פעילויות השבוע

In a seminal series of works, culminating in the monumental ”What Determines an Algebraic Variety?“ János Kollár, Max Lieblich, Martin Olsson, and Will Sawin prove that a normal projective algebraic variety of dimension at least 2 over an uncountable field of characteristic 0 can be reconstructed, in a precise sense, solely from its underlying topological space. The results of KLOS are specific to char. 0 and to normal varieties. Castle and O‘Gorman, using the model theoretic machinery of Zilber‘s Restricted Trichotomy, extend these results to all quasi projective varieties (of dimension at least 2) in all characteristics, in the case where the underlying field is algebraically closed and uncountable. In the talk, I will present the results and try to sketch the strategy of proof of the new result.

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak says that the L^2 mass of eigenfunctions of the Laplacian in hyperbolic manifolds equidistributes, as the eigenvalues tend to infinity. We consider a special class of such functions, Hecke—Maass forms, that are central in number theory. The conjecture has been established for these functions in dimension 2 and 3, but in dimension 4 there is a new challenge: one needs to rule out concentration of measure along certain large totally geodesic submanifolds. We will discuss our recent result in which we overcome this difficulty for a particular sequence of eigenfunctions known in number theory as lifts. This is a joint work with Alexandre de Faveri.

TBA

The Keller cylinder DG ring encodes homotopies between DG ring homomorphisms f_0, f_1 : A \to B.

Recently we discovered the higher cylinder DG rings Cyl_q(B), which assemble into the simplicial cylinder DG ring Cyl(B). For q=1 this recovers Keller‘s original construction.

The sets SHom_q(A,B) of DG ring homomorphisms A \to Cyl_q(B) form the simplicial Hom set SHom(A,B). Our main result is that when A is a semi-free DG ring, the simplicial set SHom(A,B) is a Kan complex.

We prove several results about the fundamental groupoid SHom_{\leq 1}(A,B), including invariance under quasi-isomorphism B‘ \to B, and that the automorphism groups are abelian. We also indicate some applications of this work.

Typed notes: https://drive.google.com/file/d/1sMzwoC_DGCotOfak8o8wYpmttgZELf6l/view

arXiv eprint: https://arxiv.org/abs/2602.11943