Abstracts

Mini course

Asbjørn Nordentoft (Université Paris-Saclay)

In these three lecture I will discuss various equidistribution questions related to closed geodesics on hyperbolic surfaces.  

In the first lecture I will explain how to prove Duke’s theorem; the length packets of closed geodesics on the modular curve equidistribute as the length goes to infinity. The proof is number theoretic and uses (among other things) spectral theory of automorphic forms.

The second lecture  will be concerned with a construction due to Duke-Imamouglu-Toth of certain hyperbolic orbifolds with boundary given by closed geodesics. The construction relies on continued fractions. I will then explain an extension due to Peter Humphries and myself to modular curves of non-trivial genus and explain the geometric difficulties that arise. This construction relies on geometric coding of geodesics. 

In the third lecture I will explain an alternative and completely  geometric definition due to Ser Peow Tan and myself (work in progress). If time permits I will explain some higher dimensional analogues.

Research Talks

Ara Basmajian (CUNY)
Title: Counting problems on Hecke surfaces
Abstract: Most of this talk will focus on the modular group and more generally the Hecke groups. The Hecke groups are $(2,q,\infty)$ triangle groups for  $q \geq 3$; the modular group corresponds to $q=3$. We consider counting problems associated to various classes of geodesics. These counting problems are part of more general phenomena that intertwine the geometry and topology of curves on surfaces with number theoretic and combinatorial  considerations. We'll report on joint work with Robert Suzzi Valli, Blanca Marmolejo and Robert Suzzi Valli, and with Mingkun Liu. 

Nhat Minh Doan (VAST/NUS)

Title: Optimal independent system for the congruence subgroups $\Gamma_0(p)$ and $\Gamma_0(pq)$

Abstract: Can the congruence subgroup $\Gamma_0(N)$ be represented using a set of freely independent generators whose Frobenius norms satisfy the growth condition $O(N)$? Joint work with Sang-hyun Kim, Mong Lung Lang, and Ser Peow Tan, we provide a positive answer to this question when $N$ is either a prime number $p$ or the product of two odd primes, denoted as $pq,$ with the condition that $|\sqrt{p}-\sqrt{q}|<\sqrt{2}$.

In the special case when $N$ is either a prime number $p$ or the square of a prime $p^2$, we can even ensure that these generators have either 0 or $N$ in their $(2,1)$ components. This result confirms a conjecture made by Kulkarni. While proving this, we also establish that $\Gamma_0(N)$ possesses a fundamental domain that is a special polygon, as defined by Kulkarni, encoded using a generalized Farey sequence, with denominators that are less than $\sqrt{\frac{4N}{3}}$.

Viveka Erlandsson (Bristol)

Title: Counting geodesics of given commutator length

Abstract: It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface S is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved independently by both by Phillips-Sarnak and Katsura-Sunada. A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface) and obtain their asymptotic growth. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto. 

Junehyuk Jung (Brown)

Title: Zelditch’s trace formula and effective Bowen’s theorem. 

Abstract: In 1989, Zelditch considered the trace of an invariant operator composed with a pseudo-differential operator. The resulting trace formula turned out to be extremely useful in studying the distribution of closed geodesics on hyperbolic surfaces. I will demonstrate the simplest case of the proof, and discuss how things can be generalized to higher dimensional hyperbolic manifolds. This is a joint work with Insung Park.

Min Lee (Brown)

 Title: Murmurations of holomorphic modular forms in the weight aspect 

Abstract: In April 2022, He, Lee, Oliver, and Pozdnyakov made an interesting discovery using machine learning – a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation 'murmurations of elliptic curves.' Naturally, one might wonder whether we can identify a common thread of 'murmurations' in other families of L-functions. In this talk, I will introduce a joint work with Jonathan Bober, Andrew R. Booker and David Lowry-Duda, demonstrating murmurations in holomorphic modular forms.

Seonhee Lim (SNU)

Title: Euclidean algorithms are Gaussian over quadratic imaginary fields

Abstract: Baladi and Vallée showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with a bounded denominator based on the transfer operator methods. We extend the result and spectral techniques to the Euclidean algorithm over imaginary quadratic fields by studying the dynamics of the nearest integer complex Gauss map, which is piecewise analytic and expanding but does not have full branch inverse maps. A finite Markov partition with a regular CW-structure enables us to associate the transfer operator acting on a direct sum of $C^1$-spaces, from which we obtain the Gaussian distribution as well as residual equidistribution. (This is joint work with Jungwon Lee and Dohyeong Kim.)

Hideki Miyachi (Kanazawa):
Title : The negative exponential decay of the Teichmueller distance between grafting and Teichmueller rays

Abstract : This is a joint work with Zhiyang LV and Yi Qi. In this talk, I will show that for arrational measurad lamination, the associated Teichmueller ray and the grafting ray are asymptotic with negative exponential decay. This result is a refinement of a result by S.Gupta.

Junho Peter Whang (SNU)

Title: On congruence classes of local systems

Abstract: We introduce the notion of congruence for local systems on manifolds, motivated by considerations in Diophantine analysis, invariant theory, and topology. Restricting to integral SL2-local systems on surfaces, we observe a finiteness result for the number of congruence classes with prescribed invariants. We also present a natural composition law among congruence classes, and pose relevant problems.