Weekly seminar links: Mathematical Physics

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Mathematical Physics

The research interests of the faculty in
mathematical physics include quantum singularity theory, Gromov-Witten theory,
Seiberg-Witten invariants, and Lie groups.

**1. ****Quantum Singularity Theory and Gromov-Witten
Theory (FAN Huijun, GUO Shuai)**

Huijun Fan et al. [Fan-Jarvis-Ruan, *Ann. of Math. (2)* 2013] constructed the
quantum singularity theory (now called FJRW theory) as an analogue of the
Gromov-Witten theory, associating a cohomological field theory to each pair of
a nondegenerate quasi-homogeneous polynomial and an Abelian group of symmetries.
It is related to the Saito-Givental theory and the Gromov-Witten theory via the
Landau-Ginzburg mirror symmetry and the Landau-Ginzburg/Calabi-Yau
correspondence, respectively. In the same paper, they also proved two
conjectures of Edward Witten, stating that ADE-singularities are self-dual and
that the total potential functions of ADE- singularities satisfy corresponding
ADE-integrable hierarchies.

In addition, Shuai Guo et al. [Guo-Zhou, *Adv. Math. *2015] proved the KKV
conjecture for Gromov-Witten invariants and Gopakumar-Vafa invariants, and gave
a method to calculate the invariants which do not satisfy the positivity
condition.

**2.
Seiberg-Witten Invariants (DAI Bo)**

By using the wall-crossing formula for
Seiberg-Witten invariants, Bo Dai et al. [Dai-Ho-Li, *J. Topol.* 2016] gave lower bounds for the minimal genus of an
embedded surface representing a fixed cohomology class in a four dimensional
compact manifold with $b_2^+=1$. The bounds depend only on the structure of the
cohomology ring and are optimal in certain cases. This is the most general
result for the minimal genus problem on four dimensional manifolds with $b_2^+=1$
and without symplectic structures.

**3. Lie Groups and Applications (AN Jinpeng)**

Jinpeng An et al. [An-Yu-Yu, *J. Differential Geom.* 2013] solved two
Lie group problems raised by Robert Langlands, and applied their results to
spectral geometry, proving that there exist pairs of simply connected closed
Riemannian manifolds which are isospectral and non-homeomorphic. A part of the
work motivated James Arthur to raise new problems in the Langlands Program.

Jinpeng An investigated Diophantine
approximation problems related to homogeneous dynamics on SL(3,R)/SL(3,Z). In
particular, he proved in [An, *Duke Math.
J.* 2016] that two-dimensional weighted badly approximable vectors form a
winning set for Schmidt's game, which gives a new proof of Schmidt's conjecture
and implies a stronger result.

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