Apr 08, 2024
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


On generalized Beauville decompositions
Junliang Shen (Yale University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Over 30 years ago, a beautiful connection between derived categories of coherent sheaves and cohomology/motives was found for abelian schemes. More precisely, the work of Beauville and DeningerMurre endowed the cohomology of an abelian scheme a canonical (motivic) decomposition which splits the Leray filtration; this structure, now known as the Beauville decomposition, is induced by algebraic cycles obtained from the FourierMukai duality. In recent years, the study of Hitchin systems (e.g. the P=W conjecture), compactified Jacobians, and holomorphic symplectic varieties suggests that there should exist an extension of the theory of Beauville decompositions for certain abelian fibrations with singular fibers, where the Leray filtration should be replaced by the perverse filtration. I will discuss some recent progress in this direction. In particular, I will present results in both the positive and the negative directions, where Lagrangian fibrations associated with hyperKähler manifolds and tautological relations over the moduli of stable curves play crucial roles. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.
 Supplements



10:30 AM  11:30 AM


Break

 Location
 SLMath: Atrium, Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Decomposing Derived Categories
Alicia Lamarche (University of Utah)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
 
 Supplements



12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Categorical nonproperness and categorical formal punctured neighborhoods of infinity in wrapped Floer theory
Sheel Ganatra (University of Southern California)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
 
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Noncommutative Balmer spectra and structure of monoidal triangulated categories
Milen Yakimov (Northeastern University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Many settings in representation theory, algebraic geometry and mathematical physics lead to monoidal triangulated categories which are generally nonsymmetric. We will present a study of their noncommutative Balmer spectra, which includes a classification of all thick ideals and a generalized Carlson's connectedness theorem. The cohomological and Balmer support maps will be linked via a notion of categorical center of the cohomology ring of a monoidal triangulated category. This is a joint work with Dan Nakano and Kent Vashaw.
 Supplements




Apr 09, 2024
Tuesday

09:30 AM  10:30 AM


Categorical absorption of singularities
Alexander Kuznetsov (V. A. Steklov Institute of Mathematics)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Zoom Link
Resolution of singularities from the categorical point of view is an operation that replaces the derived category of a singular variety by a bigger smooth and proper triangulated category. Absorption of singularities, on a contrary, replaces the derived category of a singular variety by a smaller smooth and proper triangulated category. In the talk I will introduce Pinfinity objects and show how they can be used to absorb singularities of nodal varieties.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Moduli of complexes, shifted Poisson structure and positroid varieties
Zheng Hua (University of Hong Kong)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We establish a link between open positroid varieties and moduli space of complexes of vector bundles on Kodaira cycles via the shifted Poisson structure. Using this link we solve a classification problem of extension of vector bundles over Kodaira cycle. Based on this solution we further classify the symplectic leaves of all positroid varieties with respect to the standard Poisson structure.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Quasiflag manifolds and moment graphs
Yuri Berest (Cornell University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We will discuss a new class of topological Gspaces generalizing the classical flag manifolds of compact connected Lie groups. These spaces  which we call quasiflag manifolds  are topological realizations of (derived) schemes of quasiinvariants of finite reflection groups. Many fundamental properties and geometric structures related to the flag manifolds can be extended to quasiflag manifolds. In this talk, we will focus on categorical and homotopytheoretic aspects: in particular, we will describe a new universal `gluing' construction of quasiflag manifolds in terms of simplicially enriched moment graphs. This last construction is inspired by recent developments in abstract homotopy theory (Lurie's construction of the rigidification functor for simplicial sets and his straightening/unstraightening equivalence in the theory of infinitycategories). It applies to a broad class of spaces (known as GKM spaces) whose cohomological properties can be described combinatorially in terms of moment graphs. Time permitting, we will also discuss some applications in the context of representation theory and stable homotopy theory: in particular, we will look at topological analogues (spectral refinements) of basic algebraic properties of quasiinvariants such as the Gorenstein property. (Based on joint work with A. Ramadoss and Yun Liu).
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Derived microlocal geometry and virtual invariants
Tasuki Kinjo (Kyoto University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
We will introduce a derived geometric generalization of the microlocal sheaf theory, which gives a new perspective on virtual invariants for derived moduli spaces. As an application, we will construct a Hall algebra structure on the cohomological DonaldsonThomas invariants for the canonical bundle of smooth algebraic surfaces, which gives a 3d refinement of the 2d cohomological Hall algebra due to KapranovVasserot. This talk is based on a forthcoming joint work with Adeel Khan.
 Supplements



04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Apr 10, 2024
Wednesday

09:30 AM  10:30 AM


Finite approximations as a tool for studying triangulated categories
Amnon Neeman (Universita degli Studi di Milano)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll start with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric.
And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And some other ideas, mimicking constructions in real analysis, turn out to also be powerful.
And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories.
And what makes it all interesting is (3) applications. These turn out to include the proof of an old conjecture of Bondal and Van den Bergh about strong generation, a representability theorem that leads to a short, sweet proof of Serre's GAGA theorem, a proof of a conjecture by Antieau, Gepner and Heller about the nonexistence of bounded tstructures on the category of perfect complexes over a singular scheme, as well as (most recently) a vast generalization and major improvement on an old theorem of Rickard's.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Subcategories of weakly approximable triangulated categories and uniqueness of enhancements
ALBERTO CANONACO (Università di Pavia)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
A classical theorem by Rickard roughly states that the notion of derived equivalence for (noncommutative) rings is independent of the specific type of derived category: one can consider the unbounded derived category of all (left) modules, or any of its relevant triangulated subcategories (like the category of perfect complexes). It is natural to ask if Rickard's theorem can be generalized so as to include, for instance, also the geometric case where rings are replaced by (quasicompact and quasiseparated) schemes. It will be shown that a positive answer can be given using the theory of weakly approximable triangulated categories, in combination with some results about uniqueness of dg enhancements. This is joint work, partly in progress, with A. Neeman and P. Stellari.
 Supplements




Apr 11, 2024
Thursday

09:30 AM  10:30 AM


Perverse filtrations for Jacobian fibrations
Davesh Maulik (Massachusetts Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
In this talk, I will explain how to use the FourierMukai transform to study properties of the perverse filtration for certain Jacobian fibrations, specifically for moduli of Higgs bundles on a curve. This talk is a complement to Junliang Shen’s talk, and is joint work in progress with Junliang Shen and Qizheng Yin.
 Supplements



10:30 AM  10:35 AM


Group Photo

 Location
 SLMath: Front Courtyard
 Video


 Abstract
 
 Supplements



10:35 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Filtered derived categories of curved deformations
Wendy Lowen (Universiteit Antwerpen)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Zoom Link
The curvature problem in the deformation theory of dg algebras stems from the following observations: (1) According to the Hochschild complex, deformations of a dg algebra include \emph{curved} cdga’s and in general it is not possible to realise the full cohomology by means of dg Morita deformations (KellerLowen, 2009); (2) Since the differential of a cdga does not square to zero, there is no conventional derived category and second kind derived categories in the sense of Positselski may vanish for deformations (KellerLowenNicolás, 2009).
In this talk, we propose an altogether different approach to (2) by introducing the \emph{filtered derived category} of an nth order cdg deformation of a given dga A. This turns out to be a compactly generated triangulated category in which Positselski's semiderived category embeds admissibly. Further, it allows for a semiorthogonal decomposition into n+1 copies of D(A) and for smooth A it can be interpreted as a categorical resolution in the sense of Kuznetsov of the (in general) nonexistent classical derived category of the cdga. This is joint work with Alessandro Lehmann.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Classification of noncommutative Hirzebruch surfaces
Izuru Mori (Shizuoka University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We classify noncommutative Hirzebruch surfaces in terms of locally free sheaf bimodules of rank 2 over the projective line.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Blowing down noncommutative cubic surfaces
Shinnosuke Okawa (Osaka University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In 2001 Van den Bergh defined the notion of blowup for noncommutative surfaces and showed that the blowup of a noncommutative P2 in 6 points is isomorphic to a cubic surface inside a noncommutative P3. This yields a rational map from the moduli space of noncommutative P2s equipped with 6 points to a component of the moduli space of noncommutative P3s. In this talk I will show that the abstractly defined map as such turns out to be the composition of some concrete maps. This has various consequences, including the blowdown theorem. This is a joint work in progress with Ingalls, Sierra, and Van den Bergh.
 Supplements




Apr 12, 2024
Friday

09:15 AM  10:15 AM


Deformations of stability conditions
Emanuele Macri (Université ParisSaclay)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Bridgeland stability conditions have been introduced about 20 years ago, with motivations both from algebraic geometry, representation theory and physics. One of the fundamental problem is that we currently lack methods to construct and study such stability conditions in full generality.
In this talk I would present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari and Zhao. As 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.
 Supplements



10:15 AM  10:45 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:45 AM  11:45 AM


Noncommutative abelian surfaces and generalized Kummer varieties
Laura Pertusi (Universita degli Studi di Milano)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Examples of noncommutative K3 surfaces arise from semiorthogonal decompositions of the bounded derived category of certain Fano varieties. The most interesting cases are those of cubic fourfolds and GushelMukai varieties of even dimension. Using the deep theory of families of stability conditions, locally complete families of hyperkahler manifolds deformation equivalent to Hilbert schemes of points on a K3 surface have been constructed from moduli spaces of stable objects in these noncommutative K3 surfaces. On the other hand, an explicit description of a locally complete family of hyperkahler manifolds deformation equivalent to a generalized Kummer variety is not yet available.
In this talk we will construct families of noncommutative abelian surfaces as equivariant categories of the derived category of K3 surfaces which specialize to Kummer K3 surfaces. Then we will explain how to induce stability conditions on them and produce examples of locally complete families of hyperkahler manifolds of generalized Kummer deformation type. This is a joint work in progress with Arend Bayer, Alex Perry and Xiaolei Zhao.
 Supplements



11:45 AM  12:00 PM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



12:00 PM  01:00 PM


Nspherical functors, categorification of Euler's continuants and periodic semiorthogonal decompositions
Mikhail Kapranov (Kavli Institute for the Physics and Mathematics of the Universe )

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The concept of a spherical functor involves the cones of the unit and counit of the adjunction for a pair of adjoint functors. The 2term complexes formed by the unit and counit are simplest instances of more general complexes of functors which can be seen as categorical liftings of Euler continuants, the universal polynomials expressing the numerator and denominator of a continuous fraction in terms of its coefficients. Imposing natural conditions on the totalization of these complexes, we get a concept of an Nspherical functor (with the usual case corresponds to N=4). It turns out that such functors lead to Nperiodic semiorthogonal decompositions and provide a generalization of spherical twists. Joint work with T. Dyckerhoff, V. Schechtman.
 Supplements



