Abstract:
The problem of counting closed geodesics in Riemannian Manifolds with neg- ative curvature has been of interest for many years. Recently, Eftekhary has introduced a method for counting closed geodesics in closed manifolds with arbitrary curvature. Minimal surfaces in a Riemannian manifold are higher dimensional general- izations of closed geodesics. A natural question is how to count these types of submanifolds. We introduce a function that counts minimal tori in a Rie- mannian manifold (M, g) with dim M > 4. Moreover, we show that this count function is invariant under perturbations of the metric.

Abstract:
We introduce a Heegaard-Floer homology functor from the category of oriented links in closed three-manifolds and oriented surface cobordisms in four-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction has the advantage of being independent from the decoration. Some of the basic properties of this functor as well as some examples and applications are also discussed.

Abstract:
In this talk, we construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop theory for it and use it inparticular to provide examples of thick metric attractors with intermingled basins.

Abstract:
Several years ago, Yi Shi asked if Arnold's cat map admits a sequence of periodic orbits $O_n$ uniformly distributed in the following sense:
$$\liminf_{n\to\infty}\delta(O_n)^2 \pi(O_n)>0?$$ where $\delta(O_n)$ is the minimal distance of two points in $O_n$ and $\pi(O_n)$ is the period of $O_n$.
In this talk, I will give an affirmative answer to Shi's question. In fact, I will show that such uniformly distributed periodic orbits exists for any ergodic endomorphisms on tori. This is a joint work with Daohua Yu.

Abstract:
In this talk, we use the geometric theory of dessins d’enfants to make explicit calculations on curves. In particular, an algorithmic procedure for the construction of ramified covering of curves over number fields with prescribed ramifications and for the explicit construction of Jenkins-Strebel differentials will be presented.
The calculations in the proposed method are of a purely algebraic nature, and the method is in principle applicable for the construction of a wide variety of ramified coverings of curves.
The method also gives preferred cell decompositions on the curve and allows one to express the ramified covering map algebraically and in particular is applicable to the calculation of isogenies of elliptic curves defined over number fields.
As a byproduct of our method we construct quadratic differentials explicitly on curves defined by dessins on compact oriented topological surfaces.
Jenkins- Strebel differentials are a class of quadratic differentials that are of geometric interest because they naturally endow a cell decomposition of the surface with foliations by closed curves.
That they exist in abundance is proven by variational methods but a generic quadratic differential is not Jenkins-Strebel.
The method of the computation described here enables one to exhibit Jenkins-Strebel differentials on curves defined by dessins.

Abstract:
We discuss the generalization of the rigidity of the sharp first spectral gap under Ric ≥ 0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD(0, N ) spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry-Émery manifolds along with products, certain quotients and measured Gromov-Hausdorff limits of such spaces. In precise terms, we show in such spaces, the (Neumann) first gap is equal to (pi squared) divided by (diameter squared) if and only if the space is one dimensional with a constant density function. We use new techniques, mixing Sobolev theory and singular 1D-localization which might also be of independent interest. This is a joint work with C. Ketterer from Univ. of Freiburg and Y. Kitabeppu from Kumamoto Univ.

Abstract:
In this talk, we will talk about a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the hypersurface area under certain conditions. We will prove the all time existence and convergence of the flow. As an application, we show the isoperimetric inequality for such a domain. Especially, we solve the isoperimetric problem for the star-shaped hypersurface in the Riemannian manifold endowed with a closed, non-trivial conformal vector field. This is a joint work with Dr. Pan Shujing.

Abstract:
The classical theory of holomorphic dynamics includes the theory of Kleinian groups and the iteration theory of rational maps. In the 1980s Sullivan pointed out several analogies between these two subjects. For example, the concept of a Julia set of a rational map corresponds to the concept of a limit set of a Kleinian group, and periodic orbits of a rational map corresponds to closed geodesics. Analogous to Prime Geodesic Theorems in geometry and the Prime Number Theorem in number theory, we establish the first (effective) Prime Orbit Theorems in complex dynamics outside of the uniformly expanding case. This is partially inspired by the work of Oh–Winter on the Prime Orbit Theorem for hyperbolic (i.e. uniformly expanding) rational maps. This is joint with Tianyi Zheng and Juan Rivera-Letelier.

Abstract:
Generating functions of intersection numbers on moduli spaces of curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. In this talk I will first describe how to use Schur's Q-polynomials to obtain simple formulas for these functions. I will then discuss possible extensions for more general geometric models using Hall-Littlewood polynomials. This talk is based on joint works with Chenglang Yang.

Abstract:
We introduce a local and stable phenomenon/mechanism for ergodicity of smooth dynamical systems presenting some forms of hyperbolic behaviour. This provides a new approach towards the problem of stable ergodicity of typical partially hyperbolic volume-preserving diffeomorphisms. This talk is based on joint works with Hesam Rajabzadeh and Abbas Fakhari.

Abstract:
The topology of character varieties of surface groups is well known to be of fundamental importance, in part through its relationship, via the non-abelian Hodge theory to the moduli spaces of Higgs bundles. The calculation of Serre (also known as E-) polynomials of these varieties received an important impetus with works of Hausel and Rodriguez-Villegas (2008). These arithmetic methods, as well as new geometric techniques have been successfully developed and applied to other cases, like moduli of quiver representations, and character varieties of free or free abelian groups. In this talk we will introduce another point of view in the computations of the E-polynomials of character varieties for arbitrary finitely presented groups which is based on a natural stratification coming from affine GIT and the combinatorics of partitions. This is joint work with C. Florentino and A. Zamora.

Abstract:
In this talk, we aim to identify stable and prevalent behaviors appearing in the absence of dominated splitting for linear cocycles over subshifts of finite type (SFTs). It is known by the work of Mañe in the early eighties that when the fibers are 2-dimensional, for an open and dense subset of cocycles without dominated splitting, there are periodic orbits with non-real eigenvalues of equal modulus. In attempting to extend this to higher dimensions, the main challenge is that the existence of periodic fibers with elliptic behavior is not stable in higher dimensions.
In higher dimensions, we relax the requirement for periodic points with elliptic behavior and replace it with the existence of orbits (not necessarily periodic) for which the associated fiberwise linear maps remain uniformly close to homotheties. We call them quasi-conformal orbits. Using a mechanism relying on a covering property, we show that every volume-preserving cocycle over SFTs without dominated splitting can be approximated by cocycles stably admitting quasi-conformal orbits. We will also discuss the abundance of such orbits by exploring invariant measures supported on them. This talk is based on a recent joint work with Meysam Nassiri and Zahra Reshadat.

Abstract:
The theory of equivariant moving frames, introduced by Peter Olver in the late 1990s, is a modern reformulation of Cartan's classical theory of moving frames. Over the last two decades, this constructive theory has demonstrated its large potential for various applications. Of particular interest, it offers a practical approach for solving the equivalence problems between geometric structures as well as constructing the underlying normal forms. In this talk, we give a brief overview to equivariant moving frames theory and elaborate on how it facilitates the construction of normal forms for real-analytic manifolds acted on by (pseudo)-groups. We illustrate this method by presenting some examples specifically in Cauchy-Riemann (CR) geometry setting.

Abstract:
The problem of existence of constant scalar curvature Kahler metrics (cscK) on compact Kahler manifolds has been studied for many years. In the case of complex one dimension, the answer is provided by the uniformization theorem. It states that any compact Riemann surface admits a metric of constant Gaussian curvature. One way to prove this fact is to solve a semi linear elliptic equation. In higher dimensions, cscK metrics satisfy a fully nonlinear fourth order elliptic equation. It is extremely difficult to deal with such equations partially due to lack of maximum principle. In a recent breakthrough, Chen and Cheng proved some important a priori estimates for cscK metrics. More generally they showed that in a given Kahler class, the set of metrics with uniform bounded scalar curvature and bounded relative entropy is compact in $C^{\infty}$ topology. In a joint work with Z. Lu, we generalized their results. Mainly, we replace the uniform boundedness assumption on scalar curvature with some $L^p$ boundedness.

Abstract:
In this talk, we first introduce a more geometric point of view toward the ergodic theory of dynamical systems, provided that the phase space is endowed with a metric structure. Then we show how different objects and quantities that arise naturally in this way lead to a general definition of statistical stability. Then under the condition of measure invariance, we show that ergodic maps are always statistically stable. Finally motivated by the characterization of structural stability with the hyperbolicity condition, we discuss to what extent ergodicity is a good candidate for characterization of statistical stability.

Abstract:
Kahler-Ricci flow may develop singularity at finite time. In order to continue the flow across singularity, we need to understand geometry of metrics along the flow. Bounding curvature becomes a crucial method. A famous estimate of Perelman gave a very important curvature bound for manifolds with positive first Chern class. In this talk, I will report on some progress on bounding curvature for Kahler-Ricci flow.

Abstract:
Lecture 1: Starting from Morse functions, I will briefly introduce the definition of Heegaard Floer homology. We also give a combinatorial algorithm to compute the hat version Heegaard Floer homology. This talk includes joint work with Sucharit Sarkar.

Lecture 2: By the geometrization theorem, the fundamental group determines an irreducible three-manifold except lens spaces. It follows that the Heegaard Floer homology of a three-manifold (or a null homologous knot) is determined by its fundamental group. A direct relationship between the fundamental group and Heegaard Floer homology is expected. We show that the hat version knot Floer homology of a (1,1) knot is determined by certain presentations of its fundamental group. This is joint work with Matthew Hedden and Xiliu Yang.

Abstract:
This talk first gives an introduction to the irregular Riemann-Hilbert map for the meromorphic linear system of ODEs at a k-th order pole, as an analytic locally Poisson isomorphism between the de Rham and Betti moduli spaces. It then introduces the universal quantum linear ODE with a k-th order pole, and proves that the Stokes matrices, of the quantum differential equation, give rise to an isomorphism between two associative algebras that quantize the Poisson structures on the de Rham and Betti moduli spaces respectively. Sush associative algebra isomorphism is seen as a quantization of the irregular Riemann-Hilbert map. The main idea conveyed in this talk is then to study the highly transcendental Riemann-Hilbert map via the representation theory of the involved associative algebras. For example, in the case k=2, the associative algebra involved is the Drinfeld-Jimbo quantum group, and the above results lead to a dictionary between the Stokes phenomenon at a 2nd order pole and the representation theory of quantum groups, including at the roots of unit, the Gelfand-Testlin, crystal basis and so on.

Abstract:
We note that the theory of bordism of immersions, empowered and simplified by work of Asadi-Golmankhaneh and Eccles, could be made further applicable by use of the Gaussian elimination. We then apply this to study immersion of certain low dimensional manifolds.

Abstract:
Lecture 1: Searching for Kahler-Einstein metrics on Fano manifolds is an important problem in geometric analysis. By Prof. Tian’s solution of the YTD conjecture, it is known that a K-polystable Fano manifold always admits a KE metric. In this talk I will present new variational approaches to this problem. More precisely, we show that a Fano manifold admits a KE metric if and only if its (reduced) delta invariant is bigger than 1. This talk is based on my joint work with Prof. Tian, Rubinstein, and Darvas.

Lecture 2: In this talk we will give more details for our variational approach to the Kahler-Einstein problem on Fano manifolds. More precisely, we will present a slope formula of Ding functional along subgeodesics. Using this we can relate the properness of Ding functional to the reduced delta invariant that characterizes K-polystability. This talk is based on my joint work with Darvas.

Abstract:
In this talk, first I will recall some classical results about the gradient estimate of positive p-harmonic functions on Riemannian manifolds, including the results of Kotschwar-Ni, Wang-Zhang, Sung-Wang. Then I will introduce the results about the quantitative second-order Sobolev estimate of for positive p-harmonic functions in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions in weighted manifolds under the Bakry-Émery curvature-dimension condition. This is a joint work with Jiayin Liu and Yuan Zhou

Abstract:
By Hamilton-Tian conjecture, KR flow on a Fano manifold will converge to a Q-Fano variety which admits a singular KR soliton. This gives a deformation of Fano manifold to a canonical model via the Ricci flow. In this talk, I will discuss how to determine this canonical model for a class of Fano homogeneous spaces and show that the canonical model also keeps the group structure.