## Online research seminar : Algebra, Geometry & Physics

Tuesdays 2pm

Organisers: Gaëtan Borot (HU Berlin), Yuri Manin (MPIM)

**New** connection link: https://hu-berlin.zoom.us/j/61339297016

For HU students in Maths -- or Physics P27 or P28 -- this is 2SWS and you can get credits by regular attendance and writing at least one report on a chosen talk during the term.

**Upcoming talks**

**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 2021Beware, different zoom meeting ID just for this day: 993 2439 2000(Pass code: 857807)**

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)*TBA*

**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*

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*

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)*TBA*

**27 July 2021**

Samuele Giraudo (Université Gustave Eiffel)*The music box operad*

**No seminar in August**