For a punctured surface S and a split reductive algebraic group G such as SL_n or PGL_n, Fock and Goncharov (and Shen) consider two types of moduli spaces parametrizing G-local systems on S together with certain data at punctures. They show that these spaces have special coordinate charts, hence are birational to cluster varieties. Fock and Goncharov’s duality conjectures predict the existence of a canonical basis of the algebra of regular functions on one of these spaces, enumerated by the tropical integer points of the other space. I will give an introductory overview of this topic, briefly explain recent developments involving quantum topology and mirror symmetry of log Calabi-Yau varieties, and present some open problems if time allows.

The problem of finding the maximal possible multiplicity of the first Laplacian eigenvalues has been studied at least since the 1970’s. I will present a recent work in collaboration with Simon Machado (ETH Zürich) in which we proved, for negatively curved surfaces, the first upper bound which is sublinear in the genus g. Our method also yields an upper bound on the number of eigenvalues in small spectral windows, and this upper bound is shown to be nearly sharp. We also obtain results for higher-dimensional manifolds. Our proof combines a trace argument for the heat kernel and a geometric idea introduced in the context of graphs of bounded degree in a paper by Jiang–Tidor–Yao–Zhang–Zhao (2021). Our work provides new insights on a conjecture by Colin de Verdière and a new way to transfer spectral results from graphs to surfaces.

Please join us for Pre-Colloquium tea.

Lots of problems in combinatorics and analysis are connected to incidences: given a set of points and tubes, how much can they intersect? Upper bounds for incidences have been extensively studied, but lower bounds for incidences haven’t received as much attention, and we prove results in this direction. We prove that if you choose n points in the unit square and a line through each point, there is a nontrivial point-line pair with distance <= n^{-2/3+o(1)}. It quickly follows that in any set of n points in the unit square, some three form a triangle of area <= n^{-7/6+o(1)}, a new bound for this problem. The main work is proving a more general incidence lower bound result under a new regularity condition.

Joint with Cosmin Pohoata and Dimitrii Zakharov.

Please join us for pre-seminar Tea

Confined subgroups of a Lie group or of a discrete group are generalizations of a Fuchsian group G that gives rise to a hyperbolic surface M of bounded injectivity radius. When the surface M has finite volume, the growth rate of G equals the volume growth rate of the hyperbolic plane H^2. Gekhtman and Levit established an inequality between the growth rates of G and the ambient space for a general confined subgroup of a rank-1 Lie group. In this talk, I will explain an analogous result for a confined subgroup G of a hyperbolic group, rank-1 CAT(0) group, or the mapping class group. This is established by studying the single (as opposed to the double) boundary action of G. Joint work with Ilya Gekhtman, Wenyuan Yang, and Tianyi Zheng.

In this talk, I will explain a recent result that pseudo-Anosov mapping classes are generic in every Cayley graph of mapping class groups. If time permits, I will also explain why this strategy goes well with quasi-isometries and implies genericity of Morse elements for groups quasi-isometric to (many) 3-manifold groups and special cubical groups.

Hilbert’s sixth problem asks for the axiomatic derivation of the laws of physics. Within this broad question, Hilbert singled out the derivation of the laws of fluid mechanics (like Euler’s or Navier-Stokes’ equations) by way of rigorously justifying Boltzmann’s kinetic theory for particle systems. This will be the subject of our talk. The rigorous derivation of Boltzmann’s kinetic equation was established for short times in a classic work of Lanford (1975), following inputs from Grad and Cercignani. However, Hilbert’s sixth problem requires a long time version of this result. In a recent work with Yu Deng (Chicago) and Xiao Ma (Michigan), we extend Lanford’s theorem to long times; more precisely, for as long as the solution of Boltzmann’s equation exists. The proof relies on a novel methodology that allows us to obtain effective estimates on the probability of many recollisions. 

We will discuss the following result. For every geometrically finite Kleinian group Γ < SL₂(ℂ) there is ε_Γ such that for every g ∈ SL₂(ℂ) the intersection  gΓg^-1 ∩ SL₂(ℝ) is either a lattice or a has critical exponent δ(gΓg^-1 ∩ SL₂(ℝ)) ≤ 1-ε_Γ. 

This result extends Margulis-Mohammadi and Bader-Fisher-Milier-Strover. 

We will discuss some ideas of the proof. We will focus on the applications of a new ergodic component, of preservation of entropy in a direction.

This is a 30min slow-pace pre-seminar discussion aiming to familiarise early-career researchers with basic ideas used in Thursday Analysis Seminar talk. The focus will be put on ideas, and technicalities are kept to a minimum. 

Topics include: limiting absorption principle, scattering theory, manifolds with corners. 

Please join us for pre-Colloquium Tea

Since the work of Jensen–Kato, the analysis of the Schrodinger–Helmholtz equation at low energy has been used to study wave propagation in various settings, both relativistic and nonrelativistic. Recently, Hintz has used these methods to study wave propagation on black hole spacetimes. Part of Hintz’s result is the production of asymptotics in all possible asymptotic regimes, including all joint large-time, large-radii regimes. We carry out the analogue of this analysis for the Schrodinger equation. Based on joint work with Shi-Zhuo Looi.

Consider a semi-simple Lie group, $G$ and a discrete torsion-free subgroup $\Gamma < G$. When $G$ has rank one, there is an important, well-understood connection between the growth rate of elements of $\Gamma$-orbits, bounds on the spectrum, and temperedness, or more generally properties of the unitary $L^2$ representation (due, in part, to work of Elstrodt and Patterson). In this talk I will present an extension of this connection to higher rank.  This is joint work with Tobias Weich and Lasse Wolf. 

This is a 30min slow-pace pre-seminar discussion aiming to familiarise early-career researchers with basic ideas used in Thursday Analysis Seminar talk. The focus will be put on ideas, and technicalities are kept to a minimum. 

Topics include: pseudodifferential operators, paradifferential operators.

I will give a short review of the physics-inspired approach to Donaldson theory based on topologically-twisted four-dimensional N=2 supersymmetric quantum field theory (SQFT) as propounded by Witten in 1988, and then describe some of my recent and ongoing work. One theme in particular is a generalization of Donaldson-Witten theory to describe invariants of smooth families of smooth, closed, oriented Riemannian 4-manifolds X using methods of SQFT and supergravity, and leading to physical derivations of relevant models of equivariant cohomology. Family Donaldson invariants are cohomology classes of the classifying space of orientation-preserving diffeomorphisms, i.e., elements of H*(BDiff^+(X)), and we propose path integral formulations of their cocycle representatives. Several interesting open questions arise concerning the observables and interpretation of the invariants, calling for a renewed math-physics dialog. I hope to (re?)ignite some interest in these issues through this talk. Time permitting, I will touch upon the second theme, a new perspective of topological twisting using the notion of ‘transfer of structure group’ associated with a continuous homomorphism between topological groups. This approach accounts for the global topology of the structure group of a quantum field theory, rather than just its simply connected cover, and leads to a proposal for twisting more general four-dimensional N=2 SQFTs. The talk will be based on collaborations with G. W. Moore, M. Roček, and R. K. Singh. 

Please join us for pre-Colloquium Tea

We use the Selberg zeta function to study the limit behavior of resonances of a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line. Moreover, our techniques can be used to obtain effective statements. One key idea is to introduce an intermediate zeta function that captures Archimedean and non-Archimedean information (while the Selberg resp. Ihara zeta function concerns only Archimedean resp. non-Archimedean properties). This is a joint work with Jialun Li, Carlos Matheus, and Zhongkai Tao.

In this talk, I will explain how paradifferential calculus can be applied to construct quasiperiodic-in-time solution for PDEs (“KAM for PDEs”). Due to the loss of regularity caused by “small divisors”, these problems are traditionally resolved using Nash-Moser/KAM type iterative schemes. One step of the Nash-Moser scheme is to reduce a non-autonomous linear operator that involves “small divisors” into constant coefficient form, for which a Nash-Moser/KAM reducibility argument is necessary, yielding a complicated “Nash-Moser within Nash-Moser” formalism. However, it is discovered that paradifferential calculus can be used to completely avoid such formalism, yielding “fixed point” style proof. In particular, it is possible to reduce the nonlinear equation itself into constant coefficient form (modulo smoothing remainder), not just its linearization. This is because paradifferential operators share all the algebraic structures of (pseudo)differential operators while gain back regularities due to J.-M. Bony’s paralinearization process. I will use the existence problem for quasiperiodic-in-time solution of fully nonlinear hyperbolic systems with one spatial variable as illustrative example. This talk is based on joint works with Thomas Alazard.