Weekly Bulletin

In this talk I'll present a method, using abstract spectral theory, for counting points in a group orbit. That is, given a discrete group of isometries acting on the (n+1) dimensional upper half-space, our method allows one to count the number of points in an orbit a distance T from an observer. In particular, the method does not rely on an explicit pre-trace formula, and thus can be used in infinite volume. Further, I will present how to extend the method to obtain effective error rates when counting Apollonian (and more generally Kleinian) sphere packings. This is based on joint work with Alex Kontorovich, and joint work with Dubi Kelmer and Alex Kontorovich.

The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. On the other hand, for any angle, there exist non-​constant harmonic functions on the plane that are bounded everywhere outside the angle. The situation changes for discrete harmonic functions on the standard square lattice. We will show that if a discrete harmonic function is bounded on a large portion of the lattice then it is constant. Simple counter-​example shows that in higher dimensions such improvement is no longer true. The lecture is based on a joint work with L. Buhovsky, A. Logunov and M. Sodin. We will also discuss some recent applications and generalizations of the result.

Algebraic K-theory of smooth schemes (over a regular noetherian base scheme) is representable within Morel-Voevodsky's motivic homotopy category, wherein the affine line is contractible. For rigid analytic spaces, Ayoub developed an analogous theory wherein the closed unit ball B^1 is contractible. Within Ayoub's category, Morrow's continuous K-theory and Kerz-Saito-Tamme's analytic K-theory are not representable for two reasons: First, they are not B^1-invariant and, secondly, the mapping objects are not pro-spaces. In this talk, I will sketch the construction of a rigid analytic motivic homotopy category with coefficients in condensed spectra. By design, the rigid affine line is contractible in this category and it is canonically enriched over the category of condensed spectra. I will explain how this yields that -- after passing from pro-spaces to condensed spaces -- continuous K-theory and analytic K-theory shall be representable. Furthermore, we can identify the representing object with the image of the representing object of algebraic K-theory under a canonical analytification functor. In future work, I intend to employ this representability in order to study Adams operations, similarly to Riou's work on Adams operations on higher algebraic K-theory. In the long run, this might be useful for studying the problem of lifting algebraic cycles as studied by Bloch-Esnault-Kerz who linked this problem to the Hodge conjecture for abelian varietes.

I consider an optimization problem arising in orthogonal group synchronization, in which we want to estimate orthogonal matrices from (potentially noisy) relative measurements. The naïve least-squares estimator over orthogonal matrices requires solving a nonconvex program that, in general, has many spurious local minima. We show that adding a small number of degrees of freedom (specifically, relaxing to optimization over slightly “wider” Stiefel manifold matrices) makes the nonconvexity benign in that every second-order critical point is a global minimum and, in fact, yields an optimal solution to the original unrelaxed problem. In the noiseless measurement case, our results are tight and solve a previous conjecture in synchronization over Stiefel manifolds. The key proof innovation is a new randomized perturbation direction. Joint work with Nicolas Boumal; https://arxiv.org/abs/2307.02941.

We consider a coupled system between kinetic and fluid equations, describing a cloud of particles immersed within a gas. In the "thick spray" regime, the volume fraction for the particles is not negligible compared to that of the fluid: it raises many difficulties for the study of such system, which seems to present losses of derivatives. In particular, and contrary to some other fluid-kinetic couplings, its mathematical study has almost remained absent. I will review some recent progress on thick spray equations, and show that one can actually build a Cauchy theory in Sobolev regularity (at least for a compressible viscous fluid) when the initial data satisfies a Penrose type stability condition (being in fact necessary and sufficient for well-posedness). This is based on joint works with Aymeric Baradat (CNRS, Université Lyon 1) and Daniel Han-Kwan (CNRS, Nantes Université).

We model the interaction between a slow institutional investor and a high-frequency trader as a stochastic multiperiod Stackelberg game. The high-frequency trader exploits price information more frequently and is subject to periodic inventory constraints. We first derive the optimal strategy of the high-frequency trader given any admissible strategy of the institutional investor. Then, we solve the problem of the institutional investor given the optimal strategy of the high-frequency trader, in terms of the resolvent of a Fredholm integral equation, thus establishing the unique multi-period Stackelberg equilibrium of the game. Our results provide an explicit solution which shows that the high-frequency trader can adopt either predatory or cooperative strategies in each period, depending on the tradeoff between the order-flow and the trading signal. We also show that the institutional investor's strategy is more profitable when the order-flow of the high-frequency trader is taken into account. This talk is based on a joint work with Rama Cont and Alessandro Micheli.

I will talk about an ongoing project of computing the Fourier expansions of cuspidal paramodular eigenforms, particularly in low weight. This is joint work with Eran Assaf.

The Teichmüller space of bordered surfaces can be described via metric ribbon graphs, leading to a natural symplectic structure introduced by Kontsevich in his proof of Witten's conjecture. I will show that many tools of hyperbolic geometry can be adapted to this combinatorial setting, and in particular the existence of Fenchel–Nielsen coordinates that are Darboux. As applications of this setup, I will present a combinatorial analogue of Mirzakhani's identity, resulting in a completely geometric proof of Virasoro constraints as well as Norbury's recursion for the counting of integral points. Time permitting, I will describe how to count simple closed geodesics in this setting, and how its asymptotics compute Masur–Veech volumes of the moduli space of quadratic differentials.