Weekly seminar links: 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.