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See the schedule of the first week here.
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See the schedule of the second week here.
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.
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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 \(O(B^{n-1/\min(k,n-k)})\).
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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.
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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.
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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: 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.
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Title: Division!
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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 10, 2026.
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