Date/Time : 19 Nov 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Arpan Kabiraj, Indian Institute of Technology Palakkad

In 80’s, Goldman introduced a Lie bracket on the linear span of the free homotopy classes of directed closed curves on a surface called the Goldman Lie bracket. In this talk we will discuss some properties of Goldman Lie bracket using hyperbolic geometry. We will use these properties to give characterizations of simple closed curves. This is a joint work with Moira Chas.

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Date/Time : 18 Nov 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Fernando Camacho Cadena, Strasbourg University

A hyperbolic structure on a surface S can be deformed by cutting along a curve c without self—intersections, twisting, and re-gluing. Such a deformation can also be seen as the Hamiltonian flow of the length function of c on the Teichmüller space of S, equipped with the Weil—Petersson symplectic form. In this talk, I will discuss Hamiltonian flows associated to length functions of self—intersecting curves, and their generalizations to higher rank Teichmüller spaces. This is partially joint work with J. Farre and A. Wienhard.

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Date/Time : 11 Nov 2025 13:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Amir Mohammadi, University of California, Berkeley

There is a profound connection between homogeneous dynamics and number theory, particularly in the study of Diophantine approximation. A landmark example is Margulis’s resolution of the Oppenheim conjecture in the mid-1980s using tools from dynamics, followed by further strengthenings by Eskin, Margulis, and Mozes based on Ratner’s seminal work. In this talk, we will present effective results in this direction, with an emphasis on obtaining polynomial rates. This is based on joint works with Elon Lindenstrauss, Zhiren Wang, and Lei Yang.

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Date/Time : 22 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Richard Canary, University of Michigan

Margulis proved asymptotic counting results for the number of closed geodesics less than a fixed length in the quotient of a uniform lattice in a rank one Lie group. Patterson and Sullivan developed a technology to generalize his results to convex cocompact and geometrically finite subgroups of rank one Lie groups.

An element of a higher rank Lie group has a vector-valued translation length function, which gives rise to more than one reasonable notion of the length of an element. Anosov groups of higher rank semi-simple Lie groups are the natural analogue of convex cocompactness in rank one. Sambarino proved counting results for all well-behaved length functions in this setting.

Relatively Anosov subgroups of higher rank Lie groups are one generalization of geometrically finite subgroups of rank one Lie groups. In this talk we obtain counting results in this setting. The crucial new difficulty is that the relevant flow spaces are non-compact, whereas they were compact in the Anosov setting. These results are based on work with Zhang and Zimmer on Poincare series of relatively Anosov groups and a rather general formulation of Patterson-Sullivan theory which was developed with Blayac, Zhu and Zimmer.

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Date/Time : 01 Oct 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Homin Lee, Korea Institute for Advanced Study

In this talk, we will discuss about smooth higher rank lattice actions on manifolds with positive entropy. For instance, when lattices in SL(n,R) act on an n-dimensional manifold with positive entropy with n at least 3, we will see that the lattice is commensurable with SL(n,Z) and a topological information on manifolds. This is a joint work with Aaron Brown.

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Date/Time : 25 Sep 2025 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Keiji Oguiso, The University of Tokyo

My talk is based on my joint work with Professor Koji Fujiwara (Kyoto University) and Professor Xun Yu (Tianjin University).

Main result of my talk is the finiteness of the Neron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic under the assumption that the Picard number greater than or equal to 6 (which is optimal to ensure the finiteness). In this talk, after recalling the notion of hyperbolicity of group due to Gromov and its importance in mathematics, I would like to explain first why the non-elementary hyperbolicity of K3 surface automorphism group is the problem of the Neron-Severi lattices and next how one can deduce the above-mentioned finiteness, via the study of genus one fibrations on K3 surfaces and recent work of Kikuta and Takatsu on geometrically finiteness.

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Date/Time : 27 Aug 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Guolei Zhong, Institute for Basic Science, Korea

Let X be a smooth projective variety and G an abelian subgroup of automorphisms of X over the field of algebraic numbers. Suppose that G is of positive entropy. It was proved by Dinh and Sibony that the group is then free abelian of rank no more than dim(X)-1. In the past decade, from algebraic dynamics, the geometry of the extremal case when the rank achieves the maximal has been well-understood by Zhang. In this talk, we study the maximal rank case in the aspect of arithmetic dynamics. More precisely, we construct a canonical height function associated with the group, corresponding to a nef and big divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi–Silverman conjecture for each group element. Some further discussion towards its application and the existence of Zariski dense orbits will also be presented. This is based on a joint work with Fei Hu.

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Date/Time : 21 Aug 2025 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wooyeon Kim, Korea Institute for Advanced Study

We study an effective version of Ratner’s equidistribution theorem in the space of affine lattices. For d>1, let H be a minimal horospherical subgroup of SL(d,R), embedded in ASL(d,R), with respect to a one-dimensional diagonal flow a(t). We show that the a(t)-push-forwards of a segment of H-orbit become equidistributed at a polynomial rate in the space of affine lattices, under a suitable Diophantine condition on the initial point. This extends the previous result of Strömbergsson for d=2 to higher dimensions.

Our proof exploits the SL(d,Z)-monodromy action on each fiber torus T^d induced by the H-orbits, and uses techniques from Bourgain-Furman-Lindenstrauss-Mozes on random walks in the torus setting, adapted here to a fiberwise context.

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Date/Time : 20 Aug 2025 13:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Nguyen Tat Thang, Institute of Mathematics, Vietnam Academy of Science and Technology

Let $f:\R^n\to \R^m$ be a real polynomial mapping, $\Sing(f)$ be the set of singular points of $f$ in $\R^n$, and $K_0(f)=f(\Sing(f))$. A {\it bifurcation set} of $f$ is the smallest set of values in $\R^m$ outside which $f$ is a locally trivial fibration. A regular value $\lambda\in f(\R^n)\setminus K_0(f)$ is called a {\it typical value at $\infty$} of $f$ if there is an open neighborhood over which $f$ is a trivial fibration. Otherwise, $\lambda$ is called an {\it atypical value at $\infty$} of $f$. In this talk, we characterize atypical values at infinity of a real polynomial function of three variables by a certain sum of indices of the gradient vector field of the function restricted to a sphere with a sufficiently large radius. This is an analogy of a result of Coste and de la Puente for real polynomial functions with two variables. We also give a characterization of atypical values at infinity of a real polynomial map whose regular fibers are 2-dimensional surfaces. This is a joint work with Masaharu Ishikawa.

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