UGC seminar

Arithmetic statistics is a fairly new subarea of number theory. I will give an overview of some of the typical questions and techniques in the area, and will then discuss some of my own results.

To a finitely generated group one associates a number of fundamental numerical invariants, like its Alexander or lower central series invariants. For groups of geometric nature (like fundamental groups associated to hyperplane arrangements or various parameter spaces) these groups encode subtle information of geometric and combinatorial nature. I will discuss how ideas of algebro-geometric nature led to fundamental regularity results on Alexander invariants of groups. I will provide a general introduction to this circle of ideas and present numerous examples.

In algebraic geometry, we study algebraic varieties: geometric objects that can be described by polynomials. They can be inspected using instruments which I call invariants. An invariant assigns to each variety an object (this can be a number or a group element, but also a group or a vector space, or even a category) that is invariant under isomorphisms of varieties. A “good” invariant moreover remembers some of the relationships that are present between different varieties; more formally, one can say that such an invariant satisfies a “descent property”. In this talk, I will discuss how descent properties can help us understand invariants of varieties better, and potentially construct new ones.

September 26 at 16:00 in Pangea
September 27 at 11:00 in Pangea
September 27 at 15:15 in Atlas

Abstract: The exceptional phenomena in the title of these lectures refers to those smooth manifolds which, in some way or other, are connected to the exceptional Lie groups G_2, F_4, E_6, E_7, E_8. In Cartan's list of symmetric spaces there are 12 examples of symmetric spaces for the exceptional Lie groups. The classical symmetric spaces are very well understood through the work of Borel, Bott, Hirzebruch, and Samelson. In comparison the exceptional symmetric spaces are not at all well understood. The aim of this project is to get a much better understanding of the exceptional symmetric spaces. The main examples used in the lectures will be the four Rosenfeld projective planes, FII, EIII, EVI, EVIII, in Cartan's list. They are symmetric spaces for F_4, E_6, E_7 E_8 of dimensions 16, 32, 64, 128 respectively.

There are two main lines in the approach. The first is to use K-theory and representation theory as the primary source of topological invariants. This takes full advantage of the very powerful methods, both conceptual and computational, of representation theory. The second is to use ideas from the theory of compact Lie groups acting on manifolds. In its simplest form this gives a very useful filtration of these symmetric spaces. There are also much more sophisticated outputs.

The first lecture will be used to set the context and introduce the exceptional symmetric spaces. It will also give some idea of how these methods work and the results they give. Roughly speaking, the second lecture will focus on the use of K- theory and representation theory, and the third will focus on the ideas coming from the theory of transformation groups.

How many circles are tangent to three given circles in the plane? These kinds of questions are part of enumerative geometry, a branch of algebraic geometry concerned with counting objects. In this talk I will give a gentle introduction to the subject of counting curves, introducing some of the tools like moduli spaces and intersection theory. In the last part I will talk about my own work on logarithmic Gromov-Witten invariants and the logarithmic double ramification cycle, special kind of curve counts involving tangency constraints.

Which are the simplest hypersurfaces in projective space? Traditionally, these are those defined by low-degree polynomial equations: quadrics, cubics, quartics, and so forth. Over fields of positive characteristic, however, the shape of the defining equation also plays a dramatic role. The aim in this talk is to explain a circle of ideas that identifies a class of hypersurfaces in positive characteristic that, due to the special shape of their defining equations, are much simpler than they first seem. I will intimate a philosophy for understanding them, formulate a few precise results, and pose some questions I would like resolved.

Scri, or null-infinity, is nowadays a fundamental concept for discussing the asymptotic structure of space time. On asymptotically flat manifolds, whose asymptotic structure is modelled on the conformal compactification of Minkowski space time, scri is topologically a line bundle over a sphere. There is, however, another way to compactify Minkowski spacetime based on projective geometry. This construction leads to richer regions in the time/space like regions of the boundary at infinity than the conformal one but the null region is of codimension 2. Roughly, we've lost the fibres of the line bundle. Ashtekar's exploration of the notion of asymptotic flatness at spacelike infinity also suggests that it is also meaningful to consider a line bundle over the other regions of infinity in the projective compactification. In this talk, we consider this from a geometric perspective: what are these line bundles? Is it possible to equip them with geometric structures in a natural way? This talk is based on joint ongoing work with Yannick Herfray (Univ. Tours)

Twisted sheaves are a variant of the notion of sheaves and behave similar in many ways. In this talk, I will define enumerative invariants of twisted sheaves on surfaces. These are needed to mathematically formulate the 30-year-old S-duality conjecture of Vafa-Witten. I will also show how the enumerative theory of twisted and untwisted sheaves is closely related, using deformation theory. This work is joint with A. Gholampour, Y. Jiang and M. Kool.
I will also discuss more recent work, generalising twisted invariants to dimensions 3 and 4, with possible applications to the period-index problem.

A cubic hypersurface in projective n-space defined over the rationals is given by the vanishing locus of an integral cubic form in n+1 variables. For n at least 4, it is conjectured that these varieties satisfy the Hasse principle. Recent work of Browning, Le Boudec, and Sawin shows that this conjecture holds on average, in the sense that the density of soluble cubic forms is equal to that of the everywhere locally soluble ones. But what do these densities actually look like? We give exact formulae in terms of the probability that a cubic hypersurface has p-adic points for each prime p. These local densities are explicit rational functions uniform in p, recovering a result of Bhargava, Cremona, and Fisher in the n=2 case, as well as the fact that all cubic forms are everywhere locally soluble when n is at least 9. Consequently, we compute numerical values (to high precision) for natural density of cubic forms with a rational point for n at least 4, and a conjectural value for n=3. This is joint work with Lea Beneish.

After giving a brief introduction to Calabi-Yau manifolds and classical mirror symmetry, I will focus on some of the special properties arising for torus fibered CY 3-folds.
My goal is to first describe how mirror symmetry can be used to obtain ``Gopakumar-Vafa polynomials", that capture the multiplicities of singular fibers in terms of the Chern classes of bundles that determine the structure of the torus fibration.
This will lead us to conjecture subtle constraints on the possible multiplicities that, from a physical perspective, are related to the cancellation of Dai-Freed anomalies in six-dimensional supergravity.
I will outline how these constraints connect to cobordism groups, to the APS eta-invariant on lens spaces and a quadratic refinement of the corresponding torsion pairing in cohomology.

In this joint work in progress with Helge Ruddat and Bernd Siebert, we employ a particular type of Log Smooth Degeneration (LSD) to study the Geometry of Enumerative Mirror Symmetry (GEMS).

Mirror Symmetry is a broad conjecture that predicts that symplectic invariants of a Kähler manifold correspond to algebro-geometric invariants of a mirror-dual complex algebraic variety. This is generally proven by computing both sides.

In this work, we take the first steps towards a full enumerative correspondence that canonically identifies the invariants of both sides. To do so, we employ the Intrinsic Mirror Construction of Gross-Siebert. Then the enumerative correspondence passes through an intermediary tropical manifold and tropical invariants thereof.

Moduli spaces of smooth, or stable nodal, curves have been an important subject of algebraic geometry, topology, and mathematical physics. There is a natural way to relate this moduli space with the moduli space of abelian varieties. Over the moduli space of smooth curves M_g, the relative Picard scheme gives a family of principally polarized abelian varieties (p.p.a.v.). Taking this perspective, one can obtain many interesting results on M_g. When curves degenerate into nodal curves, the relative Picard scheme is no longer compact. One can compactly such family and get relative compactified Picard schemes which is no longer a family of p.p.a.v This geometry brings many new geometry and topology. I will try to introduce some new questions on this direction. It will be a gentle overview of a lecture series which I will deliver this block.

Topological data analysis (TDA), and in particular, persistent homology, is a method used to extract higher geometric information from complex datasets and has proven highly successful in various scientific domains. In this talk, I will provide an introduction to TDA and explain its applications in studying fMRI scans and brain data, and stock prices. I will then discuss the statistical challenges that arise in interpreting the results of TDA and explain how this might be related to fractals. This is joint work with Roel Gisolf and Fernando A. N. Santos.

Polygraphic homology was introduced in the 2000s by Métayer and his collaborators as a means to understand some combinatorial results on presentation of monoids. However, it was quickly understood that polygraphic homology, and more generally the theory of polygraphs (also known as computads), was a powerful tool that allowed to revisit the homology of monoids, groups and even categories, both on a practical and theoretical level. Furthermore, the theory of polygraphs can be naturally understood as a part of higher category theory, however from a slightly different perspective than the standard approach on the subject, which is of interest by itself. In this talk, I will give a broad introduction to polygraphic homology and the theory of polygraphs. I will start from the ground up by giving some motivation and simple examples and then present the subject from a more abstract perspective.

Linear programming, the task of optimizing a linear function over a set of linear inequalities, is one of the most fundamental problems in optimization which connects to many areas of mathematics (combinatorics, high dimensional probability, convex geometry, tropical geometry, …). Despite almost a century of study, many important open problems remain:

Why does the simplex method work well in practice?
Is the diameter of a polyhedron bounded by a polynomial in the dimension and number of inequalities?
Is there a strongly polynomial algorithm for solving linear programs?

In this talk, I will provide the background to these questions, overview progress over the last few decades, including my own contributions.

Log geometry was introduced in the late 1980s by Fontaine-Illusie, Deligne-Faltings and K. Kato, among others, mainly to address two fundamental and interrelated problems in algebraic geometry: compactification and degeneration. Roughfly speaking, a log scheme is a scheme which, in addition, keeps track of a "boundary".
On the other hand, the problem of extending fppf torsors has been largely encountered in the literature, without giving a satisfactory solution in the classical setting. It turns out that the additional structure that one considers to define a log scheme allows to enlarge the category of fppf torsors by defining log torsors. In particular, this provides a new framework for studying the problem of extending torsors, presenting a fresh perspective on the question.
In this talk, we will be discussing about log schemes, log torsors, models of torsors.... We intend to end up with concrete computations to illustrate our contribution on the question.

Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, and supply chain networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and introduce two novel importance sampling (IS) methods to enhance the Monte Carlo (MC) estimator's performance. The key challenge in the IS framework is choosing an appropriate change of probability measure to significantly reduce variance, and which often requires insights into the underlying problem. To address this, we propose a generic approach to obtain an efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control (SOC) formulation. When solving the resulting SOC problem, we tackle the curse of dimensionality in two ways: the first is a learning-based method that uses a neural network to approximate the value function and where the IS parameters are determined via a stochastic optimization algorithm. The second is a dimensionality reduction technique based on the Markovian Projection concept, where we map the problem to a significantly lower dimensional space, while preserving the marginal distribution of the original SRN system. We then solve a Hamilton-Jacobi-Bellman equation for this reduced model, obtaining IS parameters that we can apply back to the full-dimensional SRN. Analysis and numerical experiments demonstrate that both proposed IS strategies substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

This expository talk is an invitation to revisit the quadratic reciprocity law as formulated by Euler and Legendre in the 18th century and proved by Gauss in the first half of the 19th century. We discuss three different proofs: a combinatorial, a Galois-theoretical and an analytical. All three can be traced back to one (or several) of the eight proofs of Gauss but have since then been simplified considerably thanks to the joint efforts of many well-known mathematicians. The employed methods had a profound influence on twentieth century mathematics and many of them are nowadays a vital part of the undergraduate program in mathematics.

Resonances of Riemannian manifolds play an important role in many areas of mathematics, e.g., analysis, dynamical systems, mathematical physics, and number theory. It has been long known that resonances of hyperbolic surfaces enjoy fundamentally different properties depending on whether the considered hyperbolic surface is compact or of finite area (but noncompact) or of infinite area. While the former two situations are fairly well understood, the latter still offers a lot of surprises despite intensive contemporary research efforts. A particularly successful approach to study the localization and distribution of resonances for hyperbolic surfaces of infinite area are transfer operator methods. Roughly speaking, this is a methodology that allows us to connect the spectral theory of hyperbolic surfaces with their dynamical properties. I will provide a gentle overview of this methodology, focusing on insights and heuristics, and discuss some recent results on the distribution of resonances.

Recently, Connes and Moscovici introduced an operator whose eigenvalues are conjectured to be related to zeros of the Riemann zeta function. It is given by a familiar differential expression, which however needs to be interpreted in a peculiar way. It turns out that this operator bears resemblance to certain geometric partial differential equations that „switch" from elliptic to hyperbolic and which arise e.g. in General Relativity. I will explain how that resemblance arises and what implications it has for (large enough) eigenvalues of the Connes—Moscovici operator.

The Liquid Tensor Experiment is a project that formally verified the main theorem of liquid vector spaces by Clausen and Scholze, following up on a challenge by Scholze on Buzzard's XenaProject blog. In this talk I will explain the thrust of this main theorem and share some mathematical insights gained from my protracted battle with its proof.

Zeros of the Riemann zeta function and eigenvalues of quantized chaotic Hamiltonians appears to have something in common. Namely, they both seem to be ruled by random matrix theory and consequently should exhibit "repulsion" in the sense that small gaps between elements are very rare. More mysteriously, while zeros of different L-functions (i.e., generalizations of the Riemann zeta function) are "mostly independent" they also exhibit subtle repulsion effects on zeros of other L-functions.
We will give a survey of the above phenomena. Time permitting we will also discuss repulsion between eigenvalues of "arithmetic Seba billiards", a certain singular perturbation of the Laplacian on the 3D torus R^3/Z^3. The perturbation is weak enough to allow for arithmetic features from the unperturbed system to be brought into play, yet strong enough to provably induce repulsion.

A variety is called rational if it admits a birational map to projective space. One of the major programs in algebraic geometry seeks to understand when a given variety is rational, and what the obstructions to rationality are. From this perspective, conic bundles are a geometrically rich class of varieties. I will talk about joint work with Sarah Frei, Lena Ji, Bianca Viray and Isabel Vogt, where we study curve classes on certain conic bundle threefolds over arbitrary fields of odd characteristic. We then use the description of these classes to study the rationality of such varieties.

In this talk we look at mathematical questions which arise naturally from imaging things that are invisible to the naked eye, ranging from models inspired by quantum mechanics to astronomy. We will explore how such inverse problems bridge different mathematical disciplines such as partial differential equations, geometry, and dynamical systems while simultaneously connecting these classical mathematical fields to modern topics in data science.

Lie algebras have played an important role throughout the development of homotopy theory. I'll start with the Whitehead bracket, which gives a Lie algebra structure on homotopy groups that extends the usual commutator on the fundamental group of a space. Then we'll see how Quillen used Lie algebras to completely describe algebraic topology "with rational coefficients". Time permitting, we'll then move on to the more modern development of Lie algebras "up to homotopy" and their role in describing topological spaces.

Artin's primititive root conjecture states that any integer g which is not a unit nor a square generates the cyclic multiplicative group (Z/pZ)* for infinitely many primes p. Similarly one may formulate such a problem over global function fields, which was first considered by Herbert Bilharz in the 1930s who achieved partial results. In this talk I will report on joint work with Ezra Waxman where we complete the proof of Artin's primitive root conjecture for function fields over a finite field and generalise the proof to a version of the conjecture for function fields of arbitrary transcendence degree.

In this talk I will explain the motivation and ideas of unstable periodic homotopy theory, where ``unstable'' refers to the study of topological spaces. Very roughly speaking, this theory is concerned with the homotopy type of the mapping spaces out of CW-complexes with finitely many cells. I will start with a review of the rational homotopy theory of topological spaces, whose methods can be generalized to unstable periodic homotopy theory. If time permits, I will discuss one or two examples of problems in this subject that I have been working on for my thesis project.

The 1 : 2 : 4 resonance is one of the four definite resonances of genuinely first order and thus known to be non-integrable. The frequency ratios provide unfolding parameters (but note that the dynamic phenomena can also occur in a single system of 6 or more degrees of freedom). The indefinite versions of the resonance do not require the equilibrium to be a local extremum of the Hamiltonian.

Normalization yields a normal form approximation and the resulting (non-integrable) system can be reduced to 2 degrees of freedom. The non-trivial isotropies of the two coupled 1 : ±2 resonances prevent the reduced phase space from being a smooth manifold but the dynamics on the singular part is in fact easier to understand. On the regular part of the reduced phase space the distribution of equilibria turns out to be determined by a single polynomial of degree 4. These are the relative equilibria that determine the behaviour of the 3 normal modes when passing through the resonance.

Topological data analysis is a burgeoning field of research that uses methods from topology and geometry to infer qualitative properties of complex datasets by analyzing their shape. In this talk, I will present an application of topological data analysis to the surveillance of critical mutations in the evolution of the coronavirus SARS-CoV-2. I will explain the underlying geometric idea, how it connects with biology, its implementation in the CoVtRec pipeline, and some concrete results from the analysis of current pandemic data. This is joint work with Bauer, Bleher, Carrière, Hahn, Neumann, Patiño-Galindo, Rabadán, and the KIT Steinbuch Centre for Computing.

One of the first highlights in the theory of manifolds is the result (1936) of Whitney stating that every smooth manifold can be embedded in a sufficiently large euclidean space. This is a quantitative result: Whitney proved that every n-dimensional manifold can be embedded in (2n+1)-dimensional space. He later introduced the Whitney trick (1944) and showed that 2n is sufficient.

This became a driving question during this period: what can be said about the spaces of embeddings/immersions between two given manifolds? Whitney himself provided in 1937 a complete description (up to homotopy) of the immersions of the circle into the plane. Twenty years later, Smale developed the method of corrugations to prove the sphere eversion theorem and, together with Hirsch, provided a complete classification (in terms of bundle monomorphisms) for immersions of subcritical dimension.

In 1969, Gromov turned the method of corrugations into a general machine (applicable in the study of many different geometric structures on manifolds) called the method of flexible sheaves, initiating the modern theory of (geometric) h-principle.

My aim with this talk is to provide a historical overview of these ideas, introduce the field of h-principle, and highlight some of the (many!) open questions in the area.

Finite sets behave in many ways as if they were vector spaces over a "field with one element," F_1. Via this analogy, many combinatorial statements may be read as statements about linear algebra over F_1. This analogy is more useful than one might expect, and we will discuss F_1-linear vector bundles over a space. This will allow us to recast the Segal conjecture in algebraic topology as an F_1-linear version of the classical Atiyah-Segal completion theorem on the K theory of classifying spaces. We finish by discussing a new proof of the Z/2 Segal conjecture using equivariant bordism theory.

Algebraic K-theory is an important (and notoriously hard to compute) invariant of rings containing at the same time number theoretic, geometric, and representation theoretic information. In this talk, I will give a leisurely introduction to the modern approach to K-theory via higher group completion, and then explain how one can refine this to capture additional "symmetries" of the input ring, leading to equivariant algebraic K-theory.

This talk will consist of:
- a friendly introduction to a certain (loosely defined) class of algebras, then
- a bit about why topologists care, and what we like to prove about them.
These algebras (the Temperley-Lieb and Brauer algebras, and various cousins) have already lead quite exciting lives in statistical physics and representation theory, and are of interest in topology because they look quite a lot like the group algebra of some symmetric group. This means that they are a really good place to study `homological stability' (whatever that is) because we understand this phenomenon really well for the symmetric groups.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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].

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.

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.

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

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.

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.

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.

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.

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.

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.

TBA

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.