## Archive of AG&Phy talks

**Archives AG&Phy September 2020-February 2023(jointly HU Berlin/MPIM Bonn with Y.I. Manin)**

**29 Sept. 2020**

Maxim Smirnov (Augsburg Universität)*Residual categories of Grassmannians* [Slides]

Exceptional collections in derived categories of coherent sheaves have a long history going back to the pioneering work of A. Beilinson. After recalling the general setup, I will concentrate on some recent developments inspired by the homological mirror symmetry. Namely, I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.

**6 Oct. 2020**

Noémie Combe (MPI für Mathematik in den Naturwissenschaften, Leipzig)*The NY Gravity operad and the Deligne-Mumford stack with hidden symmetry*

A paracomplex Deligne-Mumford stack is considered. This structure introduces additional conical singularities, in the neighbourhood of which the curvature is negative. In particular, properties are investigated and we establish a relationship between the paracomplex Deligne-Mumford stack and Deligne-Mumford stack. Moreover, we show that the NY Gravity operad associated to this space (introduced by Manin and Combe) has a mixed Hodge structure. In this framework, the strong Hodge theorem extends and the pure Hodge structure (Kähler package) holds. Since we use tools from L^{2} cohomology, a parallel with the model of Atiyah-Hitchin and Donaldson is established.

**13 Oct. 2020**

Gaëtan Borot (HU Berlin)*Geometry of the combinatorial moduli space of curves I *Slides

The moduli space of complex curves has several descriptions, giving the same topological space but different geometric structures. The description in terms of metric ribbon graphs gives it a polytopal complex structure, and Kontsevich gave it an (almost everywhere) symplectic structure used in his proof of Witten's conjecture. I will revisit the associated geometry of this space (or rather of its universal cover, ie Teichmuller space) making it parallel to the Weil-Petersson geometry coming from hyperbolic metrics on surfaces: we will see how to define Fenchel-Nielsen coordinates that are Darboux for Kontsevich symplectic structure. There is in fact a flow, originally studied by Bowditch-Epstein, Mondello and Do, taking hyperbolic geometry to combinatorial geometry, and I will present stronger results about the convergence of this flow. Based on joint work with Jørgen E. Andersen, Séverin Charbonnier, Alessandro Giacchetto, Danilo Lewanski, Campbell Wheeler.

**20 Oct. 2020**

Gaëtan Borot (HU Berlin)*Geometry of the combinatorial moduli space of curves II*Slides

Mirzakhani obtained a topological recursion for the Weil-Petersson volumes of the moduli space of bordered Riemann surfaces, based on a recursive partition of unity on the Teichmuller space that extended an identity by McShane. Her idea can be generalised to study length statistics of multicurves on surfaces. I will explain that it fits in a larger theory of "geometric recursion", allowing to define recursively mapping class group invariant functions on the Teichmuller space, whose integration on the moduli space of curves can be computed by a topological recursion. This can be done both in the hyperbolic setting, or in the combinatorial setting using the results of the first talk. This can be applied to obtain uniform geometric proofs of Witten's conjecture, of Norbury's lattice point counting, and several proofs of topological recursion for Masur-Veech volumes of the moduli space of quadratic differentials.

Based on joint works with Jørgen E. Andersen, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski, Nicolas Orantin, Campbell Wheeler.

**27 Oct. 2020**

Georgios Kydonakis (MPIM Bonn)*Complex projective structures over Riemann surfaces and solutions to Hitchin's equations*

R. Gunning in 1967 defined a projective coordinate system of a compact Riemann surface as one for which the transition functions are given by Möbius transformations. Such structures are alternatively described by particular flat bundles called by Gunning at the time "indigenous bundles". In modern terminology, this pertains to the structure of an SL_{n}(C)-oper as introduced by A. Beilinson and V. Drinfeld, who, more generally, introduced G-opers for any simple simply connected complex Lie group G. We shall focus in this talk on the relationship of these structures to certain families of solutions to Hitchin's equations. Joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke.

**3 Nov. 2020**

Jörg Teschner (DESY Hamburg)*Proposal for a geometric characterisation of topological string partition functions*

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle.

**10 Nov. 2020**

Dirk Kreimer (HU Berlin)*Monodromy in physics amplitudes*

The talk discusses the structure of amplitudes which emerge in high energy physics computations. Such amplitudes appear as 'blobbed' recursions of graphs. Each of them gives rise to a multivalued function often related to (elliptic or generalized) polylogarithms. Their monodromies relate to algebraic and combinatoric properties of graphs. We will discuss this relation in some detail.

**17 Nov. 2020**

Gavril Farkas (HU Berlin)

Compactification of moduli of holomorphic differentials

The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss various compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.

**24 Nov. 2020**

Alexander Thomas (MPIM Bonn)*A geometric approach to Hitchin components via punctual Hilbert schemes*

Hitchin components are generalizations of the classical Teichmüller space. I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface.

**1 Dec. 2020**

Alexi Morin-Duchesne (MPIM Bonn)*Boundary emptiness formation probabilities in the six-vertex model at Delta = - 1/2*

_{N,m}. This observable is defined as the sum of the squares of the ground state components of the Hamiltonian for the chain of length N, restricted to components where $m$ consecutive spins are aligned. At the combinatorial point Delta = -1/2, it takes the form of a simple product of integers. This was shown by Cantini in 2012.

_{N,m}for the spin-1/2 XXZ chain. It is equal to the linear sum of the groundstate components that have $m$ consecutive aligned spins. For reasons that will be discussed, we refer to the ratio C

_{N,m}/C

_{N,0}as the boundary emptiness formation probability. We compute C

_{N,m}at the combinatorial point as a simple product of integers.

**15 Dec. 2020**

Gautier Ponsinet (MPIM Bonn)*Universal norms of p-adic Galois representations and the Fargues-Fontaine curve*

In 1996, Coates and Greenberg computed explicitly the module of universal norms associated with an abelian variety in a perfectoid field extension. The computation of this module is essential to Iwasawa theory, notably to prove "control theorems" for Selmer groups generalising Mazur's foundational work on the Iwasawa theory of abelian varieties over Z_p-extensions. Coates and Greenberg then raised the natural question on possible generalisations of their result to general motives. In this talk, I will present a new approach to this question relying on the classification of vector bundles over the Fargues-Fontaine curve, which enables to answer Coates and Greenberg's question affirmatively in new cases.

**19 Jan. 2021**

Alice Guionnet (ENS Lyon)*Topological expansions, random matrices and operator algebras theory*

Slides

In this lecture, I will discuss the remarkable connection between random matrices and the enumeration of maps and some applications to operator algebras and physics. Part of my talk will be dedicated to my collaboration with Vaughan Jones on this subject.

**26 Jan. 2021**

Bruno Klingler (HU Berlin)*Tame geometry and Hodge theory*

Slides

Originating in Grothendieck's "Esquisse d'un programme", tame geometry has been developed by model theorists under the name "o-minimal structures". It studies structures where every definable set has a finite geometry complexity. It has for prototype real semi-algebraic geometry, but is much richer. After recalling its basic features, I will describe its recent applications to Hodge theory and period maps.

**2 Feb. 2021**

Tomáš Procházka (LMU München)*W algebras and integrable structures*

Slides

_{N}family and the associated W

_{infinity}algebra. These algebraic structures show up at many places in mathematical physics. Winfinity admits two different descriptions: the traditional description starts from the Virasoro algebra of 2d conformal field theory and extends it by local conserved currents of higher spin. The description discovered more recently is the Yangian description manifesting the integrable structure of the algebra. The map between these two pictures is non-trivial, but can be understood by using the Maulik-Okounkov 'instanton' R-matrix as a bridge between these two pictures.

**9 Feb. 2021**

Karim Adiprasito (Hebrew University of Jerusalem)*Combinatorics, Lefschetz theorems beyond positivity and transversality of primes*

I will survey applications of Hodge Theory to combinatorics, and, quite suprisingly, how Hodge-Riemann relations and Lefschetz type theorems can be proven using combinatorial methods, in settings that are beyond classical algebraic geometry, at least as long as some notion of positivity is available.

I then go one step further, and ask how many triangles a PL embedded simplicial complex in R^{4} can have (the answer, generalizing a result of Euler and Descartes, is at most 4 times the number of edges). I discuss how to reduce this problem to a Lefschetz property beyond projective structure.

The main part of the talk is devoted to provide an indication how the proof works, explain the notion of transversal primes as well as Hall matching theorems for spaces of linear operators, and their connection in the Hall-Laman relations which replace our knowledge of the signature of the Hodge-Riemann relations in the Kähler case with nondegeneracy at monomial ideals.

**16 Feb. 2021**

Vivek Shende (U. Berkeley/SDU Odense)*Holomorphic curves, boundaries, skeins, and recursion*

Slides

**23 Feb. 2021**

Vasily Golyshev (MCCME, Moscow)*Non-abelian Abel's theorems and quaternionic rotation*

I will talk on a subject which has evolved over the years in discussions with M. Kontsevich, A. Mellit, V. Roubtsov, and D. van Straten. I will explain how the kernels of non-abelian Abel's theorems can be seen as a low-technology alternative to geometric Langlands, and produce a Clausen-type lift from the Markov to a quaternionic local system on a punctured genus 1 curve

**2 Mar. 2021**

Jean-Louis Colliot-Thélène (CNRS & Université Paris Saclay)*Hilbert's irreducibility theorem and jumps in the rank of the Mordell-Weil group*

Slides

Let k be a number field and U a smooth integral k-variety. Let X → U be an abelian scheme of relative dimension at least one. We consider the set U(k)_{+} ⊂ U(k) of k-rational points m ∈ U(k) such that the Mordell-Weil rank of the fibre X_{m} at m, which is an abelian variety over k, is strictly bigger than the Mordell-Weil rank of the generic fibre X_{k(U)} over the function field k(U). We prove: if the k-variety X is k-unirational, then U(k)_{+} is dense for the Zariski topology on U. If the k-variety X is k-rational, then U(k)_{+} is not a thin set in U. The second result leads us to a discussion of varieties over which Hilbert's irreducibility theorem holds.

**9 Mar. 2021**

Thomas Willwacher (ETH Zürich)*Embeddings of manifolds in euclidean space and Feynman diagrams*

Slides

The spaces of embeddings of manifolds M in R^{n} are classical objects in topology. I will discuss the computation of their rational homotopy type, for fairly general M, based on recent joint work with B. Fresse and V. Turchin. The answer can be formulated in terms of combinatorial (Feynman) diagrams.

**16 Mar. 2021**

Pedro Tamaroff (Trinity College Dublin)*Poincaré-Birkhoff-Witt theorems: homotopical and effective computational methods for universal envelopes*

Slides

In joint work with V. Dotsenko, we developed a categorical framework for Poincaré-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures, and used methods of term rewriting for operads to obtain new PBW theorems, in particular answering an open question of J.-L. Loday. Later, in joint work with A. Khoroshkin, we developed a formalism to study Poincaré–Birkhoff–Witt type theorems for universal envelopes of algebras over differential graded operads, motivated by the problem of computing the universal enveloping algebra functor on dg Lie algebras in the homotopy category. Our formalism allows us, among other things, to obtain a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras, and extend Quillen's quasi-isomorphism C(g) → BU(g) to homotopy Lie algebras. I will survey and explain the role homological algebra, homotopical algebra, and effective computational methods play in the main results obtained with both V. Dotsenko (1804.06485) and A. Khoroshikin (2003.06055) and, if time allows, explain a new direction in which these methods can be used to study certain operads as universal envelopes of pre-Lie algebras.

**23 Mar. 2021**

Gereon Quick (NTNU Oslo)*From rational points to étale homotopy fixed points*

Slides

**30. Mar. 2021**

Mark Gross (Cambridge Univ.)*Intrinsic mirror symmetry*

Slides

I will talk about the program developed jointly with Bernd Siebert to understand mirror symmetry. In particular, I will focus on the general construction of mirrors. Associated to any log Calabi-Yau pair (X,D) with maximally degenerate boundary D or to any maximally unipotent degeneration of Calabi-Yau manifolds X→S, we associate a mirror object as either the Spec or Proj of a ring. This ring, analogous to the degree 0 part of symplectic homology of X\D, is constructed using a flavor of Gromov-Witten invariants called punctured invariants, developed jointly with Abramovich, Chen and Siebert.

**6 Apr. 2021**

François Loeser (Institut de Mathématiques de Jussieu)*Limits of complex integrals and non-archimedean geometry*

Slides

**13 Apr. 2021**

Hülya Argüz (Université Versailles St-Quentin)*Donaldson-Thomas invariants of quivers with potentials from the flow tree flormula*

Slides

A categorical notion of stability for objects in a triangulated category was introduced by Bridgeland. Donaldson-Thomas (DT) invariants are then defined as virtual counts of semistable objects. We will focus attention on a natural class of triangulated categories defined via the representation theory of quivers with potentials, and explain how to compute DT invariants in this case from a smaller subset of "attractor invariants'' which are known in many cases. For this we investigate wall-crossing in the space of stability conditions, and prove a flow tree formula conjectured by Alexandrov-Pioline in this setup. This is joint work with Pierrick Bousseau.

**20 Apr. 2021**

Anna Wienhard (Heidelberg Universität)*Positivity, higher Teichmüller spaces and (non-commutative) cluster algebras*

Slides

Higher Teichmüller theory emerged as a new field in mathematics about twenty years ago. It generalizes Thurston’s view on Teichmüller space, but also draws on new connections to representation theory, Higgs bundles, theoretical physics and cluster algebras. In this talk I will provide an overview of the different approaches to higher Teichmüller spaces, and explain how a generalization of total positivity provides a (conjectural) unifying framework. It also leads to new predictions and connections to Lie groups over non-commutative rings.

**27 Apr. 2021**

Yuri B. Suris (TU Berlin)*Bilinear discretization of quadratic vector fields: integrability and geometry*

Slides

We discuss dynamics of birational maps which appear as bilinear discretizations of quadratic vector fields. The corresponding dynamical systems turn out to be integrable much more often than could be expected. Various aspects of integrability of birational dynamical systems will be discussed, along with remarkable geometric structures behind some of the particular examples.

**4 May 2021**

Yuri Manin (MPIM Bonn)*Non-associative Moufang loop of point classes on a cubic surface*

Slides

In this talk I will explain, how the problem stated in my book *Cubic Surfaces*, was recently solved by Dimitri Kanevsky, after about fifty years since its publication.

**11 May 2021**

Omid Amini (École Polytechnique)*Hybrid geometry of curves and their moduli spaces*

It is now well-understood that the Deligne-Mumford compactification of the moduli spaces of curves is not large enough for capturing several limit phenomena concerning geometry of curves and their families.

The aim of this talk is to present a hybrid refinement of the Deligne-Mumford compactification, that uses the combinatorics and geometry of graphs and their minors (in the sense of Robertson-Seymour graph minor theory), and which allows to address some of these limit questions arising in the study of Riemann surfaces and their asymptotic geometry.

Among the resulting applications, I will provide a complete solution to the problem of limits of Arakelov-Bergman measures and present a refined analysis of the degenerations of Arakelov Green functions, close to the boundary of the moduli spaces.

The talk is based on joint works with Noema Nicolussi.

**18 May 2021**

Inna Zakharevich (Cornell)*Point counting to detect non-permutative elements of K _{1}(Var)*

Slides

The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field k) modulo the relation that for a closed immersion Y → X there is a relation that [X] = [Y] + [X \ Y]. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field k is finite for a variety X we can consider the "almost-finite" set X(kbar), which represents the local zeta function of X. In this talk we will discuss how to detect interesting elements in K_{1}(Var) (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on X.

**25 May 2021**

Will Sawin (Columbia University)*Is freeness enough for counting rational points ?*

Slides

Manin's conjecture predicts the number of rational points of bounded height on a Fano variety, but for the predictions to hold, we must first remove a "thin set" consisting of rational points lying on certain special subvarieties and lifting to certain special covering spaces. It might be better if we could instead identify the bad rational points to remove by their intrinsic geometry. It may seem that rational points do not have any intrinsic geometry, but recently Peyre has given two proposals to do this, one measuring the freeness of a point and the other using all the heights. I will explain why the freeness proposal is not, alone, sufficient.

**1 June 2021**

Nathalie Wahl (University of Copenhagen)*String topology operations*

String topology for a manifold can be defined as a certain set of operations on the homology of its free loop space, that is the space of all maps from a circle into the manifold. I’ll give an overview of some of the many non-trivial string topology operations we know, both from an algebraic model (Hochschild homology) perspective, and from a more geometric perspective, as directly defined on the homology of the loop space.

**8 June 2021**

Jacopo Stoppa (SISSA)*Log Calabi-Yau surfaces and Jeffrey-Kirwan residues*

I will discuss joint work in progress with Riccardo Ontani (SISSA). We use Gross-Hacking-Keel mirror symmetry for log Calabi-Yau surfaces in order to provide a geometric interpretation for certain remarkable formulae appearing in the physical literature, in the context of supersymmetric gauge theories, which involve Jeffrey-Kirwan residues of meromorphic forms.

**15 June 2021**

Andras Szenes (Université de Genève)*The Verlinde formula and parabolic Hecke correspondences*

The Verlinde formula for the Hilbert function of the moduli space of vector bundles on a Riemann surface is one of the most fascinating results in enumerative geometry. I will review several approaches to this theorem, and then present a brand new proof (joint work with Olga Trapeznikova) based on a new look at the Drinfeld-Hecke correspondences on curves._{Chaired by Alexander Thomas.}

**22 June 2021**

Nicolò Sibilla (SISSA)*Fukaya category of surfaces and pants decompositions*

In this talk I will explain some results joint with James Pascaleff on the Fukaya category of Riemann surfaces. I will explain a local-to-global principle which allows us to reduce the calculation of the Fukaya category of surfaces of genus g greater than one to the case of the pair-of-pants, and which holds both in the punctured and in the compact case. The starting point are the sheaf-theoretic methods which are available in the exact setting, and which I will review at the beginning of the talk. This result has several interesting consequences for HMS and geometrization of objects in the Fukaya category. The talk is based on 1604.06448 and 2103.03366.

**29 June 2021**

Ashkan Nikeghbali (Zürich Universität)*Convergence of random holomorphic functions with real zeros, random matrices and the distribution of the zeros of the Riemann zeta function*

The GUE conjecture states that the ordinates of the zeros of the Riemann zeta function on the critical line should behave statistically like eigenvalues of large random matrices: more precisely they should be asymptotically distributed like a sine kernel determinantal point process. In the past two decades, a model has emerged to understand and predict the distribution of values of the Riemann zeta function on the critical line: the characteristic polynomial of random unitary matrices. It has been thought that there should exist a random holomorphic emerging as some scaling limit of the characteristic polynomial. We give a construction of this function and describe its relation to the GUE conjecture. We then show how it naturally appears in ratios in random matrix theory. We then discuss several generalisations of this construction by other authors as well as by J. Najnudel and myself.

**6 July 2021**

Marta Mazzocco (Birmingham University)*Isomonodromic deformations: confluence, reduction and quantization*Slides

In this talk we study the theory of isomonodromic deformations for systems of differential equations with poles of any order on the Riemann sphere. Our initial motivation was to generalise a theorem by Reshetikhin that the quasiclassical solution of the standard KZ equations (i.e. with simple poles) is expressed via the isomonodromic τ-function arising in the case of Fuchsian systems. Along the way of pursuing this project, we have found a number of interesting results, some of which were already known as folklore (i.e. either done is very specific examples or not really proved formally), others completely original.

**13 July 2021**

Piotr Sulkowski (Warsaw University)*Permutohedra for knots and quivers*Slides

The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be assigned to a given knot and encode the same information. I will explain that this phenomenon is generic rather than exceptional. First, I will present conditions that characterize equivalent quivers. Then I will show that equivalent quivers arise in families that have the structure of permutohedra, and the set of all equivalent quivers for a given knot is parameterized by vertices of a graph made of several permutohedra glued together. These graphs can be also interpreted as webs of dual 3d N=2 theories. All these results are intimately related to properties of homological diagrams for knots, as well as to multi-cover skein relations that arise in counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds.

**20 July 2021**

Margaret Bilu (IST Austria)*Zeta statistics*

Slides

Many questions in number theory have a natural analogue, of more geometric nature, formulated in the Grothendieck ring of varieties. For example, Poonen's finite field Bertini theorem has a motivic counterpart due to Vakil and Wood; however, despite the clear similarities between these two results, none of the two can be deduced from the other. The aim of this talk is to describe and motivate a conjectural way of comparing such statements in arithmetic and motivic statistics, by reformulating them in terms of the convergence of zeta functions in different topologies. We will finish by mentioning some concrete settings where our conjectures are satisfied. This is joint work with Ronno Das and Sean Howe.

**27 July 2021**

Samuele Giraudo (Université Gustave Eiffel)*Operads of musical phrases and generation*Slides

Operads can be used as a framework to represent musical phrases. A musical phrase is seen in this way as an operation that can be composed with some others to construct bigger phrases. We recall the T construction, a functorial construction from the category of monoids to the category of operads which is used to build the music box operad encoding musical phrases. We then describe how to use it to randomly generate some elements from small musical phrases. This is based on specific generation algorithms using colored operads and bud generating systems. The latter are generalizations of context-free grammars.

**Wintersemester 2021-2022**

**19 Oct. 2021**

*Topological recursion in Hurwitz theory*

**26 Oct. 2021**

*Super J-holomorphic curves*

J-holomorphic curves are of great interest because they allow to construct invariants of symplectic manifolds and those invariants are deeply related to topological superstring theory. A crucial step towards Gromov–Witten invariants is the compactification of the moduli space of J-holomorphic curves via stable maps which was first proposed by Kontsevich and Manin. In this talk, I want to report on a supergeometric generalization of J-holomorphic curves and stable maps where the domain is a super Riemann surface. Super Riemann surfaces have first appeared as generalizations of Riemann surfaces with anti-commutative variables in superstring theory. Super J-holomorphic curves couple the equations of classical J-holomorphic curves with a Dirac equation for spinors and are critical points of the superconformal action. The compactification of the moduli space of super J-holomorphic curves via super stable maps might, in the future, lead to a supergeometric generalization of Gromov-Witten invariants. Based on arXiv:2010.15634 [math.DG] and arXiv:1911.05607 [math.DG], joint with Artan Sheshmani and Shing-Tung Yau.

**2 Nov. 2021**

**9 Nov. 2021**

**16 Nov. 2021**

*The Gopakumar–Vafa finiteness conjecture*

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their

proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s mcompactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

This talk is based on joint work with Eleny Ionel and Aleksander Doan.

**23 Nov. 2021**

*4D/2D duality and representation theory*

**30 Nov. 2021**

*Geometry and spectrum of random hyperbolic surfaces*

**7 Dec. 2021**

profound connections between arithmetic counting problems and the spectrum of

automorphic representations. This is a joint work with A. Nevo.

**14 Dec. 2021**

**21 Dec. and 28 Dec. 2021**

**4 Jan. 2022**

*Modular bootstrap, Segal axioms and resolution of Liouville field theory*

**11 Jan. 2022**

*Two types of integrability in Liouville quantum gravity*

**18 Jan. 2022**

*Introduction to Categorification, Diagrammatics and the Hecke Category*

The Hecke category is a ubiquitous object in representation theory. It can be constructed using sheaves on the flag variety, or certain infinite-dimensional representations (in the Bernstein-Gelfand-Gelfand category O). It acts on rational representations of algebraic groups, and on other categories in modular representation theory. However, a modern perspective is to describe the category by generators and relations, which has many advantages: it allows computers and humans (even graduate students) to perform calculations, it gives access to finite characteristic settings, it permits easier constructions of actions, etcetera. The main drawback to generators and relations is not for the end-user but for the originator: it is not easy to prove that a presentation is correct. For example, there are many non-trivial presentations of the trivial group.

This talk will begin by discussing some of the philosophy of categorification and its important shadow, the Hom form. Then we introduce the Kazhdan-Lusztig problem, to motivate the question of finding a presentation for the Hecke category. First we'll address an easier problem: finding a presentation for the 2-groupoid of the symmetric group, using this as an opportunity to introduce diagrammatic methods. After the break, we'll discuss how the presentation of the Hecke category was found, and also how it was proven, in order to help those who might wish to follow a similar path.

This talk is on work (joint with Khovanov and Williamson) much of which was originally done at the Max Planck Institute in Bonn over a decade ago.

**25 Jan. 2022**

*Pressure metric in the space of Riemannian metrics*

**1 Feb. 2022**

*Higher algebras in supergravity and M-theory*

framework called "exceptional field theory" that explains these symmetries. It is based on a novel kind of geometry, in which the diffeomorphism invariance underlying general relativity is unified with various "higher-form" symmetries. I elaborate on the higher algebraic structures that these generalized diffeomorphisms display, including Lie-infinity and Leibniz-Loday algebras.

**8 Feb. 2022**

*The tensor HCIZ integral and its relation to enumerative geometry and free probability*

_{3,p}.

♦ Free probability consists in the study of non-commutative random variables, such as random matrices in the limit of infinite size. Free cumulants are one of the central objects of free probability, and it is known that certain fundamental properties of free cumulants can be recovered from the original HCIZ integral. I will present early results on the way that our generalization of the HCIZ integral allows adapting the tools of free probability to the study of large random tensors.

**15 Feb. 2022**

*On the arithmetic of a classical family of elliptic curves with complex multiplication*

I will study the family of elliptic curves C

_{N}/Q of the form x

^{3}+y

^{3}=Nz

^{3}for any cube-free positive integer N. They are cubic twists of the Fermat elliptic curve x

^{3}+y

^{3}=z

^{3}, and they admit complex multiplication by the ring of integers of the imaginary quadratic field Q(√-3). The celebrated conjecture of Birch and Swinnerton-Dyer is one of the most important open problems in number theory concerning elliptic curves. The p-part of the conjecture has been settled for these curves for all primes p not equal to 2 or 3 by K. Rubin using powerful techniques from Iwasawa theory. The aim of this talk is to study the conjecture at the remaining primes. First, I will establish a lower bound for the 3-adic valuation of the algebraic part of their central L-values in terms of the number of distinct prime divisors of N. I will then show that the bound is sometimes sharp, which gives us the 3-part of the conjecture for C

_{N}/Q in certain special cases. In addition, I will study the non-triviality and growth of the 2-part and the 3-part of their Tate-Shafarevich group.

*Categorical dynamics: from Hopfield to Pareto*

In recent work with Yuri Manin, we proposed a model of neural information networks based on functorial

assignments of resources to networks, originating in Segal's notion of summing functors and Gamma spaces, and a corresponding categorical form of Hopfield equations on networks. I will review some properties and examples of such categorical framework and dynamics, and I will show how it can applied to a form of Pareto optimization.

**15 Mar. 2022**

*A new proof of positivity of DT invariants of symmetric quivers*

**22 Mar. 2022**

*The Hecke correspondence via Yang-Mills-Higgs flow lines*

**Sommersemester 2022**

**19. Apr. 2022**

Jun-Yong Park (MPIM Bonn)*Arithmetic topology of the moduli stack of Weierstraß fibrations over global function fields*We will first consider the formulation of the moduli of fibered algebraic surfaces as the Hom space of algebraic curves on moduli stacks of curves. Cohomology with weights on these moduli naturally allows us to enumerate elliptic & hyperelliptic curves over global function fields ordered by bounded discriminant height. In the end, we formulate analogous heuristics for parallel countings over number fields through the global fields analogy. This is a joint work with Oishee Banerjee (Bonn) and Johannes Schmitt (Zürich).

**26. Apr. 2022**

*Hecke operators over local fields and an analytic approach to the geometric Langlands correspondence*

I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan, arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists, Kontsevich, Langlands, Teschner, and Gaiotto-Witten. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun_G of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.

**3. May 2022**

*The mysterious loop invariants of a knot*

A main problem in quantum topology is the Volume Conjecture which asserts that an evaluation of the colored Jones polynomial (known as the Kashaev invariant) is a sequence of complex numbers that grows exponentially at the rate of the hyperbolic volume of a knot complement. This conjecture connects the Jones polynomial with hyperbolic geometry. The loop invariants are the refinement of the above conjecture to all orders in perturbation theory, and take values in the trace field of a knot. Hence, the loop invariants have topological, but also mysteriously geometric origin. A geometric definition of them is

currently unknown. In the talk we will discuss how these invariants behave under finite cyclic covers, and give clues about their possible geometric definition. Joint work with Seokbeom Yoon.

**10. May 2022**

**17. May 2022**

free cumulants, which are fundamental quantities in non-commutative versions of probability theory.

**24. May 2022**

*The Barr-Beck Theorem in Symplectic Geometry*

_{1}) and Fuk(M

_{2}) associated to a Lagrangian correspondence L

_{12}and its transpose. These functors are often adjoint, and under the hypothesis that a certain map to symplectic cohomology hits the unit, the hypotheses of Barr-Beck are satisfied. This can be interpreted as an extension of Abouzaid’s generation criterion, and we hope that it will be a useful tool in the computation of Fukaya categories.

**31. May 2022**

*Bimodules over categories enriched over closed monoidal categories and A∞-categories*

_{∞}-category has to be extended in order to have wider applications.

We propose to extend it to the context of bimodules over V-categories. The ground category V is assumed to be additive, closed symmetric monoidal, complete and cocomplete. We propose a framework for enriched A

_{∞}-categories: we choose a V-category D, then we define an A

_{∞}-category as a tensor D-bimodule equipped with the deconcatenation comultiplication and coderivation whose square in a sense is 0.

**7. June 2022**

*Flat F-manifolds, F-CohFT and integrable systems*

_{n}equivariance is broken to S

_{n-1}-equivariance) and a boundary condition is imposed at separating nodes only (the node is formed by attaching the special point on one component and a normal one on the other). If CohFTs reduce to Dubrovin-Frobenius manifolds when restricted to genus 0, F-CohFTs reduce to flat F-manifolds. In this talk I will explain how the well known-relation between CohFTs and integrable systems can be extended to F-CohFTs, at the cost of losing the Hamiltonian structure. I will also mention that the well known Givental-Teleman reconstruction result of a semisimple CohFT from its Frobenius manifold can be generalized to semisimple F-CohFTs. All this is the fruit of a joint work with A. Arsie, A. Buryak and P. Lorenzoni.

**14. June 2022**

*Open enumerative geometry for Landau-Ginzburg models and Mirror Symmetry*

A Landau-Ginzburg (LG) model is a triplet of data (X,W,G) consisting of a regular complex-valued function W from a quasi-projective variety X with a group G acting on X so that W is invariant. An enumerative theory developed by Fan, Jarvis and Ruan gives FJRW invariants, an analogue of Gromov-Witten invariants, for LG models. We define an open enumerative theory for certain LG models, building on the FJRW point of view. Roughly speaking, our theory involves computing specific integrals on certain moduli of discs with boundary and interior marked points. One can then construct a mirror LG model to the original one using these invariants. This allows us to prove a mirror symmetry result analogous to that established by Cho-Oh, Fukaya-Oh-Ohta-Ono and Gross in the context of mirror symmetry for toric Fano manifolds. If time permits, I will explain some key features that this enumerative geometry enjoys (e.g., open topological recursion relations and wall-crossing). This is joint work with Mark Gross and Ran Tessler.

**21. June 2022**

*Veronese subalgebras and Veronese morphisms for a class of Yang-Baxter algebras*

We study d-Veronese subalgebras A^{(d)} of quadratic algebras A_{X=A(K, X, r)} related to finite nondegenerate involutive set-theoretic solutions (X, r) of the Yang-Baxter equation, where K is a field and d > 1 is an integer. We find an explicit presentation of the d-Veronese A^{(d)} in terms of one-generators and quadratic relations.

We introduce the notion of a d-Veronese solution (Y, r_{Y}), canonically associated to (X, r) and use its Yang-Baxter algebra A_{Y}= A(K, Y, r_{Y}) to define a Veronese morphism v_{n,d}:A_{Y→}A_{X}. We prove that the image of v_{n,d} is the d-Veronese subalgebra A^{(d)}, and find explicitly a minimal set of generators for its kernel. Finally, we show that the Yang-Baxter algebra A(K, X, r) is a PBW algebra if and only if (X, r) is a square-free solution. In this case the d-Veronese A^{(d) }is also a PBW algebra.

**28. June 2022**

*Renormalisation of stochastic partial differential equations*

In this talk, we will present the main ideas of the renormalisation of stochastic partial differential equations (SPDEs), as it appears in the theory of regularity structures. It is crucially based on the notion of a model that is a collection of stochastic integrals recentered around a base point and renormalised. They are used for Taylor-expanding solutions of singular SPDEs. We will discuss the transformation of the canonical model to the renormalised one and the underlying algebraic structure which are Hopf algebras on decorated trees.

**5. July 2022**

In this talk, I will show a link between an invariant of singularities (involving arc spaces) and Rogers-Ramanujan identities. These latter are identities concerning the theory of integer partitions which have a long history in classical number theory; an integer partition of an integer n being simply a decreasing sequence of positive integers whose sum is equal to n. I will then explain how this link allows to find and prove new families of partition identities. The talk is directed at a broad public. It concerns various works with C. Brucheck and J. Schepers, with P. Afsharijoo and with P. Afsharijoo, J. Dousse and Frédéric Jouhet.

**Wintersemester 2022/23**

**18. Okt. 2022**

Ralph Kaufmann (Purdue University)

Slides*Universal operations on the Tate-Hochschild complex*

_{∞}structure with all m

_{i}= 0 for i >4, as was shown by Rivera and Wang. Together with Rivera and Wang, we show that these operations are part of a universal family of operations obtained analogously as the operations on the Hochschild complex we previously defined. This allows us to identify a series of higher bracket operations of which the bi-bracket is dual to the m

_{3}operation and the tri-bracket guarantees the higher associativity. Other operations guarantee the Poisson property of the bi-bracket. We will introduce this formalism and comment on how this leads to a new type of bordification of the Chas-Sullivan string topology space.

**25. Okt. 2022**

Ran Tessler (Weizmann Institute)

Slides*New open r-spin theories*

**1. Nov. 2022**

*r-th roots: better negative than positive*

**8. Nov. 2022**

*Solutions of the Yang-Baxter equation: groups, algebras and braces*

In 1992, Drinfel'd suggested the study of set-theoretic solutions of the Yang-Baxter equation. The seminal papers of Etingof, Schedler and Soloviev, and Gateva-Ivanova and Van den Bergh studied the structure group G(X,r) and structure monoid M(X,r) for the subclass of involutive non-degenerate solutions and their monoid algebras. These algebraic structures encode the combinatorial structure of the solution (X,r) and are of importance as their monoid algebra is a quadratic algebra. In recent joint works with I. Colazzo, E. Jespers, L. Kubat and C. Verwimp, we study the structure monoid for the larger class of left non-degenerate solutions. Furthermore, we obtain results on the finiteness properties of the associated quadratic algebras.

In the second part of the talk, we discuss skew left braces. These algebraic structures generate and govern non-degenerate set-theoretic solutions and were recently introduced by W. Rump, and L. Guarnieri and L. Vendramin. Intuivitely, a skew left brace is a set with two group operations that are related via a skew left distributivity condition.

We discuss some recent works, joint with E. Jespers, L. Kubat and L. Vendramin. In particular, we discuss radicals of skew left braces. Last, to illustrate that the study of skew left braces is a melting pot of different techniques, we present a recently unexpected connection (by A. Smoktunowicz) between pre-Lie algebras and skew left braces.

Throughout the talk, we will mention open problems and avenues for further research.

**15. Nov. 2022**

*The shapes of complementary subsurfaces to simple closed hyperbolic multi-geodesics*

**22. Nov. 2022**

*Local wild mapping class groups*

**29. Nov. 2022**

*Loop-tree duality for integrals on the moduli space of graphs*

**6. Dez. 2022**

**13. Dez. 2022**

*On the topology of character varieties of once-punctured torus bundles*

Let M be a complete hyperbolic 3-manifold of finite volume. The seminal work of Thurston and Culler-Shalen established the SL(2,C)-character variety of the fundamental group of M as a powerful tool in the study of the topology of M. This talk focusses on the particular class of manifolds that are hyperbolic once-punctured torus bundles. These are generally very well understood. Yet, there are some interesting open questions regarding their character varieties, especially concerning their topology and how much topological information can be obtained from them about the bundles.

This talk gives a quick overview of Culler-Shalen theory, introduces the manifolds in the title, and explains some work with Youheng Yao (arXiv:2206.14954) concerning their character varieties.

**3. Jan. 2023**

*Phantom minimal series and the Peter--Weyl theorem for loop groups*

Let G be a complex reductive group. The celebrated Peter--Weyl theorem decomposes the algebra of functions on G as a G x G module with respect to left and right translations. In this talk we introduce a natural analogue for the loop group G((z)). A key role is played by a family of G((z)) representations at negative level, the phantom minimal series. These are dual, in a precise but somewhat subtle homological sense, to the more familiar positive energy representations at positive level. Time permitting, we will discuss the existence of phantom minimal series for many related vertex algebras, and some interesting analytic properties of their characters.

**10. Jan. 2023**

*Quantum cohomology and derived categories of coadjoint varieties*

We will discuss properties of quantum cohomology, both small and big, of coadjoint varieties of simple algebraic groups and how they relate to the structure of Lefschetz collections in the derived categories of these varieties.Some general conjectures pertaining to this will be formulated. The talk is based on the joint works with Alexander Kuznetsov and Nicolas Perrin.

**17. Jan. 2023**

**24. Jan. 2023**

*Museum of visual Asymptotic Representation Theory*

**31. Jan. 2023**

**7. Feb. 2023**

Slides

*Interacting particle systems and random walks on Hecke algebra*

**14. Feb. 2023: cancelled**

**21. Feb. 2023**

Algebra, Geometry & Physics: a homage

Not at HU, just in MPIM Lecture Hall and online:

Algebra, Geometry & Physics: a homage

**https://zoom.us/j/91253700254**

1.30-1.45pm: Mikhail Kapranov (IPMU Tokyo)

*Historical remarks*

*Reduction theory and periods of modular forms*

Among Manin's most beautiful and influential contributions to number theory was his study of periods of modular forms, in particular the theory of modular symbols and his algebraicity theorem for the periods of cusp forms, both of which are related to the theory of continued fractions. After reviewing this material, I will turn to the inverse problem of determining a cusp form from its periods and will describe a complete solution for the case of the full modular group. This result, which I found more than 25 years ago and had

always intended to dedicate to Manin, depends on a simple but surprising lemma about continued fractions. But for the solution of the corresponding result for other Fuchsian groups I needed to establish a rather beautiful statement about reduction theory that I discovered experimentally and checked in many cases, so I kept postponing the publication and dedication until this question could be resolved and I could give the complete result. This still has not happened, and I have decided to belatedly present what I already know.

*The last lecture: Computability questions in the sphere packing problem*

I will dedicate this last lecture of Manin's "Algebra, Geometry, and Physics" seminar

to present his last work (and our last joint work) on computability questions arising

naturally in the context of the sphere packing problem. I will show how our

previous results on Kolmogorov complexity and the asymptotic bound for error correcting

codes can provide some insight on this problem.