Tag - Algebraic geometry

Emanuele Macrì: Deformations of t-structures

14th November, 2024

Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

Jacob Lurie: Rationalized Syntomic Cohomology

10th April, 2024

A few years ago, Bhatt-Morrow-Scholze introduced an invariant of p-adic formal schemes called syntomic cohomology, which has a close relationship to (étale-localized) algebraic K-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed that, after inverting p, syntomic cohomology admits a concrete description in terms of more familiar invariants, such as de Rham and crystalline cohomology. In this talk, I'll explain an alternative perspective on their result, which avoids the use of K-theoretic methods.

Sally Gilles: The v-Picard Group of Stein Spaces

3rd April, 2024

In this talk, I will present a computation of the image of the Hodge-Tate logarithm map (defined by Heuer) in the case of smooth Stein varieties. When the variety is the affine space, Heuer has proved that this image is equal to the group of closed differential forms. In general, we will see that the image always contains such forms but the quotient can be non-trivial: it contains a ℤp-module that maps, via the Bloch-Kato exponential map, to integral classes in the proétale cohomology.

Pravesh Kothari: Sum-of-Squares Proofs, Efficient Algorithms, and Applications

18th March, 2024

Any non-negative univariate polynomial over the reals can be written as a sum of squares. This gives a simple-to-verify certificate of non-negativity of the polynomial. Rooted in Hilbert's 17th problem, there's now more than a century's work that finds the right multivariate generalizations called Positivstellensatz theorems (due to Krivine, Stengle, and Putinar). Beginning in the late 1980s, researchers (initially independent) in optimization, quantum information, and proof complexity theory found an algorithmic counterpart, the sum-of-squares algorithm, of these results. Over the past decade, this algorithmic theory has matured into a powerful tool for designing efficient algorithms for basic problems in algorithm design.

In this talk, I will outline a couple of highlights from these recent developments:

1) Algorithmic Robust Statistics: In the 1960s, Tukey and Huber observed that most statistical estimators are brittle -- they break down with almost no guarantee if the model postulated for data has minor misspecification (say because of 1% outliers). In response, they initiated the field of robust statistics. Over the past five years, a new blueprint, based on the sum-of-squares algorithm, has emerged for efficient robust statistics in high dimensions with new connections to finding efficiently verifiable certificates of concentration and anti-concentration properties of high dimensional probability distributions.

2) The Kikuchi Matrix Method: Finding (or proving that there is none) a solution that satisfies 99% of a given system of k-sparse linear equations (i.e., k non-zero coefficients in each equation) over finite fields is a basic NP-hard problem and thus, unlikely to admit efficient algorithms. In the early 2000s, motivated by whether the hard instances are "pathological", researchers explored whether "semirandom" equations - arbitrary systems with right-hand sides generated uniformly and independently at random - could admit efficient algorithms that output efficiently verifiable certificates of unsatisfiability.

Recently, a restricted class of sum-of-squares proofs was at the heart of efficient algorithms for such semirandom sparse linear equations. Surprisingly, these algorithms have led to new progress in extremal combinatorics and coding theory.

Tess Bouis: Motivic Cohomology of Mixed Characteristic Schemes

14th February, 2024

I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-A1-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after A1-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.

Leonid Positselski: Semi-infinite algebraic geometry of quasi-coherent torsion sheaves

8th February, 2024

This talk is based on the book Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes—quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. I will start with some examples serving as special cases of the general theory, such as the tensor structure on the category of unbounded complexes of injective quasi-coherent sheaves on a Noetherian scheme with a dualizing complex. Then I will proceed to explain the setting of a flat affine morphism of ind-schemes into an ind-Noetherian ind-scheme with a dualizing complex, and the main ingredient concepts of quasi-coherent torsion sheaves, pro-quasi-coherent pro-sheaves, and the semiderived category. In the end, I will spell out the construction of the semi-tensor product operation on the semi-derived category of quasi-coherent torsion sheaves, making it a tensor triangulated category.

Shubhodip Mondal: Dieudonné Theory via Prismatic F-gauges

7th February, 2024

In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using classifying stacks, one can define Dieudonné module of a finite locally free group scheme as a prismatic F-gauge (prismatic F-gauges have been recently introduced by Drinfeld and Bhatt-Lurie), which gives a fully faithful functor from finite locally free group schemes over a quasi-syntomic algebra to the category of prismatic F-gauges. This can be seen as a generalization of the work of Anschütz-Le Bras on "prismatic Dieudonne theory" to torsion situations.

Shizhang Li: On Cohomology of BG

29th November, 2023

Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a p-power order commutative group scheme over a perfect field of characteristic p, in terms of its Dieudonné module.

Arend Bayer: Non-commutative abelian surfaces and generalized Kummer varieties

22nd November, 2023

Polarised abelian surfaces vary in 3-dimensional families. In contrast, the derived category of an abelian surface A has a 6-dimensional space of deformations; moreover, based on general principles, one should expect to get 'algebraic families' of their categories over 4-dimensional bases. Generalized Kummer varieties (GKV) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have 4-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over 3-dimensional subvarieties.

I present a construction that addresses both issues. We construct 4-dimensional families of categories that are deformations of Db(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from ℤ/2-actions on derived categories of K3 surfaces.

Juan Esteban Rodriguez Camargo: The Analytic de Rham Stack

8th November, 2023

In this talk, we introduce the analytic de Rham stack for rigid varieties over ℚp (and more general analytic stacks). This object is an analytic incarnation of the (algebraic) de Rham stack of Simpson, and encodes a theory of analytic D-modules extending the theory of D̂-modules of Ardakov and Wadsley. We mention how a very general six functor formalism can be construct in this set up, as well as other features such as Kashiwara equivalence and Poincaré duality for smooth maps.