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 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jia Jia, Tsinghua University

We study the moduli space of higher rank marginally stable pairs (E,s), where E is a torsion free coherent sheaf of rank r on a smooth projective surface and s = (s_1, …, s_r) is a collection of r sections of E. Fixing the Chern character of E, the moduli space is realised as a subscheme of an appropriate Quot–scheme that parametrises quotient sheaves with the corresponding Hilbert polynomial.

We establish a precise link between these moduli spaces and the stable minimal models determined by E and its sections, together with the (analytic) minimal log resolutions of the base surface. Using the birational geometry of such minimal models, we analyse in detail the components of the Hilbert–Chow morphism from the moduli space to the Hilbert scheme of Cartier divisors on the surface. This is a work in progress.

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Date/Time : 30 Jul 2025 14:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that for two real numbers r,s ≥ 1, and for two smooth closed manifolds M and N, the C^r diffeomorphism group of M is elementarily equivalent to the C^s diffeomorphism group of N if and only if r=s and M is diffeomorphic to N. In the case of r=s=0, we can even weaken the hypothesis so that the manifolds are only assumed to be compact, dropping the smoothability or closedness hypotheses. This strengthens (1) Whittaker’s theorem (1963) on the C^0 case, and (2) Takens–Filipkiewicz theorems (1982) on C^p case with an integer p. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS)

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Date/Time : 10 Jul 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nimish Shah, The Ohio State University

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds.

We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.

This is a joint work with Pengyu Yang

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Date/Time : 10 Jul 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Si-Jong Kwak, Korea Advanced Institute of Science & Technology

The gonality conjecture of a smooth projective curve raised by Green and Lazarsfeld. has been one of the leading problems in geometry and syzygy. The conjecture says that the gonality of a curve can be encoded in the Betti table if a given curve is embedded by the line bundle of sufficiently large degree. It was proved by Ein-Lazarsfeld in 2016.

In this talk, we generalize this conjecture to the secant varieties of a curve. The gonality sequence d_1, d_2, . . . , d_g are encoded in syzygies of the secant varieties of a curve. This is a joint work with J. Choe and J. Park.

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Date/Time : 02 Jul 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : An Huang, Brandies University

What is the p-adic counterpart of the derivative operator, or the Laplacian? I shall describe a number of Non-Archimedean pseudo-differential operators arising in part from considerations in p-adic string theory. I shall explain the connections to Tate’s thesis in genus zero, and then in genus one, explain the computation of the Green’s function on the Tate curve. If time permits, questions regarding the spectrum of p-adic strings shall be mentioned.

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Date/Time : 23 Jun 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Chris Leininger, Rice University

I will discuss joint work with Autumn Kent in which we establish the existence of purely pseudo-Anosov surface subgroups of mapping class groups. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere. As a corollary we obtain the first examples of closed atoroidal surface bundles over surfaces.

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Date/Time : 18 Jun 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Khanh Le, Rice University

For a hyperbolic surface X with non-empty boundary, the ortho-length spectrum of X is the multi-set of lengths of (ortho)-geodesic arcs meeting the boundary of X perpendicularly at both endpoints. The spectrum was introduced by Basmajian for hyperbolic n-manifolds in the context of studying co-dimensional one embedded geodesic submanifolds. The ortho-length spectrum has been studied extensively in relation to geometric identities, hyperbolic geometry of surfaces and inverse geometry. In this work, we initiate a new investigation into the arithmetic properties of the ortho-length spectrum. In particular, we consider hyperbolic surfaces whose cosh-of-length of ortho-geodesics are all integers, ortho-integral surfaces. We prove that there are only finitely many ortho-integral surfaces for a fixed topological type. We also constructed infinitely many pairwise non-commensurable ortho-integral surfaces. In the talk, I will motivate the consideration of such surfaces and discuss some ideas and tools going into the proof. If time permits, I will also discuss some further questions that we would like to investigate. This is joint work with Minh Doan.

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