Date/Time : 28 Oct 2026 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Guillaume Tahar, Guangdong Technion – Israel Institute of Technology

On a topological sphere endowed with a flat metric with conical singularities, the curvature gap quantifies the obstruction to realize a partition of the set of conical singularities into two sets of equal total angle defect. Unless its curvature gap is equal to zero, such a flat sphere cannot contain a simple closed geodesic. Drawing on the Delaunay decompositions of flat surfaces, we give a quantitative generalization of the latter statement. We prove that when the curvature gap is nonzero, there is an explicit upper bound on the lengths of simple geodesic trajectories in the flat sphere. If time allows, we will give extensions of this result to affine structures on punctured spheres and applications to dynamics of germs of biholomorphism tangent to the identity in complex dimension two. This is a work in collaboration with Xavier Buff, Arnaud Cheritat and Kai Fu.

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Date/Time : 28 Oct 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Guillaume Tahar, Guangdong Technion – Israel Institute of Technology

The geodesic flow on the tangent bundle of a Riemann surface endowed with a complex affine structure (or, equivalently, a meromorphic connection) is a dynamical system that exhibits remarkable similarities with the real integral curves of polynomial vector fields in two variables and with the dynamics of germs of biholomorphisms tangent to the identity in complex dimension two. We will present the geometric interpretation of the singularities of the connection, including, time permitting, irregular singularities, and show that many geometric properties of an affine surface can be read from the topology of the bifurcation locus in its tangent bundle. We will also discuss applications to irrational triangular billiards and the classification of geodesic flows up to real affine and holomorphic conjugacy. This is joint work with Xavier Buff and Arnaud Chéritat.

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Date/Time : 09 Sep 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Miri Son , National University of Singapore

Recently, Fisher and Melnick classified SL(n,R)-actions on n-dimensional manifolds for n≥3. In this talk, we generalize their result by classifying smooth and real-analytic SLnR-actions on m-dimensional manifolds for 3≤n≤m≤2n-3. This problem is related to the Zimmer program, since any action of a Lie group restricts to an action of its lattices.

This classification relies on linearization of SL(n,R)-actions near a fixed point. The analytic case was proved by Guillemin-Sternberg and Kushnirenko. In the smooth case, the key is the linear representation of SL(n,R). We discuss the smooth case, which is ongoing joint work with Insung Park.

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Date/Time : 02 Sep 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yubin Shin, National University of Singapore

Let G be a semisimple Lie group, Gamma be a lattice in G, and let A = {a_t} be a one-parameter diagonal subgroup. Given a point x in G/Gamma and a curve phi: I -> U^+ in the expanding horospherical subgroup of A, we consider the family of expanding translates a_t phi(I)x as t -> infinity. The equidistribution problem asks for conditions under which these translates eventually spread uniformly throughout G/Gamma with respect to the Haar probability measure.

Motivated by applications to Diophantine approximation and hyperbolic geometry, Shah initiated the study of this problem and obtained results in several important special cases involving SL_n(R) and a single copy of SO(n,1). Lei Yang later established equidistribution for analytic curves satisfying an algebraic condition when G = SO(n,1)^k. In this talk, I will discuss an extension of Lei Yang’s result from analytic curves to smooth curves when G = SO(n_1,1) x … x SO(n_k,1).

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Date/Time : 20 Aug 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Minsung Kim, Pohang University of Science and Technology

In chaotic systems, the mixing property is known for the fast decay of correlation. It is called rapid mixing if the correlation function decays super-polynomially. The mixing mechanism for hyperbolic systems and their compact group extensions was studied by Dolgopyat in a series of his papers in the late 90s’.

In this talk, we prove rapid mixing for almost all random walks generated by m larger equal than 2 translations on an arbitrary nilmanifold. For several classical classes of nilmanifolds, we show m=2 suffices. This provides a partial answer to the question raised in the work of Dolgopyat (’02) about the prevalence of rapid mixing for random walks on homogeneous spaces.

This is joint work with Dmitry Dolgopyat and Spencer Durham.

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Date/Time : 20 Aug 2026 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yongnam Lee, Institute for Basic Science

Motivated by DeVleming’s work on moduli of surfaces in P^3 and Chen-Hu-Jiang’s work on moduli of threefolds with volume 2 and geometric genus 4, we study the deformation of pairs of P^3 and hypersurfaces using the classification of Q-Gorenstein degenerations of P^3 with canonical singularities. We prove that if a degenerating threefold has canonical singularities, then the moduli space is smooth at the corresponding pair. Consequently, we find some boundary divisors of the moduli of smooth hypersurfaces. Finally, using the double cover method, we derive some information on the moduli space of threefolds X with canonical singularities with the same volume and geometric genus as a double cover of P^3 branched over a hypersurface. This is joint work with Jungkai Chen and Phin-Sing Soo.

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Date/Time : 12 Aug 2026 16:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Wenfei Liu, Xiamen University

The set V of volumes of stable surfaces, as a subset of positive rational numbers, satisfies the descending chain condition. Equivalently, any nonempty subset of V attains the minimum.

In this talk, I will report a joint work with Jihao Liu, showing that the minimal volume of a stable surface with Picard rank 1 is 1/6351, uniquely realized by a normal rational stable surface previously constructed from P^2 with 4 lines by Valery Alexeev and Wenfei Liu.

In the proof of the minimum, we use various filters to narrow down the possible basket of singularities on such a surface.

The last and very crucial one, the so-called pluricanonical product-dimension filter, was suggested to us by the AI!

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Date/Time : 28 Jul 2026 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Tianqi Wang, Yale University

We introduce notions of ending currents for properly discontinuous isometric actions of hyperbolic groups on Gromov hyperbolic spaces, generalizing Thurston’s ending laminations from the setting of Kleinian surface groups. In contrast with the 3-dimensional picture, we construct an example of geometrically finite representations of a closed surface group into $SO^+(4,1)$ whose parabolicity appears only at a non-simple closed geodesic, and an example of hyperbolic group extensions where different notions of ending currents, which coincide in the 3-dimensional setting, become distinct. This is joint work with Yair Minsky.

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Date/Time : 24 Jul 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Tsuyoshi Kato, Kyoto University

We associate to each proper discrete metric space X a Lie algebra that acts by locally generated derivations on an infinite tensor product of matrix algebras indexed by the points of X. We compute the homology of this Lie algebra with trivial scalar coefficients when X is the integer lattice in n-dimensional Euclidean space.

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