Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will focus on the case where X is a threefold, as there are several open questions regarding its singularities. I will describe the structure of the smooth points of this Hilbert scheme and, time permitting, discuss the structure of the mildly singular points. This is all joint (ongoing) work with Joachim Jelisiejew and Alessio Sammartano.
Abstract: I will present a convergent algorithm for the computation of mean curvature flow of surfaces with fixed boundaries. Our analysis hinges upon the one recently developed by Kovács, Li, Lubich, and collaborators for closed surfaces, which in turn use Huisken's equation for the evolution of the mean curvature and the normal vector. We extend their ideas to surfaces with boundaries by formulating appropriate boundary conditions for both the mean curvature and the normal vector. These boundary treatments are essential for the well posedness of the discretization and for proving convergence. To effectively handle boundary conditions for the normal vector, we introduce a nonlinear Ritz projection into the analysis. We prove that this projection is well posed and achieves optimal approximation orders. As a result, we derive optimal $H^1$ error estimates for the surface position, velocity, mean curvature, and normal vector. Towards the end I will present some numerical experiments which illustrate the behavior of the convergent algorithm. This is joint work with Bárbara S. Ivaniszyn and M. Sebastián Pauletti.
Abstract: I will report on forthcoming work, joint with Filipazzi, Greer, Mauri, and Svaldi, on boundedness results for abelian fibrations. We will discuss a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded. Conditional on generalized semiampleness conjectures, this bounds the number of deformation classes of hyperkahler varieties in a fixed dimension, whose second Betti number is at least 5.
Abstract: One aspect of the Langlands programme is to isolate cuspidal representations "distinguished" by some subgroup. Such representations are conjecturally obtained through some transfer from some form of the subgroup. This distinction is defined in terms of nonvanishing of "period integrals".
The Gan-Gross-Prasad conjecture relates special values of certain L-functions to period integrals on classical groups. Jacquet and Rallis proposed an approach through the relative trace formula in the form of a fundamental lemma and a transfer conjecture, which were largely proven in the work of Yun and Zhang respectively.
We will introduce the problem of distinction and periods and the arithmetic Gan-Gross-Prasad setting, which aims to generalize Gross-Zagier formula, relating Neron-Tate heights of Heegner points on modular curves, to special values of derivatives of certain L-functions. Smithling-Rapoport-Zhang have proven arithmetic transfer in one specific example of ramified unitary groups, but did not construct said transfer explicitly. In this talk, I will be introducing a new conjecture aiming to realize the arithmetic transfer explicitly in all ramified cases. This is joint work with Wei Zhang.
4176 Campus Drive - William E. Kirwan Hall
College Park, MD 20742-4015
P: 301.405.5047 | F: 301.314.0827