Taeheyong Kim (Technion Haifa), May 26, 2022
Title: Entropy, large deviations, and applications
Abstract: The theory of large deviations is concerned with the exponential decay of probabilities of remote tails of sequences of probability distributions. Entropy is concerned with the complexity of dynamical systems. In this talk, we will focus on the relationship between large deviation and entropy for some dynamical systems. We will then explore some applications of this relationship.
Keivan Mallahi-Karai (Bremen Univ.), October 4, 2022
Title: On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$
Abstract: Studying the extreme value distributions for sequences of random variables is a classical subject in probability theory. Motivated by a work of Athreya and Margulis, we study an analogous extreme value problem for the cusp excursion of the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function for the normalized deepest cusp excursions of the unipotent flow and find closed analytic formulas for the distribution function in certain ranges and establish its asymptotic behavior. The talk is based on a joint work with Maxim Kirsebom.
Taehyeong Kim (Seoul Univ./Technion), September 23, 2022
Title: Entropy rigidity and its applications to Diophantine approximation
Abstract: In this talk, we mainly focus on dynamical entropy on homogeneous spaces and review the entropy rigidity on homogeneous dynamics following the Einsiedler-Lindenstrauss lecture note (Diagonal actions on locally homogeneous spaces). As its applications, we study the following previous results about Diophantine approximation: Lim-de Saxc -Shapira (2018) and David-Shapira (2018). Finally, we introduce the effective version of entropy rigidity and its application to Lim-de Saxc -Shpira (2018). This is joint work with Wooyeon Kim and Seonhee Lim.
Kyeongro Kim (Bremen Univ.), September 20, 2022
Title: Relation numbers and generalized Farey graphs
Abstract: Let x be a complex number. Let A be an upper triangular matrix in SL(2,C) whose non-zero entries are 1, and B(x) a strict lower triangular matrix in SL(2,C) whose (2,1)-entry is x. Then, x is called a relation number if the group generated by A and B(x) is not a rank 2 free group. The characterization problem of the relation numbers is a long-standing conjecture in the group theory. This problem has been studied by many mathematicians including Rimhak Ree. In this talk, I introduce a topological and dynamical tool, the generalized Farey graphs. This gives a new equivalent definition for relation numbers. Then, I show how the generalized Farey graph can be applied to find a sequence of relation numbers which are unknown before.
Jiyoung Han (KIAS), August 30, 2022
Title: Asymptotic behaviors of lattice counting function as dimension diverges
Abstract: Consider the number of lattice points inside a given symmetric set, for example, a ball centered at the origin whose volume is V, as a random variable on the space of unimodular lattices. In 1960, Schmidt showed that as dimension goes to infinity, the sequence of these random variables with fixed volume V converges to the Poisson distribution. His work was developed by S dergren (2011) and Str mbergsson-S dergren (2019) in diverse settings. In a joint work with Anish Ghosh and Mahbub Alam, we obtained analogs of their results on 1) the space of unimodular affine lattices; 2) the space of unimodular lattices with a congruence condition. These works heavily rely on the higher moment formulas of Siegel transforms. In this talk, we recall the idea of proofs using the method of moments and how to obtain the higher moment formulas for those two cases.
Seung-yeon Ryoo (Princeton Univ.), August 30, 2022
Title: Quantitative nonembeddability of nilpotent Lie groups and groups of polynomial growth into superreflexive spaces
Abstract: It is known that simply connected nonabelian nilpotent Lie groups and not virtually abelian groups of polynomial growth fail to embed bilipschitzly into superreflexive Banach spaces. We quantify this fact in two ways. First, we provide a lower bound on the distortion of balls in the aforementioned groups into superreflexive spaces. In particular, we show that the Lp-distortion, (11. If time permits, I will discuss conjectures on the distortion and compression rate when the target space is L1.
Damian Osajda (University of Wroclaw), August 16, 2022
Title: Tits Alternative for groups acting properly on 2-dimensional CAT(0) complexes
Abstract: We prove that groups acting properly on 2-dimensional CAT(0) complexes with a bound on the orders of cell stabilisers satisfy the Tits Alternative, that is, their finitely generated subgroups are either virtually free abelian or contain a (non-abelian) free subgroup. This is joint work with Piotr Przytycki (McGill).
Dongryul Kim (Yale Univ.), August 9,10, 2022
Title: Dynamics of self-joinings of hyperbolic manifolds 1,2
Abstract: Part 1. Anosov representations can be regarded as a higher-rank version of convex cocompact Kleinian groups. From this point of view, this series of two talks introduces the notion of self-joinings of hyperbolic manifolds and the dynamics they possess. Mainly, we discuss how the classical result of Sullivan, relating critical exponent and Hausdorff dimension at infinity, is formulated in these higher-rank circumstances. This is joint work with Yair Minsky and Hee Oh.
part 2. Anosov representations can be regarded as a higher-rank version of convex cocompact Kleinian groups. From this point of view, this series of two talks introduces the notion of self-joinings of hyperbolic manifolds and the dynamics they possess. Mainly, we discuss how the classical result of Sullivan, relating critical exponent and Hausdorff dimension at infinity, is formulated in these higher-rank circumstances. This is joint work with Yair Minsky and Hee Oh.
Plinio Murillo (KIAS), November 12, 2019
Title: Growth of systole of arithmetic hyperbolic manifolds
Abstract: The systole of a Riemannian manifold M is the length of a shortest non-contractible closed geodesic in M. In this talk, we will discuss how to produce hyperbolic manifolds with large systole, and the impact in the topology of such manifolds. This talk will be an enlarged version of a previous talk at the 2019 KMS Annual Meeting, but the previous talk is not required at all.
Ronggang Shi (Fudan Univ.), November 11, 2019
Title: Hausdorff dimension of divergent trajectories on product spaces
Abstract: For a one-parameter subgroup action on a finite volume homogeneous space, the set of points admitting divergent on average trajectories is strictly less than the manifold dimension. The question of calculating the precise dimension is the main challenge in the current time. We introduce a method that can be used to answer this question for certain product systems.
Junehyuk Jung (Texas A & M and Rice Univ.), September 25, 2019
Title: Topology of the nodal set of spherical harmonics on S^3
Numerical simulation of Alex Barnett has shown that nodal sets of large degree N random wave on the 3-dimensional space are very different from those on the 2-dimensional space: only one giant component shows up in the graphics (although Nazarov-Sodin show that there is an increasing number of components as the degree tends to infinity). P. Sarnak posed the problem of computing the expected genus of the giant component and proposed that it has maximal order N3. Together with S. Zelditch, I prove that these properties hold for the real and imaginary parts of random equivariant spherical harmonics of degree N. This is joint work with S. Zelditch.
Kiho Park (Univ. Chicago), September 6, 2019
Title: Thermodynamic formalism of fiber-bunched GL(2,R)-cocycles
Abstract: We study the singular value potentials of fiber-bunched GL(2, R)-cocycles. We show that the singular value potentials of irreducible GL(2, R)-cocycles have a unique equilibrium state. Among the reducible cocycles, we provide a characterization for cocycles whose singular value potentials have more than one equilibrium state.
Min Lee (Univ. Bristol), July 05, 2019
Title: Effective equidistribution of rational points on expanding horospheres
Abstract: I will present an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least 3 dimensions. Their proof exploits measure classification results. We pursue an alternative approach, based on Fourier analysis, additive twists of automorphic forms, spectral theory, and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical subgroup.
Wenyuan Yang (Peking Univ.) May 14, 2019
Title: Martin boundary covers Floyd boundary
Abstract:In this talk, we discuss a relation between two boundaries for a finitely generated group: Martin boundary associated with a finitely supported symmetric random walk, and Floyd boundary obtained from a conformal scaling of Cayley graphs. We prove that the identity map over the group extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. Applications are given to the class of relatively hyperbolic groups. This is joint work with I. Gekhtman, V. Gerasimov and L. Potyagailo.