The talks are held on every Friday 5-6 PM on Zoom. Meeting ID TBA.
Friday, October 9: Ákos Nagy
Title: The asymptotic geometry of -monopoles
Abstract: -monopoles are special Yang-Mills-Higgs configurations on -manifolds. Donaldson and Segal conjecture that one can construct invariants of noncompact -manifolds by counting -monopoles. The first steps in achieving this goal is understanding the analytic properties of these monopoles, such as asymptotic and decay properties. In this talk, I introduce the proper analytic setup for the problem, and present our results about the asymptotic forms of -monopoles, with structure group being , on Asymptotically Conical -manifolds. This is a joint project with Daniel Fadel and Gonçalo Oliveira.
Friday, October 16: Jiayin Pan
Title: Nonnegative Ricci curvature and escape rate gap
Abstract: Let be an open -manifold of nonnegative Ricci curvature and let . We show that if has escape rate less than some positive constant , that is, minimal representing geodesic loops of escape from any bounded balls at a small linear rate with respect to their lengths, then is virtually abelian.
Friday, October 23: Fedya Manin
Title: Filling random cycles
Abstract: I will explain an average-case isoperimetric inequality for certain combinatorial models of random cycles in cubes and spheres. For example, take a knot built by connecting a sequence of uniformly random points in the unit cube (a model introduced by Ken Millett). The minimal area of a Seifert surface for this knot is on the order of with high probability. For all the models analyzed, this growth rate is the same, depending only on the codimension of the cycles; I suspect that Fourier analysis can be used to explain this apparent coincidence.
Friday, October 30: Yeping Zhang
Title: Quillen metric, BCOV invariant and motivic integration
Abstract: Bershadsky, Cecotti, Ooguri and Vafa constructed a real valued invariant for Calabi-Yau manifolds, which is now called the BCOV invariant. The BCOV invariant is conjecturally related to the Gromov-Witten theory via mirror symmetry. In this talk, we prove the conjecture that birational Calabi-Yau manifolds have the same BCOV invariant. We also build an analogue between the BCOV invariant and the motivic integration. The result presented in this talk is a joint work with Lie Fu.
Friday, November 6: Siqi He
Title: The Behavior of Solutions to the Hitchin-Simpson Equations
Abstract: The Hitchin-Simpson equations defined over a Kähler manifold are first order, non-linear equations for a pair of connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchin-Simpson equations with norms of these 1-forms unbounded. In addition, we will discuss the relationship between this behavior with the Taubes’ Z2 harmonic spinor and Hitchin’s WKB problem.
Friday, November 13: Junrong Yan
Friday, November 20: Qin Deng