** Title: ** On a theory of generalized Jacobi forms and its applications in enumerative theories

** Abstract: ** Motivated by the studies on the Gromov-Witten theory of elliptic curves,
I will explain a differential-geometric approach in contructing a theory of quasi-Jacobi forms.
This theory is then applied to study the Landau-Ginzburg/Calabi-Yau correspondence, which relates
the Fan-Jarvis-Ruan-Witten theory of certain singularities to the Gromov-Witten theroy, for the case of elliptic curves.
Moreover, it explains the holomorphic anomaly phenomenon in the generating series of the Gromov-Witten theory. If time permits,
I will also mention some applications of this theory in the studies of Gromov-Witten/Hurwitz correspondence for elliptic curves,
and geometric quantization.

** Title: ** Hurwitz numbers and the cut-and-join equation

** Abstract: **
Hurwitz numbers are classical objects in enumerative geometry. In this introductory talk,
I will review the geometric and algebraic definitions of Hurwitz numbers, and explain why these two definitions are equivalent.
Then I will introduce the generating function of Hurwitz numbers using symmetric functions,
which turns out to be a solution of the cut-and-join equation. Its link with integrable hierarchies via Boson-Fermion correspondence will also be gently addressed.

** Title: ** The minimal equal length of a pair of simple closed curves in a once punctured torus as the torus runs over its relative Teichmuller space

** Abstract: ** We consider the problem of minimizing the equal length of a pair of simple closed geodesics of given topological type in a once punctured
hyperbolic torus with fixed geometric boundary data as the torus runs over its relative Teichmuller space. For specific pairs with symmetry,
we are able to determine the minimizing torus and hence the minimal length. It is natural to compare the minimal lengths for inequivalent pair
of the same intersection number. As computer experiments show, there is a conjecture that the specific pair of slopes (1/0, 1/n) has its minimal
length smaller than any other pair of slopes (1/0, m/n),
regardless of the geometric boundary data. In joint work with Da Lei, we are able to establish a stronger result as the geometric boundary
of the torus is a conic point and the cone angle is approaching 2π.

** Title: ** On the Biconfluent Heun and its connection with Painleve IV equation

** Abstract: **
In this talk, we first apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions of certain Hill equation are
Liouvillian solutions. That are solutions obtainable by suitable differential field extensions construction. We establish a full correspondence between
solutions of non-oscillatory type equations and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained for this
Hill equation whose potential owns four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation.
We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions
in complex plane that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lame and Whittaker-Hill equations, discovered
by Whittaker and Ince almost a century ago. In the second part, we discuss special solutions of Painleve IV equation, that comes from the pioneering works of
Okamoto and Noumi. We report that the linear equation that gives raise to $P_{IV}$ via isomonodromy deformation in the classical works of Garnier and
Jimbo-Miwa also possesses special properties with the same parameter space as the $P_{IV}$. This is a joint work with Yik-Man Chiang and Chun-Kong Law.

** Title: ** Frobenius manifolds and root systems

** Abstract: ** Frobenius manifolds are devised to give a coordinate-free way to describe equations of associativity (or WDVV equations)
arising from 2D topological field theory. One such manifold can be viewed as a space parametrizing a family of Frobenius algebras,
endowed with some additional conditions. In this talk I will construct Frobenius structures on the \mathbb{C}^{\times}-bundle
of the complement of a toric arrangement associated with a root system, by making use of a family of torsion free and flat connections on it.
This gives rise to a trigonometric version of Frobenius algebras in terms of root systems.

** Title: ** Holomorphic Chern-Simons theory at large N

** Abstract: ** We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity.
We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model.
At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type
I version of Kodaira-Spencer theory in complex
dimension 5 and show that it can only be coupled to holomorphic Chern-Simons theory with gauge group SO(32) at quantum level.
This coupled system is conjectured to be a supersymmetric localization of type I string theory.

** Title: ** Singularities: from L^2 Hodge theory to Seiberg-Witten geometry

** Abstract: ** Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus, satisfying a general asymptotic condition.
We establish a version of twisted L^2 Hodge theory for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property.
It can be viewed as a generalization of Kyoji Saito's higher residue theory and primitive forms for isolated singularities.
In the second part of the talk, I will explain a connection between primitive period maps and 4d N=2 Seiberg-Witten geometry.

** Title: ** Geometric and algebraic approach to the notion of F-algebroids

** Abstract: ** Motivated by Dubrovin's duality in Frobenius manifolds we introduce the notion of F-algebroids
as a link between the notion of F-manifolds (defined by Hertling-Manin) and the notion of Lie algebroids.
In this talk we will describe the geometric structure of F-algebroids
and discuss an algebraic approach of these objects as a generalization of Lie-Rinehart pairs of algebras.
This is based on joint works with A. Torres-Gomez and J. Gutierrez.

** Title: ** Quantum cohomology for isotropic Grassmannians

** Abstract: ** We will discuss the big quantum cohomology ring of isotropic Grassmannians IG(2,2n). After introducing the basic notions we will show that these rings are regular.
In particular, by “generic smoothness”, we will give a conceptual proof of generic semisimplicity of the big quantum cohomology for these Grassmannians.
We will also relate certain decomposition of the ring with a exceptional collection of the derived category of IG(2,2n).
This is based on joint work with A. Mellit, A. Kuznetsov, N. Perrin and M. Smirnov.

- Dafeng Zuo, Di Yang