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See the schedule of the first week here.
Title: Arithmeticity and Thinness of Hypergeometric Groups
Abstract:
The monodromy groups of hypergeometric differential equations—often referred to as hypergeometric groups—are subgroups of general linear groups. Arithmetic and thin groups have become central objects of study due to their relevance in number theory, geometry, and even quantum computing. In this talk, I will give a gentle introduction to hypergeometric groups and present recent progress in their classification.
Title: Sato-Tate equidistribution for the Satake parameters of \(\mathrm{GSp}_4\).
Abstract:
I will start motivating the study of the Satake parameters of automorphic representations from geometric and arithmetic point of views. Those are complex numbers parametrising the local component of the automorphic representations. In the generic case, the Satake parameters are believed to satisfy the generalised Ramanujan conjecture, however the study of the Satake parameter of a fixed automorphic representation is a very difficult question. On the other hand their study within a “harmonic” family is amenable to trace formulas. This allows us to show that the Satake parameters of the cuspidal, generic, spherical spectrum of \(\mathrm{GSp}_4\) equidistribute with respect to the Sato-Tate measure in the level aspect. Time permitting I will give some elements of proof.
Title: Invariant measures for totally elliptic surface group representations
Abstract:
Totally elliptic representations are distant cousins of the classical Fuchsian representations arising in Teichmüller theory. Yet many of their features stand in strong contrast to those of their Fuchsian counterparts. For instance, the mapping class group action on components of totally elliptic representations is highly chaotic. In this talk, we will show that this action is uniquely ergodic, up to finite orbits which occur only in exceptional cases. The proof combines techniques from symplectic geometry and measure theory.
See the schedule of the second week here.
Title: Expanders and character varieties of random groups.
Abstract: I will describe joint work with O. Becker and P. Varju in
which we consider one-relator groups \(\Gamma_w\) on two letters, where the
relator is a word \(w\) of large length
chosen at random. If \(G\) is a
semisimple complex algebraic group, such as \(\mathrm{SL}(2,C)\), we study the character
variety of \(\Gamma_w\) with values in
\(G\).
Under GRH, we show that it is finite and a single Galois orbit with
overwhelming probability. This gives rise to somewhat mysterious
Zariski-dense rigid groups. We also discuss what happens for more an
arbitrary number of generators and relators. The techniques require
establishing uniform expansion for finite groups of Lie type of bounded
rank up to a dimension zero set of exceptions, which we achieve as a
by-product of a uniform spectral gap for linear groups recently obtained
by O. Becker and the speaker.
Title: Quantum Ergodicity for locally symmetric spaces of large volume
Abstract:
A cornerstone of quantum choas is the Quantum Ergodicity theorem, which states roughly that on a closed Riemannian manifold with ergodic geodesic flow the L2 mass of most Laplacian eigenfunctions equidistributes in the high frequency limit. An analogous question, where instead of the frequency it is the geometry which varies, was first considered by Anantharaman-Le Masson for eigenfunctions of large regular graphs, provided they converge, in a certain sense, to the regular tree, and have a uniform spectral gap. Several works have since proved Quantum Ergodicity in this large volume regime, most notably by Le Masson-Sahlsten in the setting of closed hyperbolic surfaces. In this talk, I will present joint work with Marshall, Matz, and Peterson, which extends the Quantum Ergodicity theorem in the large volume aspect to the setting of general locally symmetric spaces. The proof makes extended use of the theory of spherical functions and benefits from some recent structural observations made by Hippi in his thesis.
Title: Density of shapes of periodic tori in \(SL(n,\mathbb{R})/SL(n,\mathbb{Z})\)
Abstract:
Compact orbits of the diagonal group of \(SL(n,\mathbb{R})\) in \(SL(n,\mathbb{R})/SL(n,\mathbb{Z})\) are embeddings of flat tori of dimension n-1. In a joint work with Nihar Gargava and Jialun Li, we prove that up to rescaling, these shapes are dense in the space of modules of flat tori. The dense family of shapes is extracted from some specific orders and their suborders. We rely on results of Bourgain on equidistribution of large multiplicative cosets in the rings \(\mathbb{Z}/N\mathbb{Z}\) as well as a “banana Lemma” due to Dani-Raghavan.
Title: Second moment formulas beyond error terms
Abstract:
In this talk I’ll give a nice number theoretic observation, and highlight some geometric results which arise from analyzing a second moment formula of the Siegel—Veech transform in dimension 2.
Title: One-forms and minimal surfaces on random hyperbolic 3-manifolds
Abstract:
Although at the moment there does not seem to exist a model for a random hyperbolic 3-manifold, we discuss how Heegaard splittings and an ad-hoc notion of a random cover of an arithmetic hyperbolic 3-manifold of simplest type can be used to give some insight into properties of the spectrum of hyperbolic 3-manifolds on coexact one-forms and their relation to the existence of many stable minimal surfaces of the same genus, as inspired by the work of Boulanger and Courtois.
Title: Uniform observability for Schrödinger equations on arbitrary Riemannian covers of closed negatively curved surfaces
Abstract:
In this talk, we study observability for the Schrödinger equation on arbitrary Riemannian covering spaces \(\pi:X\to M\), where the base space \(M\) is a closed negatively curved surface. Our main result shows that, for every time \(T>0\), the Schrödinger equation is observable from the preimage \(\pi^{-1}(\Omega)\) of any nonempty open subset \(\Omega\subset M\), with a control constant \(C(M,\Omega,T)\) that is uniform with respect to the covering map \((\pi,X)\).
The proof is divided into high-frequency and low-frequency regimes. In the high-frequency regime, we identify functions on \(X\) with sections of flat Hilbert bundles over \(M\). We then extend the long-time propagation and semiclassical control estimates of Dyatlov-Jin-Nonnenmacher to all unitary flat Hilbert bundles over \(M\), with constants uniform in the choice of bundle. In the low-frequency regime, we combine the spectral inequality of Deleporte-Lagacé-Rouveyrol with an abstract observability inequality of Green-Kleinhenz to control the low-frequency remainder. This yields the desired observability inequality on \(\pi^{-1}(\Omega)\). We will also discuss applications of the resulting uniform semiclassical control estimates to spectral geometry.
This is joint work with Xin Fu, Westlake University, and Yunlei Wang, Louisiana State University.
Title: Spectral gaps of random hyperbolic surfaces.
Abstract:
Joint work with Davide Macera and Joe Thomas. The first non-zero eigenvalue, or spectral gap, of the Laplacian on a closed hyperbolic surface encodes important geometric and dynamical information about the surface. We study the size of the spectral gap for random large genus hyperbolic surfaces sampled according to the Weil-Petersson probability measure. We show that there is a c>0 such that a random surface of genus g has spectral gap at least 1/4-O(g^-c) with high probability. Our approach adapts the polynomial method for the strong convergence of random matrices, introduced by Chen, Garza-Vargas, Tropp and van Handel, and its generalisation to the strong convergence of random permutation representations of surface groups by Magee, Puder and van Handel, to the Laplacian on Weil-Petersson random hyperbolic surfaces.
Title: Counting sublattices
Abstract:
For \(2 \le k \le n-2\), the \(L^2\) norm of the function on the space of lattices \(SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R})\) that counts the number of rank \(k\) sublattices of covolume bounded by B>0 diverges. This motivates computing the truncated \(L^2\) norm by the Maass-Selberg relation. The goal of this talk is to explain how this can be done, as well as to present the following application. We show that for generic lattices, the leading error term of such count is \(O(B^{n-1/7+\varepsilon})\) for any \(\varepsilon>0\), which is in general an improvement from previously known \(O(B^{n-1/\min(k,n-k)})\).
Title: Toward a twistless converse theorem
Abstract:
What makes a sequence of complex numbers (under certain conditions) the Fourier coefficients of an automorphic form? The answer is given by the converse theorem. The converse theorem for automorphic forms has a long history, beginning with the work of Hecke and Weil: relating the automorphy of classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. A natural question is: how many twists are actually required? In this talk, I will report recent progress on this question.
This talk is based on ongoing joint work with Andrew R. Booker, Nina Zubrilina, and Claude.
Title: Randomness in the spectrum of the Laplacian: from flat tori to hyperbolic surfaces of high genus
Abstract:
I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh.
Title: The geodesic flow on random negatively curved surface covers
Abstract:
Let \(X\) be a closed hyperbolic surface. Magee, Puder and van Handel have recently proved that for almost all covers \(X_{n}\rightarrow X\) of large degree \(n\), the first nontrivial eigenvalue of the Laplacian on \(X_n\) converges to 1/4, which is the bottom of the spectrum of the hyperbolic plane. By a result of Ratner, this implies a uniform rate of exponential mixing for the geodesic flow on typical covers.
In variable curvature, such an approach is not available, and one should tackle the dynamical problem directly. We show that for a typical large degree cover of a closed Anosov surface (i.e. with chaotic geodesic flow), the geodesic flow still admits a uniform rate of exponential mixing.
Title: The entropy spectrum of a hyperbolic surface
Abstract:
The study of closed geodesics on hyperbolic surfaces sits at the crossroads of geometric topology and dynamics and often has links to geometric probability and number theory. This talk will discuss new quantitative bounds for the growth of the number of closed geodesics on surfaces with geodesic boundary. The main theme is that coarse geometric data, such as boundary length, width, or diameter, already strongly constrain geodesic growth and entropy.
The talk will also introduce the entropy spectrum of a hyperbolic surface, consisting of the entropies of its subsurfaces. This leads to the study of gaps in the entropy spectrum, rigidity phenomena, and discreteness results that reveal new geometric features of subsurface dynamics.
This is joint work with Ara Basmajian.
Title: Resonances on geometrically finite graphs
Abstract:
We initiate the study of resonances on “geometrically finite graphs” as a graph-theoretic and non-archimedean analogue of geometrically finite hyperbolic surfaces. We prove the meromorphic continuation of the resolvent and characterize the resonances as those eigenvalues for which the associated eigenfunctions satisfy certain growth properties. It is a result of our methods that there are only finitely many resonances and they may be computed via a finite-dimensional linear algebra problem. Particularly interesting examples arise from algebraic curves over finite fields where the distribution of resonances is highly structured due to deep results in automorphic forms. On the other hand, the distribution of resonances on random geometrically finite graphs exhibit interesting phenomena. In addition to their intrinsic interest, we hope that studying resonances on geometrically finite graphs may act as a testing ground for new approaches to long-standing conjectures concerning resonances on geometrically finite hyperbolic surfaces such as the Phillips-Sarnak conjecture and the Jakobson-Naud conjecture.
This is joint work with Christian Arends and Tobias Weich.
Title: Hypotheses on random 3–manifold.
Abstract:
Various models of random surfaces of very different origin have been studied and they exhibit striking common properties : hyperbolicity, expansion, local (“Benjamini–Schramm”) convergence to a universal object (the hyperbolic plane).
For 3–manifolds the picture is a lot murkier : due to rigidity phenomena, random models have to be discrete in nature if one is interested in hyperbolic geometry or topology. On the other hand triangulations of dimension 3 are much more complicated than in dimension 2 and studying random manifold triangulations seems to be well beyond our current understanding. Other models using features specific to 3-dimensional toplogy (random Heegaard splittings, random knots for instance) have been studied in some depth but their very definitions seem to go against the universal properties shared by 2-dimensional random manifolds.
In the talk I will present these facts in some detail, and I will try to present a list of expectations for random models of 3-dimensional manifolds and some justifications for them.
Title: Nodal domains under perturbation: counting and topology
Abstract:
It is known that the Laplacian on a compact Riemannian manifold has a discrete spectrum with smooth eigenfunctions. The nodal set is the zero set of a given eigenfunction and divides the manifold into nodal domains, bounded by the eigenvalue index (Courant). I will discuss how this nodal domain count behaves under perturbations of the geometry, showing that it is upper semicontinuous: perturbations can merge nodal domains but never create them. The mechanism is local: near each nodal critical point, the nodal set untwines near each singular point, and an Euler count on the resulting forest controls the change, which then assembles globally via a dual incidence graph. I will then treat localised, possibly topology-changing perturbations, and give applications: metrics that are Courant-sharp up to any prescribed level, and new cases of Payne’s nodal line conjecture. Based on joint works with Saikat Maji and Mayukh Mukherjee.
Title: The saturation probability of random lattice sphere packings
Abstract:
A sphere packing is an arrangement of spheres of equal size in n-dimensional Euclidean space with the property that the corresponding balls do not intersect except possibly along their boundaries. A sphere packing is called saturated if there is no room to insert more spheres of the same size in the packing. In this talk the focus will be on the class of lattice sphere packings which are sphere packings where the spheres are centered at the points of a full rank lattice. The main goal is to discuss the probability that a random lattice sphere packing in high dimensions (where the lattice is sampled from the invariant measure on the space of n-dimensional lattices) is saturated. This is joint work with Matthew de Courcy-Ireland.
Title: Fine-scale statistics of directions to the points in a lattice orbit in certain homogeneous spaces
Abstract:
I will report on work in progress, joint with Samuel Edwards, where we generalize results by Marklof-Strömbergsson on the fine-scale statistics of directions to lattice points in Euclidean space, and results by Marklof-Vinogradov on the fine-scale statistics of directions in hyperbolic n-space to the points in the orbit of a lattice in SO(n,1). Among the applications of our main theorem, I will discuss in particular the case of rational points in a quadric hypersurface.
Title: Division!
Abstract:
Euclid’s algorithm for division allows us to divide two numbers, keep track of remainders, and recover GCDs. I will discuss other algebraic settings: some rings are known to be Euclidean (meaning they have this algorithm), some are known not to be; many are unknown. These questions are intimately connected with the lattice structure of the number field.
Title: On the supremum of random cusp forms
Abstract: A random ensemble of cusp forms for the full modular group is introduced. For a weight-k cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be \(~\sqrt{\log{k}}\), in line with the conjectured bounds. In addition, the exponential concentration of the supremum around its median is established. Contrary to the compact case, the global expected supremum, attained around the cusp, grows like \(k^{1/4}\). This talk is based on a joint paper with B. Huang, S. Lester and N. Yesha.
This page was updated on June 30, 2026.
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