For the topics of the courses, see the “Overview” page. An updated list of short talks and their abstracts can be found below the schedule.
Time  Monday 30 Sep 
Tuesday 1 Oct 
Wednesday 2 Oct 
Thursday 3 Oct 
Friday 4 Oct 

8:00 – 8:50  Registration  
9:00 – 10:00  S. Kionke: Arithmetic Groups (1) 
W. Müller: Analytic Torsion (2) 
D. Gaboriau: Measured Group Theory (4) 
N. Bergeron: From torsion to L^{2}‑torsion (4) 
S. Kionke: Arithmetic Groups (5) 
10:00 – 10:25  Coffee (or tea) break  
10:25 – 11:25  D. Gaboriau: Measured Group Theory (1) 
S. Kionke: Arithmetic Groups (2) 
S. Kionke: Arithmetic Groups (4) 
W. Müller: Analytic Torsion (4) 
W. Müller: Analytic Torsion (5) 
11:30 – 12:30  W. Müller: Analytic Torsion (1) 
D. Gaboriau: Measured Group Theory (3) 
W. Müller: Analytic Torsion (3) 
D. Gaboriau: Measured Group Theory (5) 
N. Bergeron: From torsion to L^{2}‑torsion (5) 
12:30 – 14:00  Lunch break  
14:00 – 15:00  D. Gaboriau: Measured Group Theory (2) 
N. Bergeron: From torsion to L^{2}‑torsion (2) 
N. Bergeron: From torsion to L^{2}‑torsion (3) 
G. Harder: Special values of Lfunctions and the structure of the cohomology of arithmetic groups 

15:05 – 16:05  N. Bergeron: From torsion to L^{2}‑torsion (1) 
S. Kionke Arithmetic Groups (3) 
Excursion  Open Problem Session  
16:05 – 16:30  Tea (or coffee) break  Tea (or coffee) break  
16:30 – 18:00  Five Short Talks: Yongquan Hu Bart Michels Léo Bénard Hisatoshi Kodani Claudia Schoemann 
Five Short Talks: Daniele Dona Changliang Wang J. D. López Castaño Jun Ueki Pascal Zschumme 
Five Short Talks: Roberto Miatello Mattia Cavicchi M. V. Moya Giusti Takuma Hayashi B. Waßermann 

19:00 Dinner at “Fellini” 
Short Talk Abstracts
Download abstracts (PDF)MONDAY:
1. Yongquan Hu: Asymptotic growth of the cohomology of Bianchi groups
Abstract: Given a level N and a weight k, we know the dimension of the space of (classical) modular forms. This turns out to be unknown if we consider Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In this talk, we consider the asymptotic behavior of the dimension when the level is fixed and the weight grows. I will first explain an upper bound obtained by Simon Marshall using Emerton's completed cohomology and the theory of Iwasawa algebras. Then I explain how to improve this bound using the mod p local Langlands correspondence for GL_{2}(ℚ_{p}).
2. Bart Michels: Lower bounds for geodesic periods on hyperbolic surfaces
On compact arithmetic hyperbolic surfaces, Iwaniec and Sarnak have shown that there exist sequences of Laplacian eigenfunctions whose sup norms grow with the eigenvalue. Their large values are found at CMpoints, using the amplification method. In this talk I will explain how to obtain large values of geodesic periods and highlight some geometric features of the pretrace formulas at hand.
3. Léo Bénard: Asymptotic of twisted Alexander polynomials and hyperbolic volume
Given a hyperbolic manifold M of finite volume, we study a family of twisted Alexander polynomials of M. We show an asymptotic formula for the behavior of those polynomials on the unit circle, and recover the hyperbolic volume as the limit. It extends previous works of Müller (for M closed) and MenalFerrer–Porti. This is a joint work with Jerome Dubois, Michael Heusener (ClermontFerrand) and Joan Porti (Barcelona).
4. Claudia Schoemann: The twisted forms of a semisimple group over the integral domain of a global function field
Let K = 𝔽_{q}(ℂ) be the global field of rational functions on a smooth and projective curve C defined over a finite field 𝔽_{q}. Any finite but nonempty set S of closed points on C gives rise to an integral domain 𝒪_{S} = 𝔽_{q}[C – S] of K. Given a semisimple and almostsimple group scheme G defined over Spec 𝒪_{S} with a smooth fundamental group F(G), we aim to describe the set of (𝒪_{S}classes of) twisted forms of G in terms of some invariants of F(G) and the absolute type of the Dynkin diagram of G. This finite set is given by H^{1}_{fl}(𝒪_{S}, Aut(G)) seen as the disjoint union over P of H^{1}_{fl}(𝒪_{S}, ^{P}G^{ad}) modulo the Out(G)action, where P are the G^{ad}torsors, and turns out sometimes to biject to a disjoint union of abelian groups. This is joint work with Rony A. Bitan and Ralf Köhl.
5. Hisatoshi Kodani: On the BarNatanWitten analytic torsion associated with a representation to a noncompact Lie group
After the celebrated works by Ray and Singer, the analytic torsion has been studied by many mathematicians. In particular, the analytic torsion is of interest from the perspective of (quantum) topology, since, in some sense, Witten's work on ChernSimons theory can be regarded as a generalisation of the Theorem of Cheeger and Müller stating the equivalence of the Reidemeister torsion and the RaySinger torsion. In this talk, for several manifolds, I will give explicit computational results of the analytic torsion introduced by BarNatan and Witten in their study of ChernSimons perturbation theory with a noncompact Lie group.
TUESDAY:
1: Daniele Dona: Benjamini–Schramm convergence and invariant random subgroups
We give a brief overview of the use of the concept of Benjamini–Schramm convergence, coming from graph theory, in the world of Riemannian orbifolds: this allows a geometric reinterpretation of the notion of convergence of invariant random subgroups, and is an important starting point for the work of the “seven samurai”. The talk is based prominently on section 3 of Abert–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet.
2. Changliang Wang: Perelman's functionals on compact manifolds with conical singularities
In a joint work with Prof. Xianzhe Dai, we extended the theory of Perelman's functionals on compact smooth manifolds to compact manifolds with conical singularities. In particular, for the λfunctional, it is essentially a spectrum problem.
3. Juan Daniel López Castaño: Sunada's technique for constructing isospectral manifolds
Two compact Riemannian manifolds M_{1}, M_{2} are called isospectral, if the LaplaceBeltrami operators on them have the same set of eigenvalues counted with multiplicities (Spec(M_{1}) = Spec(M_{2})). This set is in particular interesting by many deep connections between it and the geometry of both manifolds, and spectral geometry studies such connections. The question, often posed as “Can one hear the shape of a drum?” [Kac, 1966], asks whether isospectral manifolds are necessarily isometric. In 1964 J. Milnor [Milnor, 1964] gave the first counterexample: a pair of 16dimensional flat tori which are isospectral and nonisometric. Subsequently, many other counterexamples have been constructed (see [Gordon and Wilson, 1984], [Sunada, 1985], [Vignéras,1980], [Bérard, 1993], [Bérard, 1992], [Gordon, Webb, and Wolpert, 1992], etc.).
Among these examples there are pairs of manifolds with nonisomorphic fundamental groups, locally symmetric spaces of rank one and locally symmetric spaces of higher rank, Riemann surfaces of genus ≥ 4, continuous families of isospectral manifolds, and other examples. All these examples have one property in common: the isospectral manifolds have a common Riemannian covering (M, g) which admits an isometric action by a (possibly finite) Lie group G. The isospectral manifolds are quotients of M by discrete subgroups Γ_{i} of G, thus the manifolds in these examples are locally isometric.
The goal of this talk is to present Sunada's technique for constructing isospectral manifolds [Sunada, 1985], in which G is a finite group, Γ_{1} and Γ_{2} are nonconjugate subgroups that satisfy a Sunada's conjugacy condition and M is a manifold whose fundamental groups surjects onto G. In this case, Sunada's technique allows for giving a Ginvariant Riemannian metric g such that Spec(Γ_{1}∖M, g) = Spec(Γ_{2}∖M, g). One of the many examples of isospectral, nonisometric manifolds constructed via Sunada's technique include higher rank locally symmetric spaces [Spatzier, 1990]. For some parts of the talk, I will suggest references since this one is by no means original.
4. Jun Ueki: Profinite rigidity for twisted Alexander polynomials
Recently the profinite rigidity in 3dimensional topology is of great interest with rapid progress. Nevertheless, it is still unknown whether there exists a pair (J, K) of distinct prime knots with an isomorphism π̂_{J} = π̂_{K} on the profinite completions of their knot groups.
In this talk, we formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. Key tools are Hillar's theorem and Iwasawa modules.
We examine several examples associated to Riley's parabolic representations of twobridge knot groups and give a remark on hyperbolic volumes.
5. Pascal Zschumme: Geometric construction of homology classes in Riemannian manifolds covered by products of the hyperbolic plane
We study the homology of Riemannian manifolds of finite volume that are covered by a product (ℍ^{2})^{r} = ℍ^{2} × ... × ℍ^{2} of the real hyperbolic plane. Using a variation of a method developed by Avramidi and NguyenPhan, one can show that any such manifold M possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic rdimensional submanifolds whose fundamental classes are linearly independent in the real homology group H_{r}(M;ℝ).
THURSDAY:
1. Roberto Miatello: On the distribution of Hecke eigenvalues for Hilbert modular forms
Let F be a totally real field of degree d, 𝒪_{F} its ring of integers, I an ideal in 𝒪_{F} and Γ_{0}(I)⊂ GL_{2}(ℝ)^{d} a Hecke congruence subgroup of GL_{2}(ℝ)^{d}. For 𝔭 a prime ideal in 𝒪_{F}, let T_{𝔭} be the Hecke operator acting on cusp forms in L^{2}(Γ_{0}(I)∖ GL_{2}(ℝ)^{d}) and let C_{j} be the j Casimir operators acting on the factors, for 1 ≤ j ≤ d. Let λ_{𝔭,ϖ} and λ_{ϖj} be the respective sets of eigenvalues of these operators. We study the distribution of these eigenvalues in regions Ω_{t}, as t → ∞, showing that, under a natural condition on 𝔭, the eigenvalues of T_{𝔭} (resp. of C_{j}) are distributed according to the Sato–Tate measure (resp. according to the Plancherel measure).
2. Mattia Cavicchi: Weights of the boundary motive of Shimura varieties
Let G be a reductive ℚalgebraic group giving rise to a Shimura datum. Then, the cohomology of congruence arithmetic subgroups of G with values in an algebraic representation V of G^{der} coincides with the cohomology of a natural local system μ(V) over a wellchosen associated Shimura variety S(ℂ). According to conjectures of Harder, the mixed Hodge structures appearing in such cohomology spaces H^{•}(S(ℂ),μ(V)) should contain information related to the Lfunction of the automorphic representations of G contributing to H^{•}(S(ℂ),μ(V)). It is thus important to understand the weight filtration on such spaces. In this talk, we will explain this circle of ideas. Time permitting, we will investigate how representation theory of G controls the weight filtration on the complex computing boundary cohomology of μ(V), through an invariant called corank, in the case of G = Res_{Fℚ} GSp_{4,F} (with F a totally real extension of ℚ) and of the corresponding Shimura varieties.
3. Matias Moya Giusti: Eisenstein and cuspidal cohomology of Sp_{4}(ℤ)
(In collaboration with J. Bajpai and I. Horozov)
In this talk we will present a explicit calculation of the cuspidal cohomology of Sp_{4}(ℤ) with respect to every highest weight irreducible representation of Sp_{4} and we mention some of its implications. In this work we mainly use some vanishing results of the cuspidal cohomology, the study of a spectral sequence abutting to the cohomology of the boundary of the Borel–Serre compactification and a formula for the homological Euler characteristic of Sp_{4}(ℤ).
4. Takuma Hayashi: A construction of integral models of HarishChandra modules
Recently, several people, including Michael Harris, Günter Harder, and Fabian Januszewski, have started to work on rational and integral structures of HarishChandra modules for applications to rationality and integrality of special values of automorphic Lfunctions, respectively. In this talk, I will present a general construction of integral models of HarishChandra modules.
5. Benjamin Waßermann: An L^{2}CheegerMüller theorem for hyperbolic manifolds of finite volume
Let n be an odd integer, let G = SO(n,1) and let K = SO(n). Then G/K can be identified with ndimensional hyperbolic space ℍ^{n}. Further, let ρ be an irreducible, finitedimensional complex representation of G over a vector space V_{ρ}. On the associated homogenous vector bundle ℍ^{n} × V_{ρ} ↓ ℍ^{n}, there exists a distinguished hermitian metric h_{ρ} that is Ginvariant with respect to the diagonal action.
As explained in the lecture series, this data allows us to define for any lattice Γ < G an L^{2}RaySinger torsion element τ^{(2)}_{An}(Γ,ρ) ∈ ℝ.
Similarly, from a finite CWmodel of the (not necessarily compact) quotient Γ \ ℍ^{n}, we can define an L^{2}Reidemeister torsion element τ^{2}_{Top}(Γ,ρ) ∈ ℝ. The main result of my thesis is the equality of torsion elements τ^{(2)}_{An}(Γ,ρ) = τ^{2}_{Top}(Γ,ρ) in this setup.
This equality has been proven in the case where ρ is the trivial representation and is more or less wellknown in the case that Γ is cocompact and ρ is arbitrary. In this short talk, I will explain the strategy for proving it in the very general setting, witout any further assumptions on Γ or the representation ρ aside from the mentioned ones.