Abstract: Geodesics are central objects for understanding the geometry of surfaces. In particular, the existence of closed embedded geodesics has long been a fundamental question in differential geometry. A breakthrough in establishing the best possible existence result on a two-sphere came through the use of geometric flows, nonlinear partial differential equations that deform submanifolds in a canonical way. In this talk, I will give an overview of this method and highlight how analysis, geometry, and topology come together in the proof.

Abstract: An abelian fibration is a (1-dimensional) proper family of algebraic varieties whose fibers are generally an abelian variety. A singular fiber of an abelian fibration can be thought of as a degeneration (limit) of abelian varieties. In this talk, I will talk about some techniques for studying abelian fibrations and their singular fibers. Time permitting, I will talk about my recent result of classifying singular fibers arising in some special abelian fibrations.

Abstract: In arithmetic geometry, one looks for algebraic varieties defined over Q-bar, the algebraic closure of the field of rational numbers Q. Magically, sometimes one can show that a complex manifold defined in a certain nice way is algebraic and is further cut out by polynomials with coefficients only in Q-bar. The most prominent example is a modular curve, a Riemann surface arising as the quotient of the upper half plane by an "arithmetically defined" subgroup of PSL_2(R). In this talk, I will exhibit four different ways to show some complex manifolds are defined over Q-bar: 1. by using modular forms; 2. by identifying with moduli spaces; 3. by rigidity; 4. by using non-linear differential equations.

Abstract: Quantum differential equation, or Dubrovin connection, is a key object in the theory of quantum cohomology. Usually the solution for the quantum differential equation gives the generating functions for genus 0 curve counting in a smooth variety X, and the monodromy representation for such quantum differential equation is the key to understanding the structure of the differential equation. In this talk, we will focus on the equivariant version, with X being quiver varities, like T^*P^n or T^*Gr(k,n) etc. In fact, we are going to show that the monodromy of the quantum differential equation can be described as the element in the quantum affine algebras of the corresponding quiver type. In the end, I will give the example of the qde for the Hilbert scheme of C^2 or the A_r-surfaces, and explain that why its monodromy can be described by the universal elements in the quantum toroidal algebras.