Higher Structures in Geometry and Mathematical Physics

17 avr. 2023, 07:30 → 14 juil. 2023, 19:30 Europe/Paris

Amphitheater Darboux (Institut Henri Poincaré, Paris)

Higher structures, in particular homotopy algebras, have appeared in many branches of mathematics and particularly in mathematical physics ranging from deformation theory, derived geometry, enumerative geometry, representation theory, homotopy algebras, Poisson geometry and quantization, mirror symmetry, noncommutative algebraic geometry, string theory and quantum field theory. This IHP trimester aims to promote interaction between mathematicians, physicists and groups working on related areas from different aspects. It brings together leading experts and young researchers in these subjects to address the important questions.

Scientific committee

• Kai Behrend (University of British Columbia)
• Barbara Fantechi (Scuola Internazionale Superiore di Studi Avanzati)
• Camille Laurent-Gengoux (Université de Lorraine)
• Mathieu Stiénon (Pennsylvania State University)
• Ping Xu (Pennsylvania State University)

Main activities

Introductory School at CIRM in Luminy (April 17th-21st)

https://conferences.cirm-math.fr/2697.html

Orateurs:

• Christian Brouder (Sorbonne Université) : “The physical origin of the Batalin-Vilkovisky formalism”
• Barbara Fantechi (SISSA) : “Derived deformation theory”
• Rajan Mehta (Smith College) : “A gentle introduction to graded geometry”
• Mauro Porta (Strasbourg) : “Introduction to derived algebraic geometry”
• Thomas Willwacher (ETH Zürich) : “Hopf algebras, modules, and deformation theory”

 

Three main conferences

For more details, click "Scientific Programme" in the sidebar.

• May 22nd-26th : GAP XVIII: Homotopy algebras and Higher Structures. https://ulysses8791.github.io/gap2023.html
• June 12th-16th : Higher Structures in Enumerative Geometry. https://ulysses8791.github.io/june2023.html
• July 3rd-7th : Dg-manifolds in Geometry and Physics. https://indico.math.cnrs.fr/event/7885/
• + July 10-11+Higher structures à la Glanonaise, En mémoire de notre ami et plus grand supporteur Jacky Michéa https://ulysses8791.github.io/glanon2023.html

Several mini-courses

==live stream at https://www.ihp.fr/fr/live-0==

 

• Benoit Fresse : Operads, Graph complexes, and Grothendieck-Teichmuller groups (May 3rd, May 10th, May 17th, frm 2pm to 4:30pm).  ((May 3d : General introduction, Introduction to Operads, May 10th : Rational homotopy. Sullivan models, May 17th : Applications : relation with GT. Embedding Calculus.))
• Bernhard Keller : Derived categories and Hochschild cohomology (May 15th. June 5th, June 19th, from 3:30pm to 5:30pm)
• Emily Riehl, Elements of oo-category theory, June 21st, 23rd 3-5pm.
• Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications. May 2,4,9,11,15,17 from 10am to 12am 
• Camille Laurent-Gengoux, Mathieu Stiénon, Ping Xu: Introduction to Dg-manifolds and vector fields commuting with themselves, April 24-26, 10-12am April 28 9:30-11am, 
• Ping Xu, Higher Structures and dg-manifolds, May 2nd 3-5:15 pm, May 3rd, 11-12am.
• Mathieu Stiénon, Duflo-Kontsevich Theorem by Mathieu Stiénon (II), May 4th 3-5pm, May 5th, 11am.
• Boris Tsygan : June 6 th June 9th 15:30-17:00 Gauss-Manin connection in noncommutative geometry.

Seminars : 

Please check "Ordre du Jour" for exact schedule.

 

April 28th May : Serguei Merkulov, Amphi Darboux, Gravity properad, moduli spaces M_g,n, and string topology

 

May 5th : not related to our trimester, but Anton Alekseev speaks at 4:30pm.

https://www.ihp.fr/fr/agenda/poisson-brackets-and-flat-connections

 

May 15th 11am, Vladimir Salnikov

 

June 5th  ; Lander Hermans, 

 

June 14th : FInski Siarhei, 

 

June 16th, , Andrey Lazarev.

 

June 26th,  : Dmitry Roytenberg, 

 

June 28th : Ricard Campos, Alex Takeda, Giovanni Felder 

 

July 10th :  Markus Pfaum : "Homological reduction as a tool in deformation quantization and anapplication in the quantization of  a lattice gauge model"

Other events

June 26th-27th-28th : Lectures and workshop about "Singular foliations and dg-manifolds", co-organised by Laurent-Gengoux and Leonid Ryvkin.

June 30rd : One-day workshop on "Geometry and Analysis at the foundations of QFT"

May 30th, 3pm Adrian Ocneanu will give a lecture for general public : Lois mathématiques, structure mathématique, et théorie quantique des champs

lundi 17 avril

08:00 → 16:00 Conférence au CIRM: Ecole introductive à Marseille

mardi 18 avril

08:00 → 16:00 Conférence au CIRM: Ecole introductive à Marseille

mercredi 19 avril

08:00 → 16:00 Conférence au CIRM: Ecole introductive à Marseille

jeudi 20 avril

08:00 → 16:00 Conférence au CIRM: Ecole introductive à Marseille

vendredi 21 avril

08:00 → 12:00 Conférence au CIRM: Ecole introductive à Marseille

lundi 24 avril

10:00 → 12:00 II: Introduction to dg-manifolds, and homological vector fields (I) by Camille Laurent-Gengoux

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

mercredi 26 avril

10:00 → 12:00 II: Introduction to dg-manifolds and homological vector fields (II) by Camille Laurent-Gengoux

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

vendredi 28 avril

09:25 → 10:50 II: Dg-manifolds and self-commuting vector fields (III) by Camille Laurnet-Gengoux.

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

11:00 → 12:00 Seminar: Serguei Merkulov (Luxembourg) : Gravity properad, moduli spaces M_g,n, and string topology

Weekly seminar

mardi 2 mai

10:00 → 12:00 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications.

15:00 → 16:00 Higher structures and dg-manifolds by Ping Xu: Higher structures and dg-manifolds (I) by Ping Xu

16:15 → 17:15 Higher structures and dg-manifolds by Ping Xu: Higher structures and dg-manifolds (II) by Ping Xu

mercredi 3 mai

11:00 → 12:00 Higher structures and dg-manifolds by Ping Xu: Higher structures and dg-manifolds (II) by Ping Xu

14:00 → 16:30 Benoît Fresse : Operads, Graph complexes, and Grothendieck-Teichmuller groups (I)

Benoit Fresse : Operads, Graph complexes, and Grothendieck-Teichmuller groups (May 3rd, May 10th, May 17th, frm 2pm to 4:30pm). ((May 3d : General introduction, Introduction to Operads, May 10th : Rational homotopy. Sullivan models, May 17th : Applications : relation with GT. Embedding Calculus.))

jeudi 4 mai

10:00 → 12:05 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications.

15:00 → 16:00 II: Duflo-Kontsevich Theorem for dg-manifolds by Mathieu Stiénon (I)

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

16:15 → 17:15 II: Duflo-Kontsevich Theorem for dg-manifolds by Mathieu Stiénon (II)

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

vendredi 5 mai

11:00 → 12:00 II: Duflo-Kontsevich Theorem for dg-manifolds by Mathieu Stiénon (III)

Introduction to dg-manifolds, and homological vector fields

programmCLG.pdf

mardi 9 mai

10:00 → 12:00 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications.(III)

mercredi 10 mai

14:00 → 16:30 Operads, Graph complexes, and Grothendieck-Teichmuller groups, by Benoît Fresse

Benoit Fresse : Operads, Graph complexes, and Grothendieck-Teichmuller groups (May 3rd, May 10th, May 17th, frm 2pm to 4:30pm). ((May 3d : General introduction, Introduction to Operads, May 10th : Rational homotopy. Sullivan models, May 17th : Applications : relation with GT. Embedding Calculus.))

jeudi 11 mai

10:00 → 12:00 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications.(IV)

vendredi 12 mai

11:00 → 12:00 Seminar

Weekly seminar

lundi 15 mai

09:00 → 11:00 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications. (V)

11:00 → 12:00 Seminar: Vladimir Salnikov : Z-graded manifolds I, properties of functional spaces and some applications

Weekly seminar

15:30 → 17:30 Bernhard Keller : Derived categories and Hochschild cohomology: Bernhard Keller : Derived categories and Hochschild cohomology (I)

(May 15th. June 5th, June 19th, from 3:30pm to 5:30pm)

mercredi 17 mai

10:00 → 12:00 (I): Olivier Schiffmann : Cohomological Hall algebras in dimensions one and two and applications.(VI)

14:00 → 16:30 Operads, Graph complexes, and Grothendieck-Teichmuller groups (I), by Benoît Fresse

Benoit Fresse : Operads, Graph complexes, and Grothendieck-Teichmuller groups (May 3rd, May 10th, May 17th, frm 2pm to 4:30pm). ((May 3d : General introduction, Introduction to Operads, May 10th : Rational homotopy. Sullivan models, May 17th : Applications : relation with GT. Embedding Calculus.))

lundi 22 mai

09:30 → 10:30 GAP: Registration and welcome coffee

https://personal.psu.edu/mps16/hirsutes2023/gap2023_schedule.pdf

10:30 → 11:30 GAP: Bernhard Keller (Université Paris Cité) : Singularity categories, Leavitt path algebras and Hochschild homology

11:30 → 12:30 GAP: Domenico Fiorenza, Sapienza (Università di Roma) : String bordism invariants in dimension 3 from U(1)-valued TQFTs

14:15 → 15:15 GAP: Leonid Positselski,( Czech Academy of Sciences, Prague) : The homomorphism removal and repackaging construction

15:15 → 15:30 GAP: Dominique Mouhanna, Deputy Director of IHP : Welcome Address

16:00 → 17:00 GAP: Wendy Lowen, (Universiteit Antwerpen) : Higher structures and deformations of non-commutative spaces

17:00 → 18:00 GAP: Poster Session

mardi 23 mai

09:00 → 10:00 GAP: Martin Markl, (Czech Academy of Sciences, Prague) : Operads and the blob complex

10:30 → 11:30 GAP: Julien Grivaux, (Sorbonne Univ) : Algebraic structures attached to a closed embedding

11:30 → 12:30 GAP: Shahn Majid, (Queen Mary Univ.) : Braided bimodules, quantum jet bundles and quantum geodesics

14:00 → 15:00 GAP: Kirsten Wickelgren (Duke University) : A quadratically enriched zeta function

15:30 → 16:30 GAP: Adrian Ocneanu, (Pennsylvania State University), We describe a stronger analog of the Weyl character formula, which has a similar form, but generates all the representation vectors, rather than only their weghts.

mercredi 24 mai

09:00 → 10:00 GAP: Benoît Dhérin, (Google) : Symplectic microgeometry and quantization

10:30 → 11:30 GAP: Olivier Schiffmann, (Université Paris-Saclay) : Cohomological Hall algebras associated to curves in surfaces

11:30 → 12:30 GAP: Thomas Willwacher, (ETH Zürich) : Graph complexes and moduli spaces of curves

jeudi 25 mai

09:00 → 10:00 GAP: Muriel Livernet, (Univ. Paris Cité) : Homotopy theory of spectral sequences

10:30 → 11:30 GAP: Alexander Berglund, (Stockholm University) : Higher structures in automorphisms of manifolds

11:30 → 12:30 GAP: Christoph Schweigert, (Universität Hamburg) : String-net methods for CFT correlators

14:00 → 15:00 GAP: Mikhail Kapranov, (Kavli IPMU) : Categorification of Euler’s continuants, N-spherical functors and periodic semiorthogonal decompositions

15:30 → 16:30 GAP: Pavel Mnev, (University of Notre Dame) : On the Fukaya-Morse A-infinity category

vendredi 26 mai

09:00 → 10:00 GAP: Ralph Kaufmann, (Purdue University) : Higher operations from algebra and geometry.

10:30 → 11:30 GAP: Yuri Berest, (Cornell University) : Spaces of quasi-invariants and homotopy Lie groups

samedi 27 mai

10:00 → 10:30 Higher Structures on Saturday: Maosong Xiang : The standard cohomology of regular Courant algebroids

10:30 → 11:00 Higher Structures on Saturday: Zheng Hua : Standard poisson structure on Grassmannian revisited

11:30 → 12:00 Higher Structures on Saturday: Ruggero Bandiera : Higher derived brackets and splitting principle

12:00 → 12:30 Higher Structures on Saturday: Alberto Cattaneo : BV pushforward and Yang-Mills theory

12:30 → 13:00 Higher Structures on Saturday: Francesco Bonechi : Equivariant Yang-Mills

13:00 → 13:30 Higher Structures on Saturday: Discussion

mardi 30 mai

15:00 → 16:00 Exposé grand Public, Adrian Ocneanu, Lois mathématiques, structure mathématique, et théorie quantique des champs

lundi 5 juin

11:00 → 12:00 Seminar :Lander Hermans : Deforming prestacks, a calculus of rectangles

In his foundational work Gerstenhaber furnishes the guiding example for algebraic deformation theory: for an associative algebra A he defined a dgLie bracket on its Hochschild complex and showed that it controls the deformations of A through the Maurer-Cartan equation. Algebraic geometry motivates the natural question whether a similar story exists for diagrams of associative algebras, e.g. when applied to the structure sheaf of a scheme. In this talk I will explain how the Gerstenhaber-Schack complex fulfils this role, yet also motivates to generalize from diagrams to prestacks (i.e. pseudofunctors) as the most suitable objects to start with. Inspired by the fact that the Lie-structure of the Hochschild complex arises from an underlying operadic calculus, we introduce a new L-infinity structure arising from a rectangular operadic calculus. We show it completes the story: the higher Lie brackets on the GS complex control the deformations of prestacks through the generalized MC equation. Along the way, we introduce a new type of operad, called “box operads”, which can be seen as an enriched version of Leinster’s fc-multicategories (also called virtual double categories). This is joint work with Hoang Dinh Van and Wendy Lowen.

15:30 → 17:30 Bernhard Keller : Derived categories and Hochschild cohomology: Bernhard Keller : Derived categories and Hochschild cohomology (II)

(May 15th. June 5th, June 19th, from 3:30pm to 5:30pm)

mardi 6 juin

11:00 → 12:00 Seminar : Lothar Göttsche (Trieste) :: (Refined) Verlinde and Segre formulas for Hilbert schemes of points.

This is joint work with Anton Mellit. Segre and Verlinde numbers of Hilbert schemes of points have been studied for a long time. The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert schemes of points. The Verlinde numbers are the holomorphic Euler characteristics of line bundles on these Hilbert schemes. We give the generating functions for the Segre and Verlinde numbers of Hilbert schemes of points. The formula is proven for surfaces with K_S^2=0, and conjectured in general. Without restriction on K_S^2 we prove the conjectured Verlinde-Segre correspondence relating Segre and Verlinde numbers of Hilbert schemes. Finally we find a generating function for finer invariants, which specialize to both the Segre and Verlinde numbers, giving some kind of explanation of the Verlinde-Segre correspondence.

15:30 → 17:00 Boris Tsygan: Boris Tsygan : Gauss-Manin connection in noncommutative geometry (I)

vendredi 9 juin

15:30 → 17:00 Boris Tsygan: Boris Tsygan : Gauss-Manin connection in noncommutative geometry.(II)

lundi 12 juin

09:00 → 10:00 Registration

10:00 → 11:00 Davesh Maulik : P=W conjecture for GL_n

The P=W conjecture, proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory. In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for GL_n (joint with Junliang Shen).

11:30 → 12:30 Cristina Manolache : Desingularisation of sheaves and reduced Gromov–Witten invariants

Gromov–Witten (GW) invariants of genus g, with g greater than one, do not count curves of genus g in a given space: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called “reduced GW invariants.” More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will define reduced Gromov–Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

14:00 → 15:00 Yuan-Pin Lee : QK = GV for CY3 at g=0

In this talk, I will show that on a Calabi-Yau threefold (CY3) a genus zero quantum K-invariant (QK) can be written as an integral linear combination of a finite number of Gopakumar–Vafa BPS invariants (GV) with coefficients from an explicit multiple cover formula. Conversely, all Gopakumar–Vafa invariants can be determined by a finite number of quantum K-invariants in a similar manner. The technical heart is a proof of a remarkable conjecture by Hans Jockers and Peter Mayr.

This result is consistent with the “virtual Clemens conjecture” for the Calabi–Yau threefolds. A heuristic derivation of the relation between QK and GV via the virtual Clemens conjecture and a “multiple cover formula” will also be explained. This is a joint work with You-Cheng Chou.

15:30 → 16:30 Mauro Porta : Categorical Hall algebras and their representations

Two-dimensional cohomological Hall algebras have been introduced for the first time by Schiffmann and Vasserot in 2009 and they soon proved to be an exceptional tool for the study of homology and G-theory of several kinds of moduli spaces. More recently they have been revisited with tools from derived geometry, which led to a natural categorification. A current limitation of the subject is that cohomological Hall algebras only yield positive halves of "whole" algebraic objects, such as Yangians or quantum loop groups. With Diaconescu and Sala, I have constructed categorical representations for these algebras, that generalize to a 1-dimensional setting the creation and destruction operators of Nakajima. I will survey this construction and discuss some more recent ongoing work in this direction.

mardi 13 juin

10:00 → 11:00 Emily Riehl : Do we want a new foundation for “higher structures”?

The fundamental theorem of category theory is the Yoneda lemma, which in its simplest form identifies natural transformations between represented functors with morphisms between the representing objects. The $\infty$-categorical Yoneda lemma is surprisingly hard to prove — at least in the traditional set-based foundations of mathematics. In this talk we’ll describe the experience of developing $\infty$-category theory in an alternate foundation system based on homotopy type theory, in which constructions determined up to a contractible space of choices are genuinely “well-defined” and elementwise mappings are automatically homotopically-coherently functorial. In this setting the proof the $\infty$-categorical Yoneda lemma is arguably easier than the 1-categorical Yoneda lemma. We’ll end by posing the question as to whether similar foundations would be useful for other “higher structures.” This is based on joint work with Mike Shulman and involves computer formalizations written in collaboration with Nikolai Kudasov and Jonathan Weinberger.

11:30 → 12:30 Yukinobu Toda : Quasi-BPS categories for K3 surfaces

In this talk, I will give semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces. An each summand is given by the categorical Hall product of subcategories called quasi-BPS categories, which approximate the categorification of BPS cohomologies for K3 surfaces. When the weight and the Mukai vector is coprime, the quasi-BPS category is shown to be smooth and proper, with trivial Serre functor etale locally on the good moduli space. So it gives a twisted analogue of categorical crepant resolution of the singular symplectic moduli space, and reminiscents categorical analogue of chi-independence phenomena. This is a joint work in progress with Tudor Padurariu.

14:00 → 15:00 Sarah Scherotzke : Cotangent complexes of moduli spaces

We explain how shifted symplectic structures on derived stacks are connected to Calabi-Yau structures on differential graded categories. More concretely, we will show that the cotangent complex to the moduli stack of a differential graded category A is isomorphic to the moduli stack of the Calabi-Yau completion of A, answering a conjecture of Keller-Yeung.

15:30 → 16:30 Jim Bryan : The geometry and arithmetic of banana nano-manifolds.

The Hodge numbers of a Calabi-Yau threefold X are determined by the two numbers h^{1,1}(X) and h^{1,2}(X) which can be interpreted respectively as the dimensions of the space of Kahler forms and complex deformations respectively. We construct four new examples X_N, where N \in {5,6,8,9}, of rigid Calabi-Yau threefolds (h^{2,1}=0) with Picard number 4 (h^{1,1}=4). These manifolds are of “banana type” and have among the smallest known values for Calabi-Yau Hodge numbers. We (partially) compute the Donaldson-Thomas partition functions of these manifolds and in particular show that the associated genus g Gromov-Witten potential is given by a weight 2g-2 Siegel paramodular form of index N. These manifolds are also modular in the arithmetic sense: there is a weight 4 modular form whose Fourier coefficients are obtained by counting points over finite fields on X_N. We compute this form and observe that it is the unique cusp form of weight 4 and index N. This is joint work with Stephen Pietromonaco.

mercredi 14 juin

09:30 → 10:30 Zheng Hua : Modular Poisson structures

Give a Gorenstein Calabi-Yau curve X, the moduli stack of bounded complexes of vector bundles on X admits a 0-shifted Poisson structure. For the component that is a G_m gerbe over a smooth scheme, the 0-shifted Poisson structure descends to an ordinary Poisson structure on its coarse moduli scheme, which we call a “modular Poisson structure”. In this talk we will survey some recent progress on the study of the modular Poisson structures. In particular we will give the example when X is the Kodaira cycle, where we establish a correspondence between certain torus orbits of vector bundle stack and projected open Richardson varieties on grassmannian via modular Poisson structure.

11:00 → 12:00 Wendy Lowen : Box operads and noncommutative schemes

In this talk, we discuss how the concept of algebraic deformation theory, dating back to Gerstenhaber’s deformation theory of algebras, can be applied in geometry leading to ‘noncommutative schemes’ in the sense of Van den Bergh. In the first part of the talk, we focus on models in projective geometry, like graded algebras and Z-algebras, and we describe deformations as ‘noncommutative projective schemes’ under the assumption that H^1(X,O_X) = 0 = H^2(X,O_X). In the second part of the talk, we take a local approach to schemes by deforming the structure sheaf, leading to ‘twisted presheaves’ or ‘prestacks’ in case H^2(X,O_X) is non-zero. Finally, we present an operadic structure on the Gerstenhaber-Schack complex of a prestack recently established in joint work with Hoang Dinh Van and Lander Hermans. This yields an underlying L_{infinity} structure governing deformations of prestacks.

12:00 → 13:00 Etienne Mann : Gromov-Witten theory with derived algebraic geometry

Using ideas from Toën-Schuerg-Vezzosi, one can define GW invariants via derived algebraic geometry. In this talk, we will see how some classical statements like Splitting axiom can be seen at the geometric level. An other nice application is to see a derived version of Quantum Lefchetz principle due to David Kern. Part of this work was done with Marco Robalo.

14:00 → 15:00 SEMINAR : Benjamin Hennion, Categorified Donaldson--Thomas invariants of Calabi--Yau 3-folds

Abstract: Donaldson--Thomas invariants are numerical invariants associated to Calabi--Yau varieties. They can be obtained by glueing singularity invariants from local models of a suitable moduli space endowed with a (-1)-shifted symplectic structure.

By studying the moduli of such local models, we will explain how to recover Brav--Bussi--Dupont--Joyce--Szendroi's perverse sheaf categorifying the DT-invariants, as well as a strategy for glueing more evolved singularity invariants, such as matrix factorizations.

This is joint work with M. Robalo and J. Holstein.

15:30 → 16:30 SEMINAR : Finski Siarhei : Asymptotic study of filtrations on section rings and geodesic rays of metrics

Abstract: For a complex projective manifold polarised by an ample line bundle, we study the asymptotic properties of submultiplicative filtrations on the associated section ring and show that these are related to the geometry at infinity of the space of Kähler metrics on the manifold. Among others, our study will allow us to give an asymptotic formula for the maximal dimension of a vector subspace of the space of holomorphic sections of high tensor powers of the polarising line bundle such that for any holomorphic section from our subspace the difference between the orders of annihilation along two fixed submanifolds lies in a given interval.

jeudi 15 juin

10:00 → 11:00 Tobias Dyckerhoff : Complexes of stable infinity-categories

Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples that arise in the vicinity of homological mirror symmetry, along with some results and conjectures involving them.

11:30 → 12:30 Ben Davison : Stacks of semistable sheaves on K3 surfaces

I'll explain some new results on the Borel-Moore homology of stacks of coherent sheaves on K3 surfaces, as well as intersection cohomology of coarse moduli spaces. For nonprimitive Chern classes, these spaces can be highly singular. Nonetheless, considering all multiples of a given class simultaneously, we (in joint work with Lucien Hennecart and Sebastian Schlegel Mejia) have a cohomological upgrade of the integrality theorem from DT theory, connecting the intersection cohomology of the coarse spaces with the Borel-Moore homology of the stacks. Aside from the integrality theorem itself, applications include a new description of Maulik-Toda-style GV invariants of local K3 surfaces, a proof of the Halpern-Leistner purity conjecture for the BM homology of the stack, and wall-crossing invariance results.

14:00 → 15:00 Renata Picciotto : The derived moduli of sections and virtual pushforwards

Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the "virtual structures" of the moduli problems . I will discuss a joint work with D. Karn, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections. This enriched space neatly encodes the perfect obstruction theory and virtual structure sheaves of many theories. Special cases include Gromov-Witten and quasimaps theories. To illustrate the potential of this approach, I will explain how we use local derived charts to prove a virtual pushforward formula between stable maps and quasimaps without relying on torus localization.

15:30 → 16:30 Rok Gregoric : Shifting and shearing in spectral algebraic geometry

In derived algebraic geometry, vector bundles can be shifted. This corresponds to the shearing operation on graded objects. However, over the sphere spectrum, the notions of shifting vector bundles and of shearing gradings decouple, in parallel to the existence of two variants of the affine line. In this talk, we will discuss how the picture of shifting and shearing can be reconciled by replacing the integers with more exotic grading groups. We will see how this naturally leads to considering the moduli stack of oriented formal groups, and through it to chromatic homotopy theory and the motivic tau-deformation phenomenon.

vendredi 16 juin

09:30 → 10:30 Penka Georgieva : Klein TQFT and real Gromov-Witten invariants

TQFT structures appear in relation with Gromov-Witten invariants in at least two separate occasions. In this talk I will explain how the Real Gromov-Witten theory of local 3-folds with base a Real curve gives rise to an extension of a 2d Klein TQFT. This gives strong implications for the structure of the invariants on any compact Calabi-Yau 3-fold with anti-symplectic involution.

11:00 → 12:00 Andrea Ricolfi : d-critical structure(s) on the Quot scheme of points on a Calabi-Yau 3-fold

D-critical schemes and Artin stacks were introduced by Joyce in 2015, and play a central role in Donaldson-Thomas theory. They typically occur as truncations of (-1)-shifted symplectic derived schemes, but the problem of constructing the d-critical structure on a “DT moduli space” without passing through derived geometry (which is hard) is wide open. We discuss this and related problems, and new results in this direction, when the moduli space is the Quot scheme of points on a Calabi-Yau 3-fold. Joint work with Michail Savvas.

12:00 → 13:00 Richard Thomas : (-2)-shifted symplectic virtual cycles

Behrend-Fantechi and Li-Tian showed how to produce a virtual cycle from a 2-term obstruction theory (or, in higher language, a quasi-smooth derived scheme). I will describe joint work with Jeongseok Oh that produces a virtual cycle from a 3-term symmetric obstruction theory (or, in higher language, a (-2)-shifted symplectic derived scheme). There is also a localisation formula, a K-theoretic refinement, a virtual GRR formula, etc. Earlier Borisov-Joyce used real derived differential geometry to define a real virtual homology class. When their virtual dimension is even we show our class reproduces theirs; when it is odd we show their class is torsion.

15:30 → 16:30 SEMINAR : Andrey Lazarev : Global Koszul duality (joint with M. Booth).

Koszul duality is a phenomenon observed in diverse subfields of algebra and homotopy theory. It underpins, implicitly or explicitly, rational homotopy theory and deformation theory. One important aspect of it, which is the subject of the present talk, can be formulated as an equivalence between infinity categories of dg algebras and coalgebras. The current state of knowledge (due mainly to Positselski and Keller-Lefevre) can be understood as a local theory. I will describe how to construct a global theory, in what sense it generalizes the local one and why this generalization is desirable. As an application, I will explain how to construct global moduli spaces in various moduli problems admitting a `noncommutative structure’. Among examples of such problems are moduli of flat connections in vector bundles, objects in dg categories, holomorphic structures in complex analytic bundles, and others.

lundi 19 juin

15:30 → 17:40 Bernhard Keller : Derived categories and Hochschild cohomology: Bernhard Keller : Derived categories and Hochschild cohomology (III)

(May 15th. June 5th, June 19th, from 3:30pm to 5:30pm)

mercredi 21 juin

15:00 → 17:00 Elements of oo-category theory (I), Emily Riehl

Two lectures on infinity category theory

Speaker: Emily Riehl, Johns Hopkins University

Time: June 21, and June 23, 15:00-17:00
Place: IHP

Abstract: I will explain some core basics of infinity category theory: equivalences, adjunctions, limits and colimits, with the goal of introducing stable infinity categories.

The only essential prerequisite is the notion of a strict 2-category. If folks want to look at something in advance, I'd suggest section B.1 of the "Elements" book, and in fact B.1.1 - B.1.7 is enough, with the notion of pasting diagram in B.1.4-B.1.5 being the most important.

vendredi 23 juin

15:00 → 17:00 Elelments of oo-category theory, Emily Riehl

Two lectures on infinity category theory

Speaker: Emily Riehl, Johns Hopkins University

Time: June 21, and June 23, 15:00-17:00
Place: IHP

Abstract: I will explain some core basics of infinity category theory: equivalences, adjunctions, limits and colimits, with the goal of introducing stable infinity categories.

The only essential prerequisite is the notion of a strict 2-category. For participants willing to look at something in advance, section B.1 of my "Elements in Infinite Category Theory" book, is adviced (in fact B.1.1 - B.1.7 is enough), with the notion of pasting diagram in B.1.4-B.1.5 being the most important.

lundi 26 juin

09:00 → 09:30 Higher Structures and Foliations: Hadi Nahari : On the octonionic Hopf singular foliation

10:00 → 10:30 Higher structures and foliations : Aliaksandr Hancharuk : About (infinite) Koszul-Tate resolutions

Abstract: We present a construction of arborescent Koszul-Tate resolutions. In contract to original Tate's result we do so in finitely many computations for a large class of examples. We discuss minimality of Koszul-Tate resolutions and provide conditions when such resolutions are infinite. Based on a joint work with C.Laurent-Gengoux and T. Strobl.

11:00 → 12:00 Dimitry Roytenberg : Equivalences of dg-manifolds in higher Lie theory.

According to an insight of Pavol Ševera's, integrating a dg-manifold amounts to representing its homotopy type. This leads to a natural definition of a weak equivalence of dg-manifolds. I will present a conjecture characterizing these weak equivalences in terms of data associated to the dg-manifold itself (i.e. without going to the integration), and sketch a proof in the transitive case. Time permitting, I will also discuss fibrations.

14:00 → 14:30 Rosa Marchesini : On the notion of homotopy for Lie algebroids.

The definition of homotopy as a smooth curve of Lie algebroid morphisms does not guarantee the homotopy invariance of Lie algebroid cohomology.
In order to obtain the desired property, we need to focus on a more restrictive notion that has appeared sporadically in the literature but has not yet been explored to its full potential. We motivate the choice of definition with new examples and relevant applications. The results are a joint work with my supervisor Madeleine Jotz.

15:00 → 15:30 Florian Dorsch : First steps in differentiating simplicial manifolds

The integration and differentiation processes between higher Lie groupoids and higher Lie algebroids
are active areas of research. The differentiation of a higher Lie groupoid should produce its tangent
complex, a replacement for the tangent space of a Lie group, equipped with suitable extra structure
that describes its higher Lie algebroid structure. In the talk it will be shown how a construction of
Severa for the differentiation of higher Lie groupoids may be generalized for simplicial manifolds
which lack the extra conditions making them higher Lie groupoids. The idea is that simplicial
manifolds still satisfy "local" versions of Kan conditions, which suffices to define a well-defined
notion of tangent complex and perform Severa´s construction.

mardi 27 juin

09:00 → 10:00 Sylvain Lavau. Deformations of singular foliations

The study of deformations of singular foliations cannot be streamlined
as for regular foliations, due to the presence of singularities.
However, upon replacing the singular foliation by its corresponding Lie
infinity-algebroid, we can construct a DGLA whose Maurer-Cartan elements
control the deformations of the given singular foliation. It appears
that deforming the singular foliation amounts to deforming the
corresponding Lie infinity-structure. We use this approach to study the
deformations of several kinds of singular foliations. Work in
collaboration with Camille Laurent-Gengoux.

11:00 → 12:00 Alfonso Garmendia. E-manifolds and Poisson manifolds.

Abstract: In this talk we will discuss the concept of E-maifolds (by Miranda and Scott) which are manifolds with a projective foliation. We will give some examples used in the literature, such as the case of b-, c-manifolds and 0-manifolds. We will also give some constructions on Poisson manifolds and these objects.

13:30 → 16:20 Higher Structures and Foliations : discussion

16:30 → 17:30 Higher Structures and Foliations : Omar Moshen : TBA

mercredi 28 juin

10:00 → 11:00 Mini-course : Alexei Kotov, Several topics about Q-manifolds

11:00 → 12:00 Kai Wang : The deformation theories of operated algebras and the minimal models of their operads

Abstract: The minimal models of operads play important roles in the studies of homotopy
theory of algebraic structures. For Koszul operads, the classical Koszul duality theory for
operads provides a canonical way to construct their minimal models. But the general
method to construct the minimal models of non-Koszul operads is still unknown. In this talk, we will study the deformation theory of some operated algebras whose operads are not Koszul, including Rota-Baxter algebras, differential algebras, Nijenhuis algebras etc. We will give an explicit construction of the minimal models of these operads. Then the cohomology theories and the L-infinity algebras which control the deformations of these operated algebras will be induced in a natural way.

15:30 → 16:30 David Carchedi : Derived manifolds as dg-manifolds

When two submanifolds meet non-transversely their intersection may fail to be a manifold. However, such bad intersections can be dealt with as smooth objects in the setting of derived geometry. Derived manifolds are objects constructed by iteratively taking fibered products, starting with smooth manifolds. We will show how such a theory can be derived from basic principles and explain its connection to derived algebraic geometry. Specifically, we will explain a joint result with Pelle Steffens which characterizes derived manifolds by a universal property. Then we will explain a recent result of ours that shows that the infinity category of dg-manifolds arising from the category of fibrant objects structure of Behrend-Liao-Xu is equivalent to the infinity-category of derived manifolds.

jeudi 29 juin

14:00 → 15:00 Alex Takeda : Pre-Calabi-Yau structures on the categorical formal neighborhood of infinity and the string coproduct

Pre-Calabi-Yau structures are noncommutative versions of Poisson structures, which exist on many dg/A-infinity categories of interest in mirror symmetry. In this talk, I will explain a relation between these structures and Efimov's notion of the "categorical formal punctured neighborhood of infinity", which generalizes the category of perfect complexes near the divisor at infinity in a compactification of an algebraic variety and is related, on the other side of mirror symmetry, to Rabinowitz Fukaya categories. In joint work with Manuel Rivera and Zhengfang Wang, we use both of these structures to produce certain operations on Hochschild complexes, which we propose as algebraic versions of the loop product and coproducts in string topology. After explaining this formalism, I will present some examples where these operations can be calculated to agree with the corresponding geometric definitions, sometimes even giving expressions over the integers.

15:30 → 16:30 Giovanni Felder : Restricted dynamical quantum groups and groupoid graded representations

Restricted dynamical quantum groups are the algebraic structure
underlying the restricted SOS models of statistical mechanics. After
reviewing these models as they appear in statistical mechanics, I will
explain the related representation theory of dynamical quantum groups
on groupoid graded vector spaces. The talk is based on joint work with
Muze Ren and work in progress with Jelena Anic.

16:30 → 17:30 Ricardo Campos, Homotopical relations between commutative, associative and Lie algebras

Résumé : The forgetful functor from commutative algebras to associative algebras is fully faithful, but this is no longer the case if we consider these objects homotopically, where commutativity should be seen as structure, not property.
Studying properties of this functor helps answering the question in rational homotopy theory of how much of the homotopy type of a space is captured by its algebra of singular cochains? (The experienced homotopy theorist will tell you to consider instead a commutative algebra of forms.) This is surprisingly related to the isomorphism problem for universal enveloping algebras of Lie algebras.
We show that in characteristic zero, these functors satisfy homotopical faithfulness, as well as some sort of injectivity on quasi-isomorphism type of objects. This is in essence a result in deformation theory which is naturally addressed with tools from higher Lie theory and the operadic calculus.
This joint work with Dan Petersen, Daniel Robert-Nicoud and Felix Wierstra; based on arXiv:1904.03585 and 2211.02387.

vendredi 30 juin

10:00 → 17:00 Geometry and Analysis at the foundations of QFT, workshop organised by Kasia Rejzner

Christian Brouder, Yoann Dabrowski, Viet Dang, Owen Gwilliam, Kasia Rejzner (list to be continued) will present various topics, in an informal way, about , Owen Gwilliam, various settings for infinite dimensional (graded) geometry needed in QFT, BV formalism in QFT...

lundi 3 juillet

10:00 → 11:00 Ezra Getzler : Symplectic forms on derived stacks

Abstract: Chern-Weil theory gives an explicit formula for a differential form on BGL(n) representing the Chern character. The lowest components of this form are of importance in geometry: the component of total degree 2 is the first Chern class, while the component of total degree 4 component is the shifted symplectic form on BGL(n). This formula was first obtained in the Berkeley thesis of Shulman in 1972. The extension to the derived stack of perfect complexes (essentially the generalization to graded vector spaces of BGL(n)) is more difficult: an existence proof was obtained by Toën and Vezzossi, but their approach does not lead to a formula. In this talk, I show how, using an explicit realization of this derived stack (joint work with Kai Behrend) and negative cyclic homology, we obtain an explicit formula for a differential form representing the Chern character on this derived stack, and hence also an explicit formula for the shifted symplectic form.

11:00 → 12:00 Leonid Ryvkin : Differentiation of simplicial manifolds

Kan simplicial manifolds provide with a very explicit model for $L_\infty$-groupoids. Pavol Severa proposed a procedure of differentiation for these objects, yielding an $L_\infty$-algebroid, (an NQ-manifold). In the talk I will report on joint work with Rui Fernandes, Du Li, Arne Wessel and Chenchang Zhu, where we have proven that this procedure always works.

14:00 → 15:00 Pavel Mnev : “Examples of bulk-boundary correspondences of field theories as BV pushforwards”

I will explain how one can obtain 2d Wess-Zumino-Witten model on a surface as a Batalin-Vilkovisky pushforward (perturbative fiber BV integral, or “quantized” pushforward of dg manifolds) from 3d Chern-Simons theory on the cylinder (surface)x(interval). Here Chern-Simons is treated in BV formalism with functorial cutting-gluing.

A similar construction applied to abelian Chern-Simons on a 7d cylinder yields the 6d Kodaira-Spencer gravity (a.k.a. BCOV theory). The talk is based on a joint work with Alberto S. Cattaneo and Konstantin Wernli, arXiv:2012.13983 .

15:00 → 16:00 Jae-Suk Park : Grothendieck-Galois theories over symmetric monoidal categories

16:00 → 16:30 Ruben Louis : Lie-Rinehart algebras as acyclic dg-manifolds and geometric applcations

16:30 → 18:00 Poster Session, discussions (and coffee)

mardi 4 juillet

08:45 → 09:45 Thomas Strobl : Arborescent Koszul-Tate resolutions and BFV for singular coisotropic reductions

Let I be an ideal in a commutative (associative) algebra O. Starting
from a resolution of O/I as an O-module, we construct a Koszul-Tate
resolution for this quotient, i.e.\ a graded symmetric algebra over O
with a differential which provides simultaneously a resolution as an
O-module. This algebra resolution has the beautiful structure of a
forest of decorated trees and is related to an A-infinity algebra on the
original module resolution. 

Considering O to be a Poisson algebra
and I a finitely generated Poisson subalgebra, we use the above
construction to obtain the corresponding BFV formulation. Its cohomology
at degree zero is proven to coincide with the reduced Poisson algebra
N(I)/I, where N(I) is the normaliser of I inside O, thus
generalising ordinary coisotropic reduction to the singular setting. As
an illustration we use the example where O consists of functions on
T^*(\R^3) and I is the ideal generated by angular momenta. 



This is joint work with Aliaksandr Hancharuk and, in part, with Camille
Laurent-Gengoux.

10:15 → 11:15 Oleksii Kotov : Dg bundles with some applications

Dg bundles are surjective submersions of dg manifolds, which are fiber bundles in the category of graded supermanifolds. The purpose of the talk is to show how dg-bundles appear in the context of characteristic classes, BV-BRST formalism in local field theory, and complex geometry. The talk is based on joint research with Thomas Strobl, Maxim Grigoriev and Camille Laurent-Gengoux.

11:15 → 12:15 Hsuan-Yi Liao : Homotopy fiber product and dg manifolds of finite positive amplitudes

A main motivation for developing derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both zero loci and intersections can be considered as fiber products of manifolds which may not be manifolds. To deal with this issue, we extend the category of differentiable manifolds to the category of dg manifolds of finite positive amplitudes in which "homotopy fiber products" exist. In this talk, I would like to explain properties of this larger category (it is a category of fibrant objects) and our construction of homotopy fiber products of manifolds. The talk is mainly based on a joint work with Kai Behrend and Ping Xu.

14:15 → 15:15 Madeleine Jotz : A geometrisation of N-manifolds of degree n

Lie $2$-algebroids are geometrised by linear Courant algebroids, while symplectic Lie $2$-algebroids correspond to mere Courant algebroids.
This talk begins by explaining these correspondences due to Li-Bland, Severa and Roytenberg, by establishing the underlying equivalence between [2]-manifolds and metric double vector bundles. The latter yields a dictionary between graded geometric structures on [2]-manifolds, like homological vector fields, Poisson and symplectic structures, and corresponding `classical geometric' structures on the corresponding metric double vector bundles.

Metric double vector bundles dualise to double vector bundles equipped with a (signed) involution. The latter can then be understood as $S_2$-symmetric double vector bundles -- recovering Pradines’ ‘inverse' symmetric double vector bundles.

Similarly, positively graded manifold of arbitrary degree $n$ are equivalent to $n$-fold vector bundles equipped with a (signed) $S_n$-symmetry. This talk explains more precisely how symmetric $n$-fold vector bundle cocycles are the same objects as [n]-manifold cocycles, and how symmetric vector bundles, which are indexed by $n$-cube categories, provide a new and insightful point of view on (positively) graded geometry.
This is the groundwork for understanding a possible geometrisation of Lie $n$-algebroids, like VB-Courant algebroids geometrise Lie $2$-algebroids, and Lie algebroids geometrise Lie $1$-algebroids.

This work is partly joint with Malte Heuer.

15:15 → 16:15 Marco Zambon : Reduction of Courant algebroids via graded manifolds.

Courant algebroids are certain objects in Lie theory that are used to define, for instance, Dirac structures and generalized complex structures.
We will use the correspondence between degree 2 symplectic manifolds and Courant algebroids, due to Roytenberg, to approach the reduction of Courant algebroids using graded geometry. For this purpose we will consider both graded coisotropic submanifolds and a graded version of moment maps. The resulting reduction procedure, in a particular case, recovers the work of Bursztyn-Cavalcanti-Gualtieri around 2007. This talk is based on joint work with Bursztyn, Cattaneo and Metha.

mercredi 5 juillet

08:45 → 09:45 Joost Nuiten : The Van Est isomorphism for higher stacks

A classical theorem of Van Est relates the differentiable cohomology of a Lie group to the cohomology of its Lie algebra. I will discuss a version of this result for homotopy-theoretic variants of Lie group(oid)s, such as the string group. This relates the derived category of representations of such a groupoid to the derived category of representations of its Lie algebroid. In particular, this relies on a version of Koszul duality for Lie algebroids.

09:45 → 10:15 Francesca Pratali : Dendroidal ∞-operads

The dendroidal formalism offers a powerful approach to
defining operads as specific presheaves on a tree-based category. This
combinatorial framework enables the definition of ∞-operads, wherein
composition is determined up to homotopy via presheaves that satisfy a
weak dendroidal inner Kan condition. Notably, the homotopy theory of
∞-operads has been shown to be equivalent, in both Quillen's sense and
Lurie's ∞-categorical sense, to the theories of topological and
simplicial operads.
In this talk we will review these concepts and present some open
questions in enriched settings.

10:45 → 11:45 Jon Pridham : Derived Poisson structures

Shifted Poisson structures have appeared in many guises over the past
decades. I will give an overview and examples from the perspective of
derived geometry, trying to emphasise similarities with, and differences
from, earlier and parallel approaches.

11:45 → 12:45 Christian Blohmann : The differentiation of higher elastic diffeological groupoids

First, I will review the notion of elastic diffeological spaces, on which there is a natural Cartan calculus. I will define diffeological Lie algebroids, show how they arise from elastic diffeological groupoids, and give their dual description as ringed diffeological spaces with a homological vector field. Then I will explain how to generalize this construction to higher diffeological groupoids, which yields a simple universal formula given by the coend of a cosimplicial-simplicial object in ringed diffeological spaces with a homological vector field. The project is motivated by geometric deformation theory, and is joint work with Lory Kadiyan.

jeudi 6 juillet

08:45 → 09:45 Tilmann Wurzbacher : Some remarks on multisymplectic geometry (or from Volterra to "higher" structures)

Abstract : The Hamilton-Volterra equations were derived by the end of the 19th century but their geometrisation took a long time, culminating in the dynamic Hamilton-DeDonder-Weyl equations. We illustrate geometry and dynamics by examples before discussing the question of observablesin this context.

10:15 → 11:15 Kasia Rejzner : BV formalism in the infinite dimensional setting

Batalin-Vilkovisky (BV) formalism has been originally developed in quantum field theory (QFT) and its finite-dimensional analog can be elegantly formulated in the language of dg-manifolds. The latter is relatively well understood in mathematics, but in QFT it is more natural to use an infinite-dimensional setting. This poses several technical problems, since infinite dimensional calculus comes into play. In this talk, I will review some results obtained in this direction by myself and collaborators over the last 10 years

11:15 → 12:15 Owen Gwilliams : 4-dimensional gauge theories and their holomorphic twists

Yang-Mills theories have had an enormous impact in mathematics, perhaps most famously the Donaldson and Seiberg-Witten invariants, which arise via the topological twists of supersymmetric Yang-Mills theories. In this talk I will describe derived moduli spaces that arise from holomorphic and topological-holomorphic twists of these theories and then explore what it means to quantize such spaces. Given time, we will examine how various notions of duality might carry over to this setting, such as the S-duality that Kapustin and Witten used to offer a physical view on the geometric Langlands program or the Seiberg duality that plays a key role for N=1 theories. This work is joint, in various combinations, with Chris Elliott, Eugene Rabinovich, and Brian Williams.

14:15 → 15:15 Chenchang Zhu : Higher differential geometry and symplectic structure

In this talk, we will review the higher groupoid approach towards higher differentiable stacks and introduce shifted symplectic structures on them. We will give some interesting examples and Morita equivalences for them. It is based on several joint works with Miquel Cueca Ten, Florian Dorsch, Leonid Ryvkin and Stefano Ronchi.

vendredi 7 juillet

08:45 → 09:45 Adrien Brochier : Quantum exponentiation of Hamiltonian Poisson varieties

Let G be a complex reductive algebraic group. Various authors have developed notions of (quas G-Poisson varieties equipped with a multiplicative version of a moment map, valued in the group G. One of the main motivations for that formalism is that it provides a nice combinatorial description of the canonical Atiyah--Bott Poisson structure on character varieties of surfaces. A crucial feature of these structures is that they come equipped with certain operations (fusion and Hamiltonian reduction) which in particular expresses the compatibility of the Poisson structure on character varieties with cutting and gluing of surfaces.

Exponentiation is a procedure taking an ordinary Hamiltonian variety (i.e. a Poisson variety equipped with a moment map into the dual of the Lie algebra g of G) and turning it (formally) into a "multiplicative" one. Crucially, this operation is compatible with fusion and Hamiltonian reduction on both sides, showing that the character variety of a surface is formally Poisson isomorphic to a much simpler Poisson variety. All of these structures/operations have natural interpretations in the framework of shifted Poisson structures. Exponentation, in particular, comes from a certain formal isomorphism of 1-shifted Poisson stacks g^*/G --> G/G.

In this talk, I'll describe a quantization of this construction which is essentially given by pulling back along a certain monoidal functor from quantum to classical Harish-Chandra bimodules, which quantizes the above 1-shifted Poisson map. I'll then explain how this construction is compatible with categorical/quantum analogs of fusion and Hamiltonian reduction, and I'll present some applications.

09:45 → 10:15 Jérémie Pierard de Maujouy : Generalised Frame Bundles and Einstein's Equation

We introduce the structure of generalised frame bundles, which generalises that of frame bundles with connection by keeping only the local part of the structure.
This definition is motivated by a Lagrangian field theory which we will present.
Its solution fields define on the 10-dimensional source manifold a structure of generalised frame bundle which satisfy further equations.
Some of these equations are shown to imply Einstein's equation in the case of a standard frame bundle and therefore can be used as a suitable generalisation thereof.

10:45 → 11:45 Michele Schiavina : Fried's conjecture in quantum field theory

I will discuss a field-theoretic interpretation of Ruelle zeta function, which "counts" prime geodesics on hyperbolic manifolds, as the partition function for a topological field theory (BF) with an axial-type gauge fixing condition available on the unit sphere bundle of the hyperbolic manifold. This suggests a rephrasing of a conjecture due to Fried, on the equivalence between Ruelle zeta function and analytic torsion, as gauge fixing independence in the Batalin--Vilkovisky formalism.
This is based on joint works with C. Hadfield and S. Kandel, as well as with T. Stucker.

11:45 → 12:45 Pavol Severa : Lie bialgebroids and double groupoids in the braided world

Lie bialgebroids (or, more generally, dg Gerstenhaber algebras) and their integrating objects, Poisson groupoids and double symplectic groupoids, have an interesting generalization in the braided world of quasi-Hamiltonian spaces. Examples can be constructed using moduli spaces of flat connection.

lundi 10 juillet

09:00 → 12:00 Higher Structures à la Glanonaise

9:00-9:30 Sébastien Michéa & Philippe Bonneau : Introduction

9:30-10:00 Alessandra Frabetti: Around a Glanon Groupoid

10:00-11:00 COFFEE BREAK

11:00-11:30 Gilles Halbout : TBA

11:30-12:00 Gregory Ginot : Quantization of BF-theory via higher structures

14:00 → 15:00 Seminar: Markus J. Pfaum : Homological reduction as a tool in deformation quantization and an application to the quantization of a lattice gauge model.

Weekly seminar

15:15 → 18:15 Higher Structures à la Glanonaise

15:15-15:45 Camille Laurent-Gengoux : Higher structures for lower problems

15:45-16:15 Friedrich Wagemann : Leibniz Cohomology

16:15- 17:00 COFFEE BREAK

17:00 Remote Social Meeting

mardi 11 juillet

09:00 → 12:00 Higher Structures à la Glanonaise

09:00-09:30 Pierre Bieliavsky (remotely) : TBA

09:30-10:00 Yaël Frégier : Differential Geometry in AI

10:00-10:30 Mathieu Stiénon : Duflo-Kontsevich theorem

10:30-11:30 COFFEE BREAK

11:30-12:00 Olga Kravchenko : Symétrie et antisymétrie - c'est super

12:00-12:30 Ping Xu : BV_infinity quantization of (-1) shifted derived Poisson manifolds