AA at UNILU — Algebraic Analysis and related topics

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Supported by:

University of Luxembourg
Mathematics Research Unit

Programme

The scientific programme will start on Monday, November 4 in the afternoon and last till Wednesday, November 6, 2013 around noon.

Conference dinner

The conference dinner will take place in the restaurant La Lorraine, 7 place d'Armes, L-1136 Luxembourg, on Tuesday 5 at 19:30pm. A fee of 10 € will be asked to the participants at the conference registration on Monday.

Talks:

• Damien Calaque (ETH Zürich)
• Giovanni Felder (ETH Zürich)
• Gregory Ginot (Paris 6)
• Julien Grivaux (CNRS)
• Stephane Guillermou (CNRS)
• Dominic Joyce (Oxford)
• Yakov Kremnitzer (Oxford)
• Sergei Merkulov (Luxembourg)
• Dmytro Shklyarov (Albert-Ludwigs-Universität Freiburg )
• Michel Vaquié (Toulouse)

Titles and abstracts:

• Damien Calaque, "Lagrangian structures on derived mapping stacks and classical topological field theories"
  Abstract: In this talk I will present an extension of a result of Pantev-Toen-Vaquie-Vezzosi on the construction of shifted symplectic structures on derived mapping stacks (which can be viewed as an approach to the AKSZ construction that avoids problems involving infinite dimension). The main new ingredient compare to PTVV is the presence of boundary conditions, which are necessary if one wants to understand the work of Cattaneo-Felder on the Poisson sigma-model. I will also explain how this is related to the project of constructing various extended Topological Field Theories with values in an appropriate category of Lagrangian correspondences (inspired by the work of Cattaneo-Mnev-Reshetikhin).
• Giovanni Felder, "Holomorphic modular forms for SL(3,Z)"
  Abstract: I will present an attempt to extend the theory of holomorphic modular forms to the case of congruence subgroups of SL(n,Z). I will discuss examples for n=3, related to the elliptic gamma function. A map to automorphic forms, generalizing the relation between classical modular forms and Maass forms will be described.
• Gregory Ginot, "Factorization algebras and applications to En-algebras"
  Abstract: The talk will illustrate how to study the category of En-algebras using factorization algebras and factorization homology techniques. In particular, we will describe the universal enveloping En-algebra associated to a Lie algebra, explain the notion of centralizers of factorization algebras and its application to higher Deligne conjecture, iterated Bar construction.
• Julien Grivaux, "The geometry of quantized analytic cycles"
  Abstract: In this talk, we will give an overview of the theory of quantized analytic cycles and present some new results concerning derived intersections. Then we will explain how to construct characteristic classes attached to them.
• Stephane Guillermou, "Eliashberg's $C^0$-rigidity theorem and microlocal sheaf theory"
  Abstract: Eliashberg's theorem says that the group of symplectic diffeomorphisms of a symplectic manifold is $C^0$-closed in the group of all diffeomorphisms. A classical result of the microlocal theory of sheaves says that the microsupport of any non-zero sheaf is an involutive subset of the cotangent bundle. We will explain how to deduce Eliashberg's theorem from the involutivity theorem.
• Dominic Joyce, "Categorification of Donaldson-Thomas theory using perverse sheaves"
  Abstract: This is an overview of a collection of projects joint with O. Ben-Bassat, C. Brav, V. Bussi, D. Dupont, S. Meinhardt and B. Szendroi. Pantev, Toen, Vezzosi and Vaquie introduced the notion of k-shifted symplectic structure on a derived scheme or derived stack, for all integers k, where 0-shifted symplectic structures on derived schemes are just classical algebraic symplectic structures on classical smooth schemes. They prove that derived moduli stacks of (complexes of) coherent sheaves on a Calabi-Yau m-fold have a (2-m)-shifted symplectic structure. So the case k = -1 is relevant to Donaldson-Thomas theory of Calabi-Yau 3-folds. We prove a "Darboux Theorem" for k-shifted symplectic derived schemes for all k < 0. When k = -1, this says that a -1-shifted symplectic derived scheme (which includes moduli schemes of simple (complexes of) coherent sheaves on a Calabi-Yau 3-fold) is Zariski locally equivalent to the critical locus of a regular function on a smooth scheme. Next, we define "d-critical loci" (X,s), a classical scheme X with an extra (classical) geometric structure s which records information on how X may be written locally as a critical locus. We construct a truncation functor from -1-shifted symplectic derived schemes to d-critical loci, and deduce that moduli schemes of simple (complexes of) coherent sheaves on a Calabi-Yau 3-fold are d-critical loci. A d-critical locus (X,s) has a "canonical bundle", which for moduli schemes is the determinant line bundle of the natural obstruction theory. An "orientation" is a choice of square root of this canonical bundle; this is essentially the same as "orientation data" in the work of Kontsevich-Soibelman. We prove that an oriented d-critical locus (X,s) carries a natural perverse sheaf P_{X,s} (also a D-module, and a natural mixed Hodge module), such that if (X,s) is locally modelled on Crit ( f : U --> C) then P_{X,s} is locally modelled on the perverse sheaf of vanishing cycles of f. The pointwise Euler characteristic of P_{X,s} is the Behrend function of X. For a D-T moduli scheme, the graded dimension of the hypercohomology H^*(P_{X,s}) is the corresponding Donaldson-Thomas invariant. Thus, this provides a categorification of Donaldson-Thomas invariants. We also prove that an oriented d-critical locus (X,s) carries a natural motive M_{X,s}, such that if (X,s) is locally modelled on Crit ( f : U --> C) then M_{X,s} is locally modelled on the motivic Milnor fibre of f. Applied to Calabi-Yau 3-fold moduli schemes, this is relevant to motivic Donaldson-Thomas invariants a la Kontsevich-Soibelman.
  There is quite a lot more (mostly work in progress) that I will not have time to cover in the talk, but will be happy to discuss privately, including extensions of the above from schemes to Artin stacks, applications to Lagrangian intersections (a la Behrend-Fantechi) and defining "Fukaya categories" using perverse sheaves, and further categorification of D-T moduli schemes / Lagrangian intersections using matrix factorization categories.
• Yakov Kremnitzer, "Analytic geometry as relative algebraic geometry"
  Abstract: I will report on joint work with Oren Ben-Bassat on a new approach to analytic geometry. Following the work of Deligne and Toen-Vaquie on geometry relative to a symmetric monoidal category, we develop geometry relative to the category of Banach spaces. This gives a new approach to analytic geometry in both the Archimedean and non-Archimdean settings. Working over the real numbers this gives a new approach to differential geometry as well.
  
• Sergei Merkulov, "An exotic automorphism of the Lie algebra of polyvector fields"
  Abstract: Using some new operads of compactified semialgebraic configuration spaces, we show an explicit formula for a universal action of an element of the Grothendieck-Teichmueller group as a Lie-infinity automorphism of the Lie algebra of polyvector fields on an arbitrary smooth manifold.
• Dmytro Shklyarov, "On Hodge theoretic and categorical invariants of singularities"
  Abstract: The goal of the talk is to explain how various Hodge theoretic invariants (the vanishing cohomology with its Hodge filtration, the spectrum, etc.) of an isolated critical point of a polynomial can be recovered from the category of matrix factorizations associated with the singularity.
• Michel Vaquié, "Shifted Symplectic Structures"
  Abstract: We introduce the notion of n-shifted symplectic structures, a generalization of the notion of symplectic structures on smooth varieties and schemes. This notion is an important tool to study the moduli spaces of sheaves on higher dimensional manifolds from the point of views of homotopy theory and deformation quantization. (joint work with Tony Pantev, Bertrand Toen and Gabriele Vezzosi).