My research interests are in the intersection of three different
areas: symplectic geometry, complex geometry, and representations of
infinite-dimensional Lie algebras. The starting point of my research
is the so-called Witten's conjecture proved by Kontsevich. It says
that the full generating function of intersection numbers on the
Delign--Mumford compactificatio of the moduli spaces
ℳ_{g,n} of Riemann surfaces is a tau-function of the KdV
hierarchy. It is well known (M. Sato, 1981) that the tau-functions of
the KdV hierarchy can be described in terms of the so-called Hirota
bilinear equations. Equivalently, the coefficients of the
tau-functions satisfy an infinite system of quadratic equations which
can be interpreted as the Plücker relations of the embedding of
an infinite Grassmannian in a projective space. Following Givental we
refer to this system of quadratic equations as Hirota quadratic
equations (HQEs). It turns out that the HQEs of KdV can be interpreted
as the regularity condition of a certain bilinear operator acting on
the tensor square of the tau-function. The key observation, due to
Givental (in 2003), is that the bilinear operator is defined in terms
of vertex operators whose coefficients are periods of the
A_{1} singularity (Givental, 2003). The project that I started
working on was suggested by Givental. He asked me to investigate
whether we can construct HQEs more generally in Gromov--Witten theory
of a smooth projective variety X by using the period integrals
associated with the mirror model of X. After the work of Dubrovin and
Zhang, it became clear that we have to work with targets X, such that,
their quantum cohomology is semi-simple. There are 3 major
developments that happened since 2003. First, in a joint work (in
2013) with
Bojko Bakalov, we discovered that the vertex operators with
coefficients period integrals define a representation of the lattice
vertex algebra associated to the Milnor lattice -- the lattice of
vanishing cycles or corresponding Lefschetz thimbels. Second, Hiroshi
Iritani found (in 2009) a remarkable conjecture about the Milnor lattice of the
mirror model in terms of the topological K-ring of X. Finally,
partially motivated by my joint work with Givental and Frenkel (in
2010), I was able to find certain connection formulas for the Operator
Product Expansion (OPE) of the vertex operators. The project of
constructing HQEs in GW theory of a smooth projective variety X
can be formulated now in the language of the lattice vertex algebras
associated with the lattice underlying the topological K-ring of
X. Namely, one has to construct a state in the lattice vertex algebra
satisfying certain screening equations. So far, the targets for which
the construction was worked out are the Fano orbifold lines
P_{a,b,c}.

The above paragraph describes the main focus of my research. During the years I got involved in various related projects, such as, the topological recursion of Eynard and Orantin, matrix models, mirror symmetry for manifolds with semi-simple quantum cohomology, and K-theoretic GW theory.

My work is available on the arXiv.

The journal references are available in my Publication List.

1. Chenghan Zha

Master Thesis: The period map of a two-dimensional semi-simple Frobenius manifold.

PhD Thesis: Integral structures in the local algebra of a singularity.

2. Xiaokun Xia

Master Thesis: Gamma integral structure for the blowup of P

1. Gromov--Witten theory and representations of affine Lie algebras. Lectures that I gave in 2007 in Stanford.

2. Gromov--Witten theory and integrable hierarchies. Lectures that I gave in Beijing 2009.

3. IPMU seminar on Bolibrukh's counter example for the Riemann--Hilbert problem.

4. Primitive forms and vertex operators. The first draft of a book.