Workshop
“DG-enhancements and higher category methods”

We propose a geometric setting leading to categorical braid group actions.

First we consider the quasi-coherent Hecke category QCHecke(G,B) for a reductive group G with a Borel subgroup B. We show that a monoidal action of QCHecke(G,B) on a triangulated category gives rise to a categorification of degenerate Hecke algebra representation known as Demazure Descent Data.

Next we replace the group G by the derived group scheme LG of topological loops with values in G and consider QCHecke(LG,LB). A monoidal action of the category QCHecke(LG,LB) gives rise to a categorical action of the affine Braid group.

Finally we present an example of the construction above coming from a category of equivariant matrix factorizations.

I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry. Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in these various areas of math and how it is characterised categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces. The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans. In fact, we work with the larger category of Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of analytic geometry and to compare with the standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions), the theory of Stein domains and others. If time remains I will explain how this extends to a formulation of derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.

This is joint work with Federico Bambozzi (Regensburg) and Kobi Kremnizer (Oxford).

Let g be a finite-dimensional reductive Lie algebra defined over a field k of characteristic zero. The set of all representations in g of a Lie algebra A has a natural structure of an affine k-scheme, called the representation scheme . In this talk, we will discuss a derived version of this geometric construction obtained by extending the representation functor . to the category of DG Lie algebras and deriving it in the sense of Quillen's homotopical algebra. The corresponding derived scheme . is represented by a commutative DG algebra whose homology depends only on A (and g) and called the representation homology of A in g.

We construct a canonical DG algebra map: relating the G-invariant part of representation homology of A in g to the W-invariant part of representation homology of A in a Cartan subalgebra of g. We call this map the derived Harish-Chandra homomorphism as it is a natural derived extension of the classical restriction map.

We conjecture that, for a two-dimensional abelian Lie algebra A, the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide an evidence for this conjecture, including proofs in several special cases. For a complex semisimple Lie algebra g, we compute the (weighted) Euler characteristic of . in terms of matrix integrals over a compact real form of G and compare it to the Euler characteristic of . This yields a remarkable constant term identity, which is analogous to the famous Macdonald identity for g. We explain this analogy by giving a new homological interpretation of the Macdonald identities in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

(This is joint work with G. Felder, A. Ramadoss, A. Patotski and T. Willwacher)

We construct dg-enhancements for categories of representations of semi-simple real Lie groups and the geometric categories that give rise to them.

Candidate categories of spectral noncommutative motives are enriched over . As a consequence, understanding and its ring structure sheds light on structural properties of these categories of motives. Moreover, Morava has recently speculated that these categories can be studied using a Tannakian formalism involving the co-ring (using the trace map ). This talk is about recent work with Mike Mandell which describes shows that the multiplication is nilpotent in positive degrees.

I describe how generalized matrix algebras and homotopes appear in different ways in the study of quasi-hereditary algebras. I show that for an arbitrary quasi-hereditary algebra they provide a description of the endomorphism algebra of irreducible projective objects. By means of homotopes I construct also a family of quasi-hereditary algebras numbered by Fibonacci sequences and assigned to birational morphisms of smooth surfaces.

We discuss the dg-gluing under suitable conditions on the cohomology of the gluing functor. We explain how the homotope construction appears in the problem of describing the glued abelian category of recollement and show how it works on some examples in geometry and representation theory.

I present an attempt (not yet successful, but hopefully illuminating) to give an explicit formula for the 2-shifted symplectic form on the derived stack of perfect complexes. The existence of this form was proved by Toën and Vezzosi, but their existence proof invokes the cobordism hypothesis, and thus their proof is essentially by (extremely difficult) obstruction theory.

This talk considers different incarnations of infinity local systems. It turns out these can be studied as categorifications of the cohomology of a topological space X, obtained by taking coefficients in the model category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent to each other and to homotopy locally constant sheaves on X and to representations of chains on the based loop space of X. We also give an explicit combinatorial description in terms of Maurer-Cartan elements.

I will present a refinement of the standard blow up of an ideal which has better categorical properties. In particular, it provides a step of a categorical resolution of singularities. This is a joint work with Dmitry Kaledin.

We give an interpretation of certain Maurer-Cartan elements in the formal Hochschild complex of a small -category in terms of torsion Morita deformations. For a broad class of -categories, and for formal deformations, this yields a solution of the curvature problem, that is the phenomenon that Hochschild cocycles naturally parametrize curved -deformations. We also discuss the difference with the infinitesimal deformation setup. This is a report on joint work with Michel Van den Bergh.

We will discuss some interesting motivic measures and rationality of motivic zeta function.

In this talk, I will present an ongoing work (joint with D. Gepner). We develop a type of enrichment of infinity categories well-suited to the study of stable categories appearing in the theory of Deformation Quantization (DQ) modules. As an application, we obtain an integral representation theorem for DQ-modules along the line of Orlov's or Toën's integral representation theorems.

Non-commutative Hodge theory is the study of Hodge structures on the cyclic homology groups of dg-categories C. In this talk, we will study the case C = , where X is a quasi-projective variety and G is an algebraic group. Using Halpern-Leistner's theory of derived Kirwan surjectivity, we prove the collapse of nc Hodge-to-de Rham spectral sequence in variety of situations for example when X is smooth, G is reductive and is finite dimensional. These results on the degeneration of the spectral sequence also extend to categories of singularities. We also identify the periodic cyclic homology with a version of equivariant K-homology in the sense of Atiyah and Segal. This is joint work with Dan Halpern-Leistner.

I'll describe some new examples where the derived category of one variety can be embedded inside the derived category of a second variety. These examples are all given by a single construction, and they include the quintic 3-fold sitting in a certain Fano 11-fold, and a new derived equivalence between two Calabi-Yau 5-folds. We also recover the Pfaffian-Grassmannian' derived equivalence between two non-birational Calabi-Yau 3-folds.

The proof is inspired by string theory, and uses a non-abelian gauged linear sigma model to relate both derived categories to some categories of (global) matrix factorizations. This is joint work with Richard Thomas.

We explain how model categorical techniques allow to lift derived functors and adjunctions between them to enhancements when working with schemes or stacks over a field. Presently, we apply these techniques to homological smoothness and Fourier-Mukai functors. We expect more applications.

Orlov's famous representability theorem asserts that any fully faithful exact functor between bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this talk we show that this result is false without the full faithfulness hypothesis. This is joint work with Alice Rizzardo.

For any sufficiently regular dg-category $T$ it is possible to construct a geometric stack which classify its objects. This define a functor which is adjoint to the functor which associates to any algebraic space $X$ the dg-category of perfect complexes $Perf(X)$ on $X$. The problem of reconstruction is to find additionnal structure on the dg-category $Perf(X)$ in way to recover the space $X$.

Let be a smooth projective variety over the formal punctured disk . The Griffiths-Landman-Grothendieck “Local Monodromy Theorem” asserts that the Gauss-Manin connection on the de Rham cohomology has a regular singularity at the origin and that the monodromy of this connection is quasi-unipotent. I will explain a noncommutative generalization of this result, where the de Rham cohomology is replaced by the periodic cyclic homology of a (smooth proper) DG category over K equipped with the Gauss-Manin-Getzler connection. The proof of the Noncommutative Local Monodromy Theorem is based on the reduction modulo p technique and some ideas of N.Katz and D. Kaledin. Namely, I will prove that for any smooth proper DG category over the p-curvature of the Gauss-Manin-Getzler connection on its periodic cyclic homology is nilpotent.

If time allows I will also explain a noncommutative generalization of the Katz p-curvature formula relating the p-curvature of the Gauss-Manin-Getzler connection with the Kodaira-Spencer class (which is, in the noncommutative setting, a canonical element of the second Hochschild cohomology group of the DG category)

This talk is based on a joint work with Dmitry Vaintrob.

We study super-commutative nonpositive DG rings. An example is the Koszul complex associated to a sequence of elements in a commutative ring. More generally such DG rings arise as semi-free resolutions of rings. They are also the affine DG schemes in derived algebraic geometry. The theme of this talk is that in many ways a DG ring A resembles an infinitesimal extension, in the category of rings, of the ring .

I first discuss localization of DG rings on and the cohomological noetherian property. Then I introduce perfect, tilting and dualizing DG A-modules. Existence of dualizing DG modules is proved under quite general assumptions. The derived Picard group of A, whose objects are the tilting DG modules, classifies dualizing DG modules. It turns out that is canonically isomorphic to , and that latter group is known by earlier work. A consequence is that A and have the same (isomorphism classes of) dualizing DG modules.