Past

A long time ago, Grothendieck made some conjectures. This has resulted in some things.

The Kruzkhov's semi-group of a scalar conservation law extends as a semi-group over $L^1$, thanks to its contraction property. M. Crandall raised in 1972 the question of whether its trajectories can be distributional, entropy solutions, or if they are only "abstract" solutions. We solve this question in the case of the multi-dimensional Burgers equation, which is a paradigm for non-degenerate conservation laws. Our answer is the consequence of dispersive estimates. We first establish $L^p$-decay rate by applying the recently discovered phenomenon of Compensated Integrability. The $L^\infty$-decay follows from a De Giorgi-style argument. This is a collaboration with Luis Sivestre (University of Chicago).

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

We model the trading strategy of an investor who spoofs the limit order book (LOB) to increase the revenue obtained from selling a position in a security. The strategy employs, in addition to sell limit orders (LOs) and sell market orders (MOs), a large number of spoof buy LOs to manipulate the volume imbalance of the LOB. Spoofing is illegal, so the strategy trades off the gains that originate from spoofing against the expected financial losses due to a fine imposed by the financial authorities. As the expected value of the fine increases, the investor relies less on spoofing, and if the expected fine is large enough, it is optimal for the investor not too spoof the LOB because the fine outweighs the benefits from spoofing. The arrival rate of buy MOs increases because other traders believe that the spoofed buy-heavy LOB shows the true supply of liquidity and interpret this imbalance as an upward pressure in prices. When the fine is low, our results show that spoofing considerably increases the revenues from liquidating a position. The profit of the spoof strategy is higher than that of a no-spoof strategy for two reasons. First, the investor employs fewer MOs to draw the inventory to zero and benefits from roundtrip trades, which stem from spoof buy LOs that are ‘inadvertently’ filled and subsequently unwound with sell LOs. Second, the midprice trends upward when the book is buy-heavy, therefore, as time evolves, the spoofer sells the asset at better prices (on average).

The Aldous diffusion is a conjectured Markov process on the
space of real trees that is the continuum analogue of discrete Markov
chains on binary trees. We construct this conjectured process via a
consistent system of stationary evolutions of binary trees with k
labelled leaves and edges decorated with diffusions on a space of
interval partitions constructed in previous work by the same authors.
This pathwise construction allows us to study and compute path
properties of the Aldous diffusion including evolutions of projected
masses and distances between branch points. A key part of proving the
consistency of the projective system is Rogers and Pitman’s notion of
intertwining. This is joint work with Noah Forman, Soumik Pal and
Douglas Rizzolo.                            

Morse theory has a long history with many spectacular applications in different areas of mathematics. In this talk I will explain an extension of the main theorem of Morse theory that works for a large class of functions on singular spaces. The main example to keep in mind is that of moment maps on varieties, and I will present some applications to the topology of symplectic quotients of singular spaces.
 

We discuss properties of the cosmological horizon of a de Sitter universe, and compare to those of ordinary black holes. We consider both the Lorentzian and Euclidean picture. We discuss the relation to the sphere partition function and give a group-theoretic picture in terms of the de Sitter group. Time permitting we discuss some properties of three-dimensional de Sitter theories with higher spin particles. 

Prime numbers have been looked at for centuries, but some of the most basic questions about them are still major unsolved problems. These problems began as idle curiosities, but have grown to become hugely important not only in pure mathematics, but also have many applications to the real world. I'll talk about some of these quests to find patterns in the sequence of prime numbers.

Single cell RNA-Seq data is challenging to analyse due to problems like dropout and cell type identification. We present a novel clustering 
approach that applies mixture models to learn interpretable clusters from RNA-Seq data, and demonstrate how it can be applied to publicly 
available scRNA-Seq data from the mouse brain. Having inferred groupings of the cells, we can then attempt to learn networks from the data. These 
approaches are widely applicable to single cell RNA-Seq datasets where  there is a need to identify and characterise sub-populations of cells.

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.