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 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 : 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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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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