2025

I will present details about an ongoing joint work, where we intend to prove derived equivalence between two non-birational Calabi-Yau 3-folds. I will present a summary of the status of classification of CY3s and mirror symmetric motivations that prompt us to specifically study these two 3-folds. I will start with some classical projective geometry where we construct the titular 3-folds, following which I will move on to toric geometry where we showcase preliminary evidence for their derived equivalence. I will end with a discussion of GLSMs which is a relatively modern technique in the landscape of derived categories, which we hope will lead us to our desired proof.

The derived category of a Fano variety will often have a semiorthogonal decomposition consisting of a sequence of exceptional line bundles and a remaining subcategory, which we will call the Kuznetsov category. The categorical Torelli problem asks whether knowing this category up to exact equivalence is enough to determine the variety up to isomorphism. We will explain how a technique based on a computation of the Hochschild-Serre algebra of the category shows that the answer is yes in many cases, including many weighted hypersurface and a Veronese double cone. This is joint work with Xun Lin and Shizhuo Zhang.

Virasoro constraints were recently formulated for moduli of sheaves. The main difference compared to Gromov-Witten theory lies in the linearity of the former which allowed us (B.-Lim-Moreira) to derive sheaf-theoretic constraints from the geometric vertex algebras of Joyce. Our approach applies to more general linear categories. For both sheaves and representations of quivers, I will explain that Virasoro costraints determine primary states of the vertex algebra. This is already sufficient to illustrate the formulation for any linear theory, and it leads to the proof of many results via wall-crossing.

 

In this talk, I will present a joint work with Matilde Manzaroli. I will introduce real tori (in the greater generality) as well as real equivariant torus embeddings. Then I will describe the structure of the torus orbits. It will allow us to easily compute the Betti numbers of the real loci of smooth real equivariant torus embeddings, provided that those real loci are compact. Then I will present a canonical "foliation" of their real loci by equivariant embeddings of split tori. I will express a necessary and sufficient condition for this foliation to be given by the fibre of a fibre bundle, and explain how to reduce to this case by performing toric blow-ups.

There are several approaches to studying configuration spaces of points on curves. Examples are the Deligne-Mumford and Hassett spaces or, in the case of points on a projective line, a straightforward GIT construction. Deligne and Mostow further investigated the situation where moduli spaces of weighted points on the projective line have ball quotient models. This leads to a finite set of configuration spaces of ordered or partially ordered points on P^1. In all of these cases, the GIT compactification and the Baily-Borel compactification of the ball quotient are isomorphic. These spaces have natural (partial) desingularizations, namely the Kirwan blow-up and the toroidal compactification. It follows from results of Gallardo-Kerr-Schaffler and Kiem-Moon that, in the case of ordered configurations, the Deligne-Mostow map lifts to an  isomorphism of these compactifications. This can fail for only partially ordered configurations. The main result of this talk is to explain a criterion exactly when these two compactifications coincide. This is also related to the Log Minimal Model Program. This is joint work with Yota Maeda.   

I will explain how to use the motivic volume formula and toric degenerations to study the rationality problem for complete intersections in Grassmannians, in particular Gr(2,5).

We study rationality questions for the Fano schemes of non-maximal linear spaces on a smooth complete intersection X of two quadrics, especially over non-closed fields. We start by showing that they are all geometrically rational. We then ask their rationality over k and analyze in details the case of second maximal linear spaces. In particular, we generalize results of Hassett-Tschinkel and Benoist-Wittenberg when X has odd dimension, and extend work of Hassett-Kollár-Tschinkel when X has even dimension and k = R. This is joint work with Lena Ji.

Based on the preprint Bhatt, Iyengar, Ma - Regular rings and perfectoid algebras. Notes for the lecture here.

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results. Joint work with Zaïneb Bel-Afia and Chiara Meroni.