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. The usual location is HFG611, but make sure you check the corresponding announcement email.**MI talks:**The MI-talks on pure mathematics will take place on Tuesdays 16.00-17.00. The applied ones will take place on Thursdays at 16:00-17:00. The usual location is HFG611, but make sure you check the corresponding announcement email.**How to attend online:**The biweekly invitations to the Teams 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:

February 7, 2023

**Title:** *The univalence principle (MI-talk) *

**Speaker:** Paige North

The Equivalence Principle is an informal principle asserting that equivalent mathematical objects have the same properties. For example, group theory has been developed so that isomorphic groups have the same group-theoretic properties, and category theory has been developed so that equivalent categories have the same category-theoretic properties (though sometimes other, ‘evil’ properties are considered). Vladimir Voevodsky established Univalent Foundations as a foundation of mathematics (based on dependent type theory) in which the Equivalence Principle for types (the basic objects of type theory) is a theorem. Later, versions of the Equivalence Principle for set-based structures such as groups and categories were shown to be theorems in Univalent Foundations.

In joint work with Ahrens, Shulman, and Tsementzis, we formulate and prove versions of the Equivalence Principle for a large class of categorical and higher categorical structures in Univalent Foundations. Our work encompasses bicategories, dagger categories, opetopic categories, and more.

February 28, 2023

#### Previous talks:

January 17, 2023

**Title:** *C^0 symplectic geometry*

**Speaker:** Dusan Joksimovic (Paris)

The famous C^0-rigidity theorem of Eliashberg and Gromov (which states that the group of symplectic diffeomorphisms is closed w.r.t. C^0-topology inside the group of all diffeomorphisms) is considered the beginning of a subfield of symplectic geometry which today is known as C^0 symplectic geometry. Roughly, it investigates non-smooth symplectic objects and the behavior of smooth symplectic objects with respect to the C^0 topology.

In this talk, we will introduce some basic objects in (C^0) symplectic geometry, outline some of the most important questions in the field, and give an overview of the known results.

January 12, 2023

**Title:** *Modeling the impact of Corona measures (applied MI-talk) *

**Speaker:** Martin Bootsma

In this talk, I will discuss a model we have developed for the ministry of Health, Welfare and Sports to estimate the impact of Corona ticket measures (QR-code) on the spread of SARS-CoV-2 in the Netherlands. I will discuss how the underlying assumptions lead to the model formulation and I will briefly discuss some results.

December 1, 2022

**Title:** *Phase retrieval with time-frequency structured measurements (applied MI-talk) *

**Speaker:** Palina Salanevich

Phase retrieval is the non-convex inverse problem of signal reconstruction from intensity measurements with respect to a measurement frame. This problem is motivated by practical applications, such as diffraction imaging and audio processing. The nature of the measurements in a particular application determines the structure of the measurement frame. This makes the study of the phase retrieval with structured, application relevant frames especially interesting.

In the talk, we are going to focus on phase retrieval with Gabor frames, where the measurement vectors follow time-frequency structure that naturally appears in imaging and acoustics applications. We will discuss how to achieve stable and efficient reconstruction with such measurements and how prior information about the signal class can be used to regularize the phase retrieval problem and reduce the number of measurements required for reconstruction.

November 29, 2022

**Title:** *Polar degree of projective hypersurfaces in the presence of singularities (MI-talk) *

**Speaker:** Dirk Siersma

For any hypersurface V in projective n-space P, given by f=0, the notion of polar degree is defined as the topological degree of the (projectivized) gradient mapping of the homogeneous polynomial f. This is a map from P-V to P.

We will discuss first the history of polar degree and give several examples, e.g. the determinant hypersurface has polar degree 1. The hypersurfaces with polar degree are called homaloidal and are of extra interest because the gradient map is bi-rational.

Polar degree zero is related to the question of what happens if the Hessian of f is identically zero. This was solved by Gordan and Noether in 1876.

After a long period of algebraic studies, recently topological methods gave some interesting results. Dolgacev classified in 2000 all the projective homoloidal plane curves: a short list. Huh determined in 2014 all homoloidal hypersurfaces in P with at most isolated singularities.

In this talk we will reprove Huh's results with methods of singularity theory. Moreover we will prove the Huh's conjecture that his list of polar degree 2 surfaces with isolated singularities is complete!

Finally we say something more about hypersurfaces with the non-isolated singularities.

November 24, 2022

**Title:** *beta-expansions in one and higher dimensions: an ergodic view (applied MI-talk) *

**Speaker:** Karma Dajani

In this talk we give an exposition on one of the interactions between ergodic theory and number theory. We will concentrate on the concept of $beta$-expansions, which are representations of numbers of the form $x=displaystylesum_{i=1}^{infty}displaystylefrac{a_i}{beta^i}$ with $beta>1$ a real number, and $a_iin{0,1,cdots, lceil beta rceil -1}$.

We explain first simple concepts in ergodic theory that can help us understand the asymptotic behaviour of a typical expansion. What typical is depends on the stationary measure under consideration, and each such measure highlights a particular property of points in its support, i.e. the world that the measure sees. We extend the one-dimensional ideas to higher dimensions and show how they can be used to study multiple codings of points in an overlapping Sierpinski gasket.

November 22, 2022

**Title:** *Wall-crossing for Virasoro constraints*

**Speaker:** Arkadij Bojko (ETH Zürich)

Virasoro constraints in Gromov-Witten theory were introduced by Witten and famously proved by Kontsevich, while their analogue for moduli of sheaves has been studied mostly on examples. Using the recent wall-crossing framework of D. Joyce phrased in terms of vertex algebras, I express them in joint work with M. Moreira and W. Lim universally in terms of primary states with respect to a fixed conformal vector. This new formulation is compatible with wall-crossing, which leads for example to proofs of the constraints by reduction on rank for vector bundles on curves and torsion-free sheaves on surfaces with (p,p) cohomology.

November 15, 2022

**Title:** *Quot scheme theory of surfaces*

**Speaker:** Woonam Lim (ETH Zürich)

I will give an overview of the theory of Quot schemes of surfaces and their virtual invariants. The main conjecture of this subject, due to Oprea-Pandharipande, concerns rationality of the Quot scheme partition functions. I will explain how to approach this conjecture via Hilbert scheme of points and curves.

October 13, 2022

**Title:** *Fields medal 2022: The works of Hugo Duminil-Copin and June Huh (MI-talk. Location: KBG Pangea) *

**Speaker:** Wioletta Ruszel, Carel Faber

Wioletta Ruszel will discuss the mathematical contributions of Hugo Duminil-Copin. Carel Faber will discuss the mathematical contributions of June Huh.

October 11, 2022

**Title:** *Fields medal 2022: The works of James Maynard and Maryna Viazovska (MI-talk. Location: KBG Atlas) *

**Speaker:** Lola Thompson, Rob Bisseling

Lola Thompson will discuss the mathematical contributions of James Maynard. Rob Bisseling will discuss the mathematical contributions of Maryna Viazovska.

October 6, 2022

**Title:** *Highlights of discrete mathematics (applied MI-talk) *

**Speaker:** Carla Groenland

I will give a very high-level overview of some subfields of discrete mathematics, outlining the type of questions researchers find interesting and the type of techniques that are being used. I will also mention some of the work that is being done by our colleagues in the CS department. The entire talk should be easy to follow by master’s students of every track.

September 29, 2022

**Title:** *Sharp estimates on random hyperplane tessellations (applied MI-talk) *

**Speaker:** Sjoerd Dirksen

In my talk I will consider the following question. Draw independent random hyperplanes with standard Gaussian directions and uniformly distributed shifts. How many hyperplanes are needed to tessellate a given subset of R^n into uniformly small cells of a given diameter with high probability?

I will first explain two motivating applications for this question: data dimension reduction and signal processing under coarse quantization. I will then present a generally optimal answer to the posed question, which surprisingly deviates from the answer that was conjectured in the literature. If time permits, I will show an extension of this result to a specific structured random tessellation that is designed for computationally fast data dimension reduction.

The talk is based on a recently accepted paper with Shahar Mendelson (ANU Canberra) and Alexander Stollenwerk (UCLouvain).

September 20, 2022

**Title:** *Étale cohomology and independence of ℓ (MI-talk) *

**Speaker:** Remy Dobben de Bruyn

The geometric complexity of a complex algebraic variety is measured by algebraic invariants like the fundamental group and (co)homology groups. I will explain why algebraic geometers believed that such invariants should also exist for varieties over finite fields, and sketch how this was realised in the 1960s by ℓ-adic étale cohomology (depending on an auxiliary prime number ℓ). At the end of the talk, I will discuss some results and work in progress on the foundational "independence of ℓ" problem.

June 23, 2022

**Title:** *FFTU: the fastest Fourier transform in Utrecht (applied MI-talk, also at 4pm) *

**Speaker:** Rob Bisseling

he multidimensional Fast Fourier Transform (FFT) is at the heart of many grid-based scientific computations, including weather and climate prediction, and (quantum) molecular dynamics.

In this talk, we present a parallel algorithm for the FFT in higher dimensions. This algorithm generalizes the cyclic-to-cyclic 1D parallel algorithm to a cyclic-to-cyclic multidimensional parallel algorithm, with only a single all-to-all communication step in most practical cases.

We present our multidimensional implementation FFTU which utilizes the sequential FFTW program (the "fastest Fourier transform in the West") for its local FFTs. Our experimental results for 2D, 3D, and 5D on up to 4096 cores of the new Dutch national supercomputer Snellius show that FFTU is competitive with the state-of-the-art and that for 3D it may be the fastest, not in only in Utrecht, but also beyond.

June 21, 2022

**Title:** *Localization techniques in Enumerative Geometry (MI-talk) *

**Speaker:** Sergej Monavari

A classical way to produce invariants is through Intersection Theory, usually on a smooth projective variety. We give a gentle introduction on how to use torus actions to refine invariants in several directions, in particular in K-theory, and on how to weaken smoothness and properness assumptions. As a concrete example, we explain how to extract meaningful invariants from the moduli space of zero-dimensional quotients of a locally free sheaf on a toric variety, and illustrate various closed formulas for different flavours of “higher rank Donaldson-Thomas invariants of points”, which solve a series of conjectures proposed in String Theory. This is based on joint work with N. Fasola and A. Ricolfi.

June 7, 2022

**Title:** *Computing endomorphism rings and Frobenius matrices of Drinfeld modules*

**Speaker:** Mihran Papikian (Pennsylvania State University)

Let $mathbb{F}_q[T]$ be the polynomial ring over a finite field $mathbb{F}_q$. We study the endomorphism rings of Drinfeld $mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a reciprocity law for the division fields of Drinfeld modules. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm. This is joint work with Sumita Garai.

May 24, 2022

**Title:** *Derived Hecke operators (MI-talk) *

**Speaker:** Jack Davies

Modular forms are classical objects in number theory, and Hecke operators on these modular forms are a useful organisational tool. In this talk, I will define modular forms over the complex numbers and the classical Hecke operators which act upon them. Then I will discuss a generalisation of this theory from the complex numbers to a theory for general rings, and eventually for derived rings. Finally, I would like to show some little games we can play with these kinds of derived Hecke operators, as a sample of their utility. Only basic complex analysis will be assumed for the first half, and in the second half we will use some language from modern algebraic geometry.

May 10, 2022

**Title:** *Concrete abstract computability theory (MI-talk) *

**Speaker:** Jetze Zoethout

In the 1930s, Turing, Kleene, Church and others proposed a precise, mathematical notion of a computable function on the natural numbers. For the first time in history, this allowed mathematicians to prove that certain problems are not solvable by an algorithm. Moreover, it turned out that the computability theory of the natural numbers enjoys certain "algebraic" properties that also manifest themselves elsewhere. A structure satisfying these algebraic properties is called a partial combinatory algebra (PCA), and the study of PCAs can be seen as abstract recursion theory. In this talk, I want to give you a flavor of the subject by considering some concrete examples of PCAs, and how they relate to each other.

April 26, 2022

**Title:** *The blowup formula for instanton Vafa-Witten invariants*

**Speaker:** Nick Kuhn (Max Planck Institute, Bonn)

As a consequence of the S-duality conjecture, Vafa and Witten conjectured certain symmetries concerning invariants derived from spaces of vector bundles on a closed Riemannian four-manifold. We focus on the case of a smooth complex projective surface X, where a satisfying mathematical definition of Vafa-Witten invariants has been given by Tanaka and Thomas. Their invariants are a sum of two parts, one of which can be defined in terms of moduli spaces of stable vector bundles on X. Focusing on this instanton part of the VW invariants one can ask how it changes under a basic algebraic operation: Blowing up the surface X at a point. I will report on joint work with Oliver Leigh and Yuuji Tanaka towards an answer to this question.

April 12, 2022

**Title:** *Introduction to operads (MI-talk) *

**Speaker:** Ieke Moerdijk

Operads were introduced in the 1970s in an attempt to describe the algebraic structure of loop spaces in topology, but have since made their appearance in many other parts of mathematics. The goal of this talk will be to give an introduction to the theory of operads accessible to our master students. If time permits, I will end with a new view on (Koszul) duality for operads.

March 29, 2022

**Title:** *Polarizations of abelian varieties over finite fields via canonical liftings*

**Speaker:** Stefano Marseglia

We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero, i.e., a lifting for which the reduction morphism induces an isomorphism of endomorphism rings.

This is joint work with Jonas Bergström and Valentijn Karemaker.

March 15, 2022

**Title:** *Tame topology and a finiteness theorem for variations of Hodge structures*

**Speaker:** Thomas Grimm

About 20 years ago physicists found a remarkable set of solutions to some of the fundamental equations arising from string theory. It has subsequently been conjectured by Douglas and others that this set of solutions is finite. This conjecture has since shaped the view of string theory as a predictive theory leading to a finite number of possible universes.

In this talk, I will first describe the underlying mathematical statement of this conjecture as a finiteness condition on the set of self-dual integral classes in a variation of Hodge structure. I will then argue that the conjecture can be proved using a recent remarkable result of Bakker, Klingler, and Tsimerman that connects Hodge theory and tame topology. I will give a brief introduction to tame topologies built from o-minimal structures and describe the structure that tames the period mapping capturing the variation of a Hodge structure. The finiteness theorem for self-dual classes is a generalization of a theorem for the locus of Hodge classes by Cattani, Deligne, and Kaplan.

This talk is based on joint work with B. Bakker, C. Schnell, and J. Tsimerman, see 2112.06995 [math.AG].

March 1, 2022

**Title:** *In Search of the Absolute: Zeuthen's holism and the enumerative geometry of conics*

**Speaker:** Nicolas Michel

Throughout his long and distinguished career, the Danish mathematician Hieronymous Georg Zeuthen constantly contributed to the emergence of enumerative methods in geometry. He did so alternatively as observer, as participant, as professor, or even as referee in the context of the frequent disputes which plagued the early development of this theory. His 1914 Lehrbuch der abzählenden Methoden der Geometrie, the sum of this prolonged engagement, opens on surprising pronouncements: in lieu of first theorems or definitions, we find a philosophical discussion of the "meaning of numbers" and the "relativity of the concepts of general and particular."

Drawing on Zeuthen's holistic outlook on the history and epistemology of geometry, this talk will set out to explain why, to an algebraic geometer at the onset of the 20th century, answering such technical questions of mathematics might first require philosophical detours—and what we might lose by discarding them.

February 15, 2022

**Title:** *Totally geodesic subvarieties in the Torelli locus*

**Speaker:** Carolina Tamborini

Riemannian symmetric spaces are Riemannian manifolds with special symmetry properties. They are important in various fields of geometry. In the seminar, we will be interested in the following fact: the moduli space A_g of principally polarized abelian varieties is a quotient of the Siegel space, which is a Riemannian symmetric space. This fact can be used to approach the study of the Torelli locus, which is the closure in A_g of the image of the moduli space M_g of smooth, complex algebraic curves of genus g via the Torelli map j: M_g-->A_g.

We will first introduce Riemannian symmetric spaces and their totally geodesic submanifolds. Next, we will describe the problem of studying the geometry of the Torelli locus in A_g and its relation with totally geodesic submanifolds of the Siegel space. Finally, we will explain how this is linked to a famous conjecture from Coleman and Oort.

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

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.