• Title: Equivariant quantum D-modules in positive characteristic

• Abstract: The quantum D-module is a flat connection defined from counts of genus 0 holomorphic curves, a basic object in Gromov--Witten theory whose understanding has seen progress in recent years using positive characteristic methods. We survey the relationship of the mod p quantum connection to quantum deformations of Steenrod operations on mod p cohomology, and its applications in representation theory through symplectic duality (3D mirror symmetry) of symplectic resolutions. 

• Time and Location: June 5, 10 - 12 AM, Online (Zoom information will be announced by request)

• Title: Fukaya categories for surface cluster algebras

• Abstract: Given a surface with boundary and marked points (also called stops), there is a well-known associated cluster algebra. Its cluster variables correspond to arcs in the surface and clusters to ideal triangulations. This talk will discuss two types of associated Fukaya-type categories which close relate with this cluster algebra via a type of categorification. The first is relative 3-Calabi-Yau, and (conjecturally) arises as the partially wrapped Fukaya category of a 3-fold fibered over the surface. The second is (relative) 2-Calabi-Yau, and is given by the 1-periodic version of the partially wrapped Fukaya category of the marked surface itself. These two categories are closely related: the second is the so-called cosingularity category, meaning a certain Verdier quotient, of the first. All relevant background on cluster algebras will be recalled.

• Time and Location: May 22, 4 - 5 PM, Online (Zoom information will be announced by request)

• Title: Transverse foliation for three-dimensional Reeb flows

• Abstract:  In his pioneering work, using pseudoholomorphic curves in the symplectisations, Hofer studied Hamiltonian flows restricted to an energy level which is of contact type. He then proved many cases of Weinstein's conjecture on the existence of a periodic orbit in dimension three. The theory further developed by Hofer, Wysocki  and Zehnder, and since then,  pseudoholomorphic curves in symplectisations have been used as a powerful tool to study Hamiltonian systems with two degrees of freedom. In particular, they introduced a notion of a transverse foliation, which is a singular foliation of an energy level, where the  singular set consists of finitely many periodic orbits, called bindings, and the regular leaves are punctured Riemann surfaces transverse to the flow and asymptotic to the bindings at the punctures. This allows one to reduce the Hamiltonian dynamics on the three-dimensional energy level to the study of an area-preserving surface map.  In this lecture series, I will provide a gentle introduction on the theory of Hofer, Wysocki and Zehnder and explain how to construct transverse foliations in various concrete problems such as the restricted three-body problem. This is based on joint work with Naiara de Paulo, Umberto Hryniewicz, Pedro Salomão and Alexsandro Schneider.

• Time and Location: May 12, 1-3 PM, Online (Zoom information will be announced by request)

• Title: Hall algebras of Surfaces, Stories and Aspirations 

• Abstract: I will discuss the Hall algebra of the Fukaya category studied by myself and Samuelson, covering the main results of the two papers. In particular, I will explain something about Hall algebras, why they are important and the hope for an intrinsic construction of skein theory. Towards the end I will discuss Haiden's skein relation and the connection to Legendrian geometry.

• Time and Location: Apr 30, 9:00-10:00, Online (Zoom information will be announced by request)

• Title: Quasimorphisms on the group of density preserving diffeomorphisms of the Möbius band.

• Abstract: The existence of quasimorphisms on groups of homeomorphisms of manifolds has been extensively studied under various regularity conditions, such as smooth, volume-preserving, and symplectic. However, in this context, nothing is known about groups of 'area'-preserving diffeomorphisms on non-orientable manifolds. In this talk, I will discuss about the recent study about groups of density-preserving diffeomorphisms on non-orientable manifolds, ecspetially on the Möbius band. Here, the density is a natural concept that generalizes volume without concerning orientability. This talk is based on the joint work with S.Maruyama.

• Time and Location: Apr 11, 10:00-12:00, in Room 8406.

• Note.

• Title: Introduction to Categorical Entropy.

• Abstract: A discrete dynamical system consists of two components: a set X and a self-mapping f on X. If X admits a topological or metric structure and f respects this structure, an important dynamical invariant, called topological or metric entropy, can be defined. Similarly, if X possesses a richer mathematical structure and f preserves it, one can expect the existence of a corresponding notion of entropy. Recently, Kontsevich and his collaborators introduced the concept of categorical entropy. In this introductory talk, I will present the definition of categorical entropy and discuss its relationship with the usual notion of entropy, particularly in the context of symplectic topology. Especially, I will present a symplectic manifold X and an automorphism f on X such that the corresponding categorical entropy forces f to have positive topological entropy.

• Time and Location: Feb 26, 10:00-11:30, in Room 1423.

• Note.

• Title: Geometry and dynamics of diagonal shifts in the field of formal series.

• Abstract: In the first part of the talk, we explore the basic geometry of the space of lattices and periodic orbits under specific diagonal shifts in the context of positive characteristic formal series fields. The goal of the next part is to generalize this framework in two ways. First, we will consider higher-dimensional diagonal shifts, specifically examining the dynamics of torus orbits on PGL3. Second, we will generalize the spaces of lattices to more general discrete subgroup quotients, analyzing the corresponding geometric structures. Additionally, we will present some ongoing results related to this work and discuss several open problems that arise in this area. It may include some keywords such as buildings, projective frames, and multi-dimensional continued fractions.

• Time and Location: Feb 17, 10:00-12:00, in Room 1423.

• Notes will be available after the talk. 

• Title: Positive recurrence of a certain Gibbs measure on finite-area surfaces with pinched negative curvature.

• Abstract: Thermodynamic formalism in ergodic theory studies invariant measures that maximize the pressure of a given potential function on the phase space. The simplest and most important example is the measure of maximal entropy (MME). For the geodesic flow on manifolds with pinched negative curvature, a thermodynamic formalism enables the construction of a Gibbs measure for any Hölder continuous potential. When the Gibbs measure is finite, it defines a unique pressure-maximizing measure; otherwise, no measure of maximal pressure exists. Unfortunately, Gibbs measures are not always finite for non-compact manifolds, even for the measure of maximal entropy on finite-area surfaces. In this talk, I will focus on the Gibbs measure describing the asymptotic behavior of Brownian motion on a finite-area surface with pinched negative curvature and prove that it is always finite. 

• Time and Location: Jan 22, 10:00-12:00, in Room 1423.

• Note.

• Title: Equidistribution of Expanding translates of smooth curves on the homogeneous space with an action of a product of SO(n,1)’s.

• Abstract: Equidistribution of analytic or smooth curves has been explored in homogeneous spaces with an action of SL(n, \R), one copy of SO(n,1), and a product of SO(n,1)’s by Shah, P. Yang, and L. Yang.

For the case of a product of SO(n,1)’s( let this group be G), Lei Yang specified under what condition analytic curves in the expanding horospherical subgroup with respect to a diagonalizable multiplicative one parameter subgroup in G generate a collection of parametric measures whose limit measure is the unique G-invariant measure. We generalize this result into smooth curves and specify countable obstructions to equidistribution of smooth curves in a sense that if a given curve spends only measure 0 time in the union of obstructions, then it is equidistributed.

• Time and Location: Jan 7, 11:00-12:00, in Room 1423.