Past

Whereas the spectral properties of random matrices has been the subject of numerous studies and is well understood, the statistical properties of the corresponding eigenvectors has only been investigated in the last few years. We will review several recent results and emphasize their importance for cleaning empirical covariance matrices, a subject of great importance for financial applications.

I will describe some new-ish results on the average and maximum size of the Riemann zeta function in a "typical" interval of length 1 on the critical line. A (hopefully) interesting feature of the proofs is that they reduce the problem for the zeta function to an analogous problem for a random model, which can then be solved using various probabilistic techniques.

Whereas the spectral properties of random matrices has been the subject of numerous studies and is well understood, the statistical properties of the corresponding eigenvectors has only been investigated in the last few years. We will review several recent results and emphasize their importance for cleaning empirical covariance matrices, a subject of great importance for financial applications.

Discussin of https://arxiv.org/abs/1908.09858

The Toda integrable system was originally designed as a specific model for lattice field theories. Following Kostant's insights, we will explain how it naturally arises from the representation theory of Lie algebras, and present some more recent work relating it to cotangent bundles of Lie groups and the topology of Affine Grassmannians.

(Joint work with: Jüri Lember, Heinrich Matzinger, Raul Kangro)

Principal component analysis is an important pattern recognition and dimensionality reduction tool in many applications and are computed as eigenvectors

of a maximum likelihood covariance that approximates a population covariance. The eigenvectors are often used to extract structural information about the variables (or attributes) of the studied population. Since PCA is based on the eigen-decomposition of the proxy covariance rather than the ground-truth, it is important to understand the approximation error in each individual eigenvector as a function of the number of available samples. The combination of recent results of Koltchinskii & Lounici [8] and Yu, Wang & Samworth [11] yields such bounds. In the presented work we sharpen these bounds and show that eigenvectors can often be reconstructed to a required accuracy from a sample of strictly smaller size order.

Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).
 

 

Given a finite set X, is an easy exercise to show that a binary operation * from XxX to X which is injective in each variable separately, and which is also associative, makes (X,*) into a group. Hrushovski and others have asked what happens if * is only partially associative - do we still get something resembling a group? The answer is known to be yes (in a strong sense) if almost all triples satisfy the associative law. In joint work with Tim Gowers, we consider the so-called `1%' regime, in which we only have an epsilon fraction of triples satisfying the associative law. In this regime, the answer turns out to be rather more subtle, involving certain group-like structures which we call rough approximate groups. I will discuss these objects, and try to give a sense of how they arise, by describing a somewhat combinatorial interpretation of partial associativity.
 

Right-angled Artin groups (RAAGs) were first introduced in the 70s by Baudisch and further developed in the 80s by Droms.
They have attracted much attention in Geometric Group Theory. One of the many reasons is that it has been shown that all hyperbolic 3-manifold groups are virtually finitely presented subgroups of RAAGs.
In the first part of the talk, I will discuss some of their interesting properties. I will explain some of their relations with manifold groups and their importance in finiteness conditions for groups.
In the second part, I will focus on my PhD project concerning subgroups of direct products of RAAGs.

Do you feel unable to explain why maths are cool? Are you looking for fun and affordable theorems for your non-mathematician friends? This is your topic.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?

 Lipschitz (or H\"older) spaces $C^\delta, \, k< \delta <k+1$, $k\in\mathbb{N}_0$, are the set of functions that are more regular than the $\mathcal{C}^k$ functions and less regular than the $\mathcal{C}^{k+1}$ functions. The classical definitions of H\"older classes involve  pointwise conditions for the functions and their derivatives.  This implies that to prove   regularity results for an operator among these spaces  we need its pointwise expression.  In many cases this can be a rather involved formula, see for example the expression of $(-\Delta)^\sigma$  in (Stinga, Torrea, Regularity Theory for the fractional harmonic oscilator, J. Funct. Anal., 2011.)

In  the 60's of last century, Stein and Taibleson, characterized bounded H\"older functions via some integral estimates of the Poisson semigroup, $e^{-y\sqrt{-\Delta}},$ and of  the Gauss semigroup, $e^{\tau{\Delta}}$. These kind of semigroup descriptions allow to obtain regularity results for fractional operators in these spaces in a more direct way.

 In this talk we shall see that we can characterize H\"older spaces adapted to other differential operators $\mathcal{L}$ by means of semigroups and that these characterizations will allow us to prove the boundedness of some fractional operators, such as $\mathcal{L}^{\pm \beta}$, Riesz transforms or Bessel potentials, avoiding the long, tedious and cumbersome computations that are needed when the pointwise expressions are handled.

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi-Yau variety X, and some invariants extracted from a mirror family of Calabi-Yau varieties. After the physicists Bershadsky-Cecotti-Ooguri-Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck-Riemann-Roch. I will explain the main ideas of the proof of the conjecture for Calabi-Yau hypersurfaces in projective space, based on the Riemann-Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang-Lu-Yoshikawa.
 

This is joint work with G. Freixas and C. Mourougane.

We extend the lightning Laplace solver (Gopal and Trefethen, SINUM 2019) to unbounded domains and to the biharmonic equation. Illustrating the high accuracy of such methods, we get beautiful contour plots of Moffatt eddies.

I'll give a general introduction to tensor-triangular geometry, the algebraic study of tensor-triangulated categories as they appear in topology, geometry and representation theory. Then I'll discuss an elementary idea, that of a "field" in this theory, and explain what we currently know about them.

A subset $D$ of a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ is called a "perfect difference set" if every nonzero element of $\mathbb{Z}/m\mathbb{Z}$ can be written uniquely as the difference of two elements of $D$. If such a set exists, then a simple counting argument shows that $m=n^2+n+1$ for some nonnegative integer $n$. Singer constructed examples of perfect difference sets in $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ whenever $n$ is a prime power, and it is an old conjecture that these are the only such $n$ for which $\mathbb{Z}/(n^2+n+1)\mathbb{Z}$ contains a perfect difference set. In this talk, I will discuss a proof of an asymptotic version of this conjecture.

Many computational techniques in data science involve the factorization of a data matrix into the product of two or more structured matrices. Examples include PCA, which relies on computing an SVD, recommendation systems, which leverage non-negative matrix factorization, infilling missing entries with low rank matrix completion, and finding sparse representations via dictionary learning. In our work we study a new matrix factorization problem, involving the recovery of $\textbf{A}$ and $\textbf{X}$ from $\textbf{Y} := \textbf{A}\textbf{X}$ under the following assumptions; $\textbf{A}$ is an $m \times n$ sparse binary matrix with a fixed number $d$ of nonzeros per column and $\textbf{X}$ is an $n \times N$ sparse real matrix whose columns have $k$ nonzeros and are dissociated. This setup is inspired and motivated by similar models studied in the dictionary learning literature as well as potential connections both with stochastic block models and combinatorial compressed sensing. In this talk we present a new algorithm, EBR, for solving this problem, as well as recovery guarantees in the context of a particular probabilistic data model. Using the properties of expander graphs we are able to show, under certain assumptions, that with just $N = \textit{O}( \log^2(n))$ samples then EBR recovers the factorization up to permutation with high probability. 

Can spatial fungal networks be informative for both ecology and network science?

Filamentous organisms grow as adaptive biological spatial networks. These networks are in a continuous balance of two main forces: exploration of the habitat to acquire scarce resources, and the transport of those resources within the developing network. In addition, the construction of the network has to be kept a low cost while taking into account the risk of damage by predation. Such network optimization is not unique to biological systems, but is relevant to transport networks across many domains. Thus, this collaborative project between FU-Berlin and University of Oxford represents the beginning of a research program that aims at: First, setting up protocols for the use of network analysis to characterize spatial networks formed by both macroscopic and microscopic filamentous organisms (e.g. Fungi), and determining the fitness and ecological consequences of different structure of the networks. Second, extracting biologically-inspired algorithms that lead to optimized network formation in fungi and discuss their utility in other network domains. This information is critical to demonstrate that we have a viable and scalable pipeline for the measurement of such properties as well provide preliminary evidence of the usefulness of studying network properties of fungi.

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term $\mathrm{Tr}(\Phi^4)$. By methods from quantum field theory we show that also the quartic case is exactly solvable. All cumulants can be expressed as composition of elementary functions with the inverse of another elementary function. For infinite matrices the inversion gives rise to hyperlogarithms and zeta values as familiar from quantum field theory. For finite matrices the elementary functions are rational and should be viewed as branched covers of Riemann surfaces, in striking analogy with the topological recursion of the Kontsevich model. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable.
 

Gauss noted quadratic reciprocity to be among his favourite results, and any undergrad will quickly pick up on just how strange it is despite a plethora of elementary proofs. By 1930, E. Artin had finalized Artin reciprocity which wondrously subsumed all previous generalizations, but was still confined to abelian contexts. An amicable non-abelian reciprocity remains a driving force in number-theoretic research.

In this talk, I'll recount Artin reciprocity and show it implies quadratic and cubic reciprocity. I'll then talk about some candidate non-abelian reciprocities, and in particular, which morals of Artin reciprocity they preserve.

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre, and Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions.

I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabre, A. Farina, and L. Lombardini.

The genus of a knot in a 3-manifold is defined to be the minimum genus of a compact, orientable surface bounding that knot, if such a surface exists. We consider the computational complexity of determining knot genus. Such problems have been studied by several mathematicians; among them are the works of Hass--Lagarias--Pippenger, Agol--Hass--Thurston, Agol and Lackenby. For a fixed 3-manifold the knot genus problem asks, given a knot K and an integer g, whether the genus of K is equal to g. In joint work with Lackenby, we prove that for any fixed, compact, orientable 3-manifold, the knot genus problem lies inNP, answering a question of Agol--Hass--Thurston from 2002. Previously this was known for rational homology 3-spheres by the work of Lackenby.

 I will talk about optimal estimates for stochastic integrals
in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).