina Salanevich

Single layer feedforward neural networks (SLFN) have been widely applied to solve problems such as classification and regression because of their universal approximation capability. At the same time, iterative methods usually used for training SLFN suffer from slow convergence, getting trapped in a local minimum and being sensitivity to the choice of parameters. Random Vector Functional Link Networks (RVFL) is a randomized version of SLFN. In RVFL, the weights from the input layer to hidden layer are selected at random from a suitable domain and kept fixed in the learning stage. This way, only output layers are optimized, which makes learning much easier and cheaper computationally. Igelnik and Pao proved that the RVFL network is a universal approximator for a continuous function on a bounded finite dimensional set. In this talk, we provide a non-asymptotic bound on the approximation error, depending on the number of nodes in the hidden layer, and discuss an extension of the Igelnik and Pao result to the case when data is assumed to lie on a lower dimensional manifold.

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.