Calendar

A celebrated theorem of C.L. Siegel from 1929 shows that the multiplicity of eigenvalues for the Laplace eigenfunctions on the unit disk is at most two.  We study the fourth order clamped plate problem, showing that the multiplicity of eigenvalues is uniformly bounded. Our method is based on new recursion formulas and Siegel–Shidlovskii theory. If time permits, we discuss possible applications also to nodal geometry. The talk is based on a joint work with Yuri Lvovsky.

Motivic complexes of Voevodsky have no right to two properties (1) cohomological sparsity and (2) a relationship with differential forms. This is, however, true p-adically over characteristic p by a result of Geisser-Levine, relying on previous results of Bloch-Kato-Gabber. I will explain this result, including the cast of characters involved like the logarithmic de Rham-Witt sheaves, Bloch’s higher Chow groups and algebraic/Milnor K-theory.

The talk will revolve around combinatorial aspects Prym varieties, a class of Abelian varieties that occurs in the presence of double covers. Pryms have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. As I will explain, problems concerning Pryms may be reduced, via tropical geometry, to combinatorial games on graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.

Sampling from a classical, thermal distribution is, in general, a computationally hard problem. In particular, standard Monte Carlo algorithms converge slowly close to a phase transition or in the presence of frustration. In this work, we explore whether a quantum computer can provide a speedup for problems of this type. The sampling problem can be reduced to the task of preparing a pure quantum state, the so-called Gibbs state [1]. Samples from the thermal distribution are obtained by performing projective measurements on this state. To prepare the Gibbs state, we exploit a mapping from a classical Monte Carlo algorithm to a quantum Hamiltonian whose ground state is the Gibbs state [2]. We demonstrate with concrete examples that a quantum speedup can be achieved by identifying optimal adiabatic trajectories in an extended parameter space of the quantum Hamiltonian. Our approach elucidates intimate connections between computational complexity and phase transitions. Finally, we propose a realistic implementation of the algorithm using Rydberg atoms suitable for near-term quantum devices.

[1] R. D. Somma and C. D. Batista, Phys. Rev. Lett. 99, 030603 (2007).
[2] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006).

In this talk, we discuss how to derive the equivariant SYZ mirror of toric manifolds by counting holomorphic discs. In the case of (semi-)Fano toric manifolds, those mirrors recover Givental’s equivariant mirrors, which compute the equivariant quantum cohomology. Also, we formulate and compute open Gromov-Witten invariants of singular SYZ fiber, which are closely related to the open Gromov-Witten invariants of Aganagic-Vafa branes. This talk is based on joint work with Hansol Hong, Siu-Cheong Lau, and Xiao Zheng.

–Organized by Professor Shing-Tung Yau

No additional detail for this event.

Every Newton polygon satisfying the Kottwitz conditions occurs on Shimura varieties of PEL-type in positive characteristic (Viehmann/Wedhorn). In most cases, it is not known whether these Newton polygon strata contain points representing the Jacobians of smooth curves. In some cases, this is not even known for the mu-ordinary stratum. We provide a positive answer for the mu-ordinary and almost mu-ordinary strata in infinitely many cases. For base cases, we consider the arithmetic of some of Moonen’s families of cyclic covers of the projective line. As an application, we produce infinitely many new examples of unusual Newton polygons which occur for Jacobians of smooth curves. This is joint work with Li, Mantovan, and Tang.

In game theory, we often use infinite models to represent “limit” settings, such as markets with a large number of agents or games with a long time horizon. Yet many game-theoretic models incorporate finiteness assumptions that, while introduced for simplicity, play a real role in the analysis. Here, we show how to extend key results from (finite) models of matching, games on graphs, and trading networks to infinite models by way of Logical Compactness, a core result from Propositional Logic. Using Compactness, we prove the existence of man-optimal stable matchings in infinite economies, as well as strategy-proofness of the man-optimal stable matching mechanism. We then use Compactness to eliminate the need for a finite start time in a dynamic matching model. Finally, we use Compactness to prove the existence of both Nash equilibria in infinite games on graphs and Walrasian equilibria in infinite trading networks.

No additional detail for this event.

No additional detail for this event.

will speak on: “Symplectic, or mirrorical, look at the Fargues-Fontaine curve”

Homological mirror symmetry describes Lagrangian Floer theory on a torus in terms of vector bundles on the Tate elliptic curve.  A version of Lekili and Perutz’s works “over Z[[t]]”, where t is the Novikov parameter.  I will review this story and describe a modified form of it, which is joint work with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on the torus. When the fiber of this sheaf of rings is perfectoid of characteristic p, and the holonomy around one of the circles in the torus is the pth power map, it is possible to specialize to t = 1, and the resulting theory there is described in terms of vector bundles on the equal-characteristic-version of the Fargues-Fontaine curve.

Tea at 4:00 pm – Math Common Room, 4th Floor

Talk at 4:30 pm – Hall A

Jang’s equation is a degenerate elliptic differential equation which plays an important role in the positive mass theorem. In this talk, we describe a high order WENO (Weighted Essentially Non-Oscillatory) scheme for the Jang’s equation. Some special solutions will be shown, such as those possessing spherical symmetry and axial symmetry.