Kenny Ascher
Title: Wall crossing results for moduli spaces of higher dimensional algebraic varieties
Abstract: Brendan Hassett introduced the moduli space of weighted stable pointed curves — moduli spaces parametrizing nodal curves with smooth points weighted by a rational number between zero and one — and considered the birational morphisms obtained by varying the weights. It became apparent that this technique is fruitful in understanding the relationships among various compactifications of the same moduli space. After surveying this work, I will discuss generalizations of this "wall-crossing" picture to two classes of higher dimensional moduli spaces: moduli of varieties of log general type and K-moduli of log Fano varieties, as well as some applications of this technique towards understanding explicit compactifications.
Harold Blum
Title: Moduli of Fano varieties with complements
Abstract: While the theories of KSBA-stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of K-trivial varieties remains less well understood. I will discuss a new approach to this problem in the case of K-trivial pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all slc degenerations. In the case when X is a degeneration of P^2, this approach is successful. This is joint work with K. Ascher, D. Bejleri, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.
François Charles
Title: Formal-analytic arithmetic surfaces and finiteness results for fundamental groups
Abstract: I will introduce the notion of a formal-analytic arithmetic surface, obtained by gluing germs of surfaces along arithmetic curves together with a Riemann surface with boundary. I will explain how techniques from geometry of numbers in infinite rank may be used to study these objects and how, through arithmetic analogues of results of Nori, we may use them to prove various finiteness results for arithmetic surfaces and their fundamental groups. This is joint work with Jean-Benoît Bost.
Julia Gordon (slides)
Title: Uniform bounds for oscillatory integrals
Abstract: This talk will be largely about the recent work of R. Cluckers and I. Halupczok (and our completed long-term joint project), which gives a new way to use model theory to get uniform bounds for families of oscillatory integrals, both over the p-adic fields and over the reals. I will give an introduction to the relevant results from model theory (the so-called quantifier elimination and cell decomposition theorems which go back to the work of Tarski in the 1930s and Jan Denef in the 80s), and then focus on less known applications that lead to uniform bounds for various functions that arise in representation theory.
Inder Kaur
Title: Hodge conjecture for singular varieties
Abstract. In this talk I will discuss a cohomological version of the Hodge conjecture for singular varieties. I will give a sufficient condition in terms of Mumford-Tate groups for a variety to satisfy the singular Hodge conjecture. If time allows I will give explicit examples of such varieties. This is joint work with Ananyo Dan.
Lars Kühne
Title: The uniform Bogomolov conjecture for algebraic curves
Abstract: I will present an equidistribution result for families of (non-degenerate) subvarieties in a family of abelian varieties. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, implying in particular that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Furthermore, one can deduce a rather uniform version of the Mordell conjecture by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick–Brown). All these results have been generalized beyond curves in joint work with Ziyang Gao and Tangli Ge, but I will focus on the instructive case of curves.
Andrei Okounkov (slides from colloquium held on Friday, March 31)
Title: L-function genera
Abstract: In an ongoing joint work with David Kazhdan, we have been applying certain simple geometric ideas to determine the part of the L^2-spectrum of Hecke operators that comes from the Eisenstein series. Among the geometric constructions that play a role in this approach are certain genera built from L-functions. These look like something that people may apply usefully in other contexts and will try to explain what they are and how we use them.
Rekha Thomas (slides)
Title: Algebra and Geometry of the Pinhole Camera
Abstract: A basic task in computer vision is to reconstruct 3-dimensional scenes from (noisy) camera images.
This problem has deep roots in projective geometry and is naturally amenable to tools from algebraic geometry -- both modern and classical.
In this talk I will describe some recent results on the reconstruction problem for the case of two projective cameras.
The classical theory of cubic surfaces and their 27 lines make an unexpected appearance.