More information about the seminar can be found at: https://ywfan-math.github.io/ADCD.html

**Upcoming Talk：**

**Title: **Boundedness problems in conformal dynamics

**Speaker:** Yusheng Luo (Stony Brook University)

**Date and time:** May 23 (Tue.), 09:00-11:00 am

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

In 1980s, Thurston’s formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurston’s proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurston’s boundedness theorem and the Sullivan dictionary, McMullen conjectured that certain hyperbolic components of rational maps are bounded.

In this talk, I will start with a historical discussion on a general strategy of the proof of Thurston’s boundedness theorem. I will then explain how a similar strategy could work for rational maps, and discuss some recent breakthrough towards McMullen's boundedness conjecture.

**Past Talks:**

**Title:** Categorical dynamical systems arising from sign-stable mutation loops

**Speaker: **Shunsuke Kano (Tohoku University)

**Date and time: **Apr. 25 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

A cluster modular group is a symmetry group of the theory of cluster algebras. An element is called a ''mutation loop''. I and T. Ishibashi introduced the notion of sign stability of the mutation loops as a generalization of the pseudo-Anosovness of the mapping classes. In this talk, I will define the autoequivalences of the derived category of the Ginzburg dg algebra of a quiver with potential associated with a sign-stable mutation loop, and show the calculation of their categorical entropies. If time permits, I'll explain the pseudo-Anosovness of these autoequivalences.

**Title: **Some Developments on Filtration Methods in Diophantine Geometry and Nevanlinna Theory

**Speaker: **Julie Tzu-Yueh Wang (Academia Sinica)

**Time: **Apr. 18 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

In 1994 Faltings and W\"ustholz introduced a new geometric method in the study of Diophantine approximation, called the filtration method, which involved working with ``many" sections of a line bundle and producing many linear combinations of them vanishing along appropriate divisors. This was further developed by Evertse and Ferretti. Independently, Corvaja and Zannier also worked with filtrations of the same kind, which was further refined and developed by Levin and Autissier, etc. Recently, Ru and Vojta formulated a general version of the celebrated Schmidt's Subspace Theorem that unifying many known results with filtration methods. We will introduce these developments and mention some applications of Ru-Vojta's theorem in the study of integral points and gcd theorem. We will also mention some corresponding results in Nevanlinna theory. This talk includes joint works with Erwan Rousseau and Amos Turchet and a joint work with Ji Guo and one with Yu Yasufuku.

**Title: **Motivic DT invariants of quadratic differentials

**Speaker:** Fabian Haiden (Centre for Quantum Mathematics, University of Southern Denmark)

**Time: **Apr. 11 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

The problem of counting saddle connections and closed loops on Riemann surfaces with quadratic differential (equivalently: half-translation surfaces) can be, somewhat surprisingly, reformulated in terms of counting semistable objects in a 3-d Calabi-Yau category with stability condition. Here "counting" happens within the powerful framework of motivic Donaldson-Thomas theory as developed by Kontsevich-Soibelman, Joyce, and others. For meromorphic quadratic differentials with simple zeros, this reformulation is due to the work of Bridgeland-Smith. The case of quadratic differentials without higher order poles - in particular holomorphic ones - requires entirely different methods, based on deformation of A-infinity categories and transfer of stability conditions. As an application, counts of saddle connections and closed loops are related by the wall-crossing formula as one moves around in the moduli space. Based on 2104.06018 and 2303.18249.

**Title:** Intersection of holomorphic curves with generic hypersurfaces

**Speaker: **Duc Viet Vu (University of Cologne)

**Time:** Mar. 28 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

Let f be holomorphic curve in a complex projective manifold X. I will explain why the geometric intersection of the (transcendental) curve f with a very generic divisor of a very ample line bundle L on X is of maximal growth and f almost misses general enough analytic subsets on X of codimension at least 2. This is a joint-work with Dinh Tuan Huynh.

**Title: **Dynamics on projective spaces: maps, correspondences and group actions

**Speaker: **Lucas Kaufmann (University of Orléans)

**Time:** Mar. 21 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

In the first part talk I will introduce the main objects and fundamental results on the (global) dynamics of holomorphic maps on the complex projective space. In a second part, I will discuss multivalued maps (i.e. holomorphic correspondences) and their dynamics. If time permits, I will mention some applications to the study of group actions on the Riemann sphere and to the theory of products of random matrices.

**Title: **Stability conditions on K3 surfaces and mass of spherical objects

**Speaker:** Genki Ouchi (Nagoya University)

**Time:** Mar. 14 (Tue.), 15:00-17:00

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

Huybrechts proved that a stability condition on a K3 surface is determined by the stability of spherical objects. Motivated by the study of the Thurston compactification of spaces of stability conditions expected by Bapat, Deopurlar and Licata, I would like to show that a stability condition on a K3 surface is determined by the mass of spherical objects. This talk is based on the joint work with Kohei Kikuta and Naoki Koseki.

**Title:** Calabi-Yau threefolds with c_2-contractions-revisited

**Speaker:** Keiji Oguiso (University of Tokyo)

**Time:** Tues.,15:00-17:00, Dec.20,2022

**Venue:** Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract:**

The second Chern class $c_2$ plays a special role in studying Calabi-Yau threefolds. Among other things, we discuss about KSC (Kawaguchi-Silverman's Conjecture) and the finiteness problem of real forms of Calabi-Yau threefolds with $c_2$-contractions. Though nothing appears in the title and main statements, our arguments use dynamical degrees due to Dinh-Sibony and current work on slow dynamics due to Dinh-Lin-O-Zhang in essential ways.

**Title: **Integral-affine structures and degenerations of K3 surfaces

**Speaker:** Philip Engel (University of Georgia)

**Time:** Tues.,9:00-11:00 am, Dec.13,2022

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract:**

I will discuss a correspondence between degenerations of polarized K3 surfaces and integral-affine structures on the 2-sphere containing a weighted balanced graph. Under this correspondence, natural questions emerge about the dynamics of the straight line flow on integral-affine manifolds. We will explore (1) the relationship between closed trajectories and immersed elliptic curves in the K3 surface, and (2) the possibility of a tropical Yau-Zaslow formula for counting immersed trees.

**Title**: Anosov representations, Hodge theory, and Lyapunov exponents

**Speaker: **Simion Filip (University of Chicago)

**Time:** Tues.,9:00-11:00am,Dec.6,2022

**Venue:** Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

Discrete subgroups of semisimple Lie groups arise in a variety of contexts, sometimes "in nature" as monodromy groups of families of algebraic manifolds, and other times in relation to geometric structures and associated dynamical systems. I will discuss a class of such discrete subgroups that arise from certain variations of Hodge structure and lead to Anosov representations, thus relating algebraic and dynamical situations. Among many consequences of these relations, I will explain Torelli theorems for certain families of Calabi-Yau manifolds (including the mirror quintic), uniformization results for domains of discontinuity of the associated discrete groups, and also a proof of a conjecture of Eskin, Kontsevich, Moller, and Zorich on Lyapunov exponents. The discrete groups of interest live inside the real linear symplectic group, and the domains of discontinuity are inside Lagrangian Grassmanians and other associated flag manifolds. The necessary context and background will be explained.

**Title: **Topological entropy for non-archimedean dynamics

**Speaker:** Junyi Xie (Peking University)

**Time: **Tues.,15:00-17:00,Nov.29, 2022

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract: **

The talk is based on a joint work with Charles Favre and Tuyen Trung Truong. We prove that the topological entropy of any dominant rational self-map of a projective variety defined over a complete non-Archimedean field is bounded from above by the maximum of its dynamical degrees, thereby extending a theorem of Gromov and Dinh-Sibony from the complex to the non-Archimedean setting. We proceed by proving that any regular self-map which admits a regular extension to a projective model defined over the valuation ring has necessarily zero entropy. To this end we introduce the \epsilon-reduction of a Berkovich analytic space, a notion of independent interest.

**Title:** Integral-affine structures and degenerations of K3 surfaces

**Speaker:** Philip Engel (University of Georgia)

**Time: **Tues.,9:00-11:00am,Nov. 22, 2022

**Venue: **Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

**Abstract:**

I will discuss a correspondence between degenerations of polarized K3 surfaces and integral-affine structures on the 2-sphere containing a weighted balanced graph. Under this correspondence, natural questions emerge about the dynamics of the straight line flow on integral-affine manifolds. We will explore (1) the relationship between closed trajectories and immersed elliptic curves in the K3 surface, and (2) the possibility of a tropical Yau-Zaslow formula for counting immersed trees.

** **

Title: A comparison of categorical and topological entropies on Weinstein manifolds

Speaker: Hanwool Bae (Seoul National University)

Date and time: Nov. 15, 15:00-17:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract: Every symplectic automorphism on a symplectic manifold induces an auto-equivalence on the (derived) Fukaya category, which gives rise to a categorical dynamical system. In this talk, I will first give a brief review of various Fukaya categories of symplectic manifolds with boundaries. Then, for a given symplectic automorphism, I will discuss how the categorical entropies of auto-equivalences induced by f on different Fukaya categories are compared. Then I will explain that in the case when M is a Weinstein domain and f is an exact symplectic automorphism on M that equals the identity map near the boundary of M, the topological entropy of f is greater than or equal to the categorical entropy of the corresponding auto-equivalence on the wrapped Fukaya category of M. This is based on joint work with D. Choa-W. Jeong-D. Karabas- S. Lee and Sangjin Lee.

Title: Kummer Rigidity for Irreducible Holomorphic Symplectic Manifolds

Speaker: Seung uk Jang (University of Chicago)

Date and time: Nov. 8, 9:00-11:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract:

In the preliminary part, we will go through the notion of irreducible holomorphic symplectic (IHS) manifolds, which is one of the high-dimensional generalizations of K3 surfaces. The action of holomorphic automorphisms on IHS manifolds is well-known, in terms of cohomology actions and characteristic currents (i.e., Green currents) that the automorphism induces. We will briefly go through the known theory, together with an easy, computable example originating from Arnold's Cat map.

The main talk will discuss studying holomorphic automorphisms on IHS manifolds that have the volume-class Green measures. The tools are analogous to those for the K3 surfaces, as seen in recent research by Cantat, Dupont, Filip, and Tosatti.

We will see which part of the arguments for the K3 surface can be generalized easily, and which part faces some difficulties. I will present the current status of overcoming each, and for which assumptions it is known that such automorphisms originate from a toral affine map (i.e., is 'Kummer').

Title: An upper bound for polynomial log-volume growth of automorphisms of zero entropy

Speaker: Fei Hu (University of Oslo)

Date and time: Nov. 1, 9:00-11:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract:

Let f by an automorphism of zero entropy of a smooth projective variety X. The polynomial log-volume growth plov(f) of f is a natural analog of Gromov's log-volume growth of automorphisms (of positive entropy), formally introduced by Cantat and Paris-Romaskevich for slow dynamics in 2020. A surprising fact noticed by Lin, Oguiso, and Zhang in 2021 is that this dynamical invariant plov(f) essentially coincides with the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with (X, f), introduced by Artin, Tate, and Van den Bergh in the 1990s. It was conjectured by them that plov(f) is bounded above by d^2, where d = dim X.

We prove an upper bound for plov(f) in terms of the dimension d of X and another fundamental invariant k of (X, f) (i.e., the degree growth rate of iterates f^n with respect to an arbitrary ample divisor on X). As a corollary, we prove the above conjecture based on an earlier work of Dinh, Lin, Oguiso, and Zhang. This is joint work with Chen Jiang.

Title: Equidistribution of Hodge loci

Speaker: Salim Tayou (Harvard University)

Date and time: Oct. 25, 9:00-11:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract: Given a polarized variation of Hodge structures, the Hodge locus is a countable union of proper algebraic subvarieties where extra Hodge classes appear. In this talk, I will explain a general equidistribution theorem for these Hodge loci and explain several applications: equidistribution of higher codimension Noether-Lefschetz loci, equidistribution of Hecke translates of a curve in the moduli space of abelian varieties and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint work with Nicolas Tholozan.

Title: Periodic points for systems of varieties

Speaker: Junho Peter Whang (Seoul National University)

Date and time: Oct. 18, 15:00-17:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract: Given a finite system of regular maps between algebraic varieties, it is natural to seek a procedure that determines whether or not a given (rational) point has finite orbit under repeated application of the maps. In this talk, we establish the existence of such a procedure for some simple system of varieties.

Title: b-divisors and dynamical degrees

Speaker: Charles Favre (École Polytechnique)

Date and time: Oct. 11, 15:00-17:00

Venue: Zoom Meeting ID: 897 9522 8294 Passcode: 1.17628

Abstract: Joint work with Nguyen Bac Dang. We develop an intersection theory for b-divisors and used it to get informations on the degree growth of iterates of rational maps.