# This week

#### October 20, 2017, 17:00: Sergey Arkhipov (Aarkhus) Braid relations in the affine Hecke category and differential forms with logarithmic singularities.

We recall the even and odd algebro-geometric realizations of the affine Hecke category - one via equivariant coherent sheaves on the Steinberg variety and the other in terms of some equivariant DG-modules over the DG-algebra of differential forms on a reductive group G.

The latter one has a toy analog called the coherent Hecke category. It contains certain canonical objects satisfying braid relations via convolution. The proof uses simple facts from the geometry of Bott-Samelson varieties.

Our goal is to provide a similar proof of braid relations in the affine Hecke category. It turns out that canonical braid group generators are given by certain DG-modules of logarithmic differential forms and braid relations follow immediately from a general statement which seems to be new: direct image of the DG-module of logarithmic differential forms does not depend on a resolution of singularities.

The talk is organized jointly with the Mirror Symmetry lab

# Future seminars

#### November 3, 2017, 15:30: John Alexander Cruz Morales (Universidad Nacional de Colombia) On Stokes matrices for Frobenius manifolds

In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral matrices and their relations with mirror symmetry. This is part of an ongoing project with Maxim Smirnov which extends previous work with Marius van der Put for the case of quantum cohomology of projective and weighted projective spaces to other Frobenius manifolds not necessarily of quantum cohomology type.

#### November 3, 2017, 17:00: Grigory Mikhalkin (Geneva) -

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#### Wednesday, November 15, 2017, 17:00: Maxim Smirnov (MPIM Bonn) On Lefschetz exceptional collections and quantum cohomology of Grassmannians.

Given a Lefschetz exceptional collection on a variety X one defines its residual subcategory as the orthogonal to the rectangular part of the collection. In this talk we will discuss some (partially) conjectural relations between the quantum cohomology of X and the structure of the residual subcategory in the case of ordinary and symplectic isotropic Grassmannians. The talk is based on joint works, some finished and some still in progress, with A. J. Cruz Morales, S. Galkin, A. Mellit, N.Perrin, and A. Kuznetsov.

#### November 17, 2017: Artan Sheshmani (Harvard) Nested Hilbert schemes, local Donaldson-Thomas theory, Vafa-Witten / Seiberg-Witten correspondence

We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki's wall- crossing techniques) enables one to obtain a relations between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants and some modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau following arXiv:1701.08902 and arXiv:1701.08899.

#### November 24, 2017: Grzegorz Kapustka (Jagiellonian University, Kraków) On the Morin problem

We will study the Morin problem and present a method of classification of finite complete families of incident planes in P5. As a result we prove that there is exactly one, up to Aut(P5), configuration of maximal cardinality 20 and a unique one parameter family of 19 planes. Finally I will discuss the Morin problem in higher dimensions in particular study configurations of P3 in P9 and their relations with polarised hyper-Kahler fourfolds of K3^[2] type and Beauville-Bogomolov degree 4.

TBA

# Past seminars

#### January 15, 2016: (, , Aix-Marseille) " "

We obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus and give a limit theorem for these flows http://arxiv.org/abs/0804.3970v4

#### January 22, 2016: Dmitry Doryn (IBS, Pohang) Feynman periods: numbers and geometry.

I will speak on the Feynman periods, the values of Feynman integrals in (massless, scalar) phi^4 theory, from the number-theoretical perspective. Then I define a closely related geometrical object, the graph hypersurface. One can try to study the geometry of these hypersurfaces (cohomology, Grothendieck ring, number of rational points over finite fields) and to relate it to the periods. The most interesting results come out from the study of the c_2 invariant (on the arithmetical side).

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#### March 9 (Wednesday), 2016, 18:30, room 1001, Gueo Grantcharov (FIU, Miami): On some examples of special non-Kaehler metrics

We consider two types of non-Kaehler metrics -- balanced and astheno-Kaehler. There is an opinion that a compact complex manifold can not admit both, even if they are different. We provide examples on twistor spaces and homogeneous manifolds, which partly support such an opinion.

#### March 11, 2016, ()

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#### March 25, 2016, Alexei Pirkovskii (HSE) , q-

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#### March 30, 2016, 18:30 (Wednesday), Hamid Ahmadinezhad (Bristol) Birational classification of 3-fold Fano-Mori spaces; a new outlook.

Abstract: I will give an overview of the geometry of Fano 3-folds after Mori theory. After discussing past approaches to the classification, I will highlight why such attempts seem hopeless. Building on recent advances in the geometry of Fanos, I introduce a new viewpoint on the classification problem. A main emphasis will be given to the unpredicted behaviour of the first examples of non-complete intersection biratinally rigid Fanos, discovered in a joint work with Takuzo Okada. I will also talk about the other end of the spectrum, the rational Fanos.

#### April 1, 2016, 17:00, (Glasgow University) V -

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#### April 8, 2016, 17:40, room 1001, Christian Liedtke (Technische Universität München), Good Reduction of K3 surfaces

By a classical theorem of Serre and Tate, extending previous results of Neron, Ogg, and Shafarevich, an Abelian variety over the field of fractions K of a local Henselian DVR has good reduction if and only if the Galois action on its first l-adic cohomology is unramified ("no monodromy"). We show that if the Galois action on second l-adic cohomology of a K3 surface over K is unramified, then the surface admits an RDP model'', and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction.) Moreover, we give examples where such an unramified extension is really needed. This is joint work with Yuya Matsumoto.

#### April 15, 2016, "Topology day in HSE"

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#### April 22, 2016, 17:00, room 1001, Marco Mazzucchelli (CNRS, ENS de Lyon) On the multiplicity of isometry-invariant geodesics.

Abstract: The problem of isometry-invariant geodesics, introduced by K. Grove in the 70s, is a generalization of the closed geodesics one: given an isometry of a closed Riemannian manifold, one looks for geodesics on which the isometry acts as a non-trivian translation. In this talk, after recalling the framework of the problem, we present a few new multiplicity results on certain classes of Riemannian manifolds. We will also discuss a contact-geometric generalization: the existence problem for Reeb orbits that are invariant under a strict contactomorphism. Part of the talk is based on a joint work with Leonardo Macarini.

#### April 26, 2016 (Tuesday), 18:30, room 1001, Daniil Rudenko (HSE), Goncharov conjectures and functional equations for polylogarithms

Classical polylogarithms and functional equations which these functions satisfy have been studied since the beginning of the 19th century. Nevertheless, the structure of these equations is still understood very poorly. I will explain an approach to this subject, based on the link between polylogarithms and mixed Tate motives.

A substantial part of the talk will be devoted to the explanation of this link, provided by Goncharov Conjectures. After that, I will present some results about functional equations which can be proved unconditionally. If time permits, I will finish with another application of this circle of ideas to scissor congruence theory.

#### April 27, 2016 (Wednesday), 18:30, room 1001, Alexander Tikhomirov(HSE), On the geography of the moduli space of semistable rank two sheaves on projective space

We study the Gieseker-Maruyama moduli space M(n)=M_{P^3}(2;0,n,0) of semistable rank two coherent sheaves with Chern classes c_1=c_3=0 and c_2=n>0 on the projective space P^3. It contains as an open subset the moduli space M*(n) of rank two stable vector bundles with c_1=0 and c_2=n on P^3. In 1988 L.Ein showed that the number of irreducible components of M*(n) is unbounded as n grows.However, M(n) contains also irreducible components having non-locally free sheaves as their generic points. The first example of this phenomenon for n=2 was found by G. Trautmann and J. Le Potier in 1993. The aim of this talk is to provide an explicit construction of a big number of new irreducible components of M(n) having as their generic points the sheaves with 0-dimensional and 1-dimensional singularities, respectively. We show that in both cases the number of such components is unbounded as n grows. T his is a joint work with M. Jardim and D. Markushevich.

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#### April 29, 2016 (Friday), 17:00, room 1001, Pierre Cartier (IHES, France), Jet spaces and Witt vectors, an analogy

We shall describe a new inductive construction of jet spaces of curves in a manifold . Using the analogy from Witt vectors with jet spaces in a p-adic world , developed by A. Buium , we propose an new inductive construction of the rings of Witt vectors of finite length.

#### May 6, 2016 (Friday), 17:00, room 1001, Misha Verbitsky (HSE), .

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#### May 13, 17:00, Anton Ayzenberg (HSE) - .

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#### May 20, 17:00, Anatoly Vershik (PDMI) How to classify the filtrations, --- e.g. the decreasing sequences of sigma- algebras or algebras

The Classification of the filtrations appears in the theory of dynamical systems, statistical physics, theory of stochastic processes and in classical analysis. This is the art of general problem: how to make the link between classification of finite faminlies of the pbjects and classification of infinite families. The main question: what one must add to the set of finite invariants and when we do not need the new invarinats. Classical example of the case of absense of new infinite invariants is Kolmogorov's (1933) zero-one law for Bernoulli scheme. But very often we need in the "highest" invariants. I will tell about this problem in the framework of asymptotic studies of combinatorics and probability theory. No special knowledge is needed for understanding of this talk.

#### May 27, 2016, Alexander Kolesnikov (HSE) Hessian metrics related to mass transportation problem. Applications to geometry and analysis

The Monge-Kantorovich optimal mass transporation problem has numerous applications in analysis, probability, and PDE's. We will talk about its interplay with the differential geometry, focusing the metric properties of the related Hessian space with convex potential solving the corresponding real Monge-Ampere equation. We discuss recent extensions of some classical results on the Kaehler-Einstein equation and explain the motivation coming from convex geometry and probability.

#### June 3, 2016, Alexander Shapiro (UC Berkeley) Clusters, quantum groups, and half-Dehn twists

*Abstract: *A quantum cluster (or quantum torus) is an algebra over C(q) with q-commuting generators. Various embeddings of quantum groups into quantum tori have been studied over the past twenty years in relation with modular doubles, quantum Gelfand-Kirillov conjecture, and construction of braided monoidal categories. In a recent paper by K. Hikami and R. Inoue, such an embedding of the quantum group U_q(sl_2) was used to relate the corresponding R-matrix with quantum cluster mutations and half-Dehn twists.

I plan to explain how to generalize the results of Hikami and Inoue to U_q(sl_n). The quantum group is embedded into the tensor square of the quantized categorification space of 3 flags and 3 lines in C^n. This embedding uses the combinatorics of m-triangulations and the notion of amalgamation introduced by V. Fock and A. Goncharov. I also plan to show how the conjugation by the R-matrix can be expressed via a sequence of cluster mutations. If time permits, I will speculate on potential applications and generalization to quantum groups of other types.

This is based on a joint work in progress with Gus Schrader.

#### June 10, 2016, Friday, 17:00: Vladimir Rubtsov (HSE) "Polynomial Poisson sructures, Elliptic Algebras, Heisenberg group and Cremona transformations"

We shall discuss some aspects of polynomial Poisson structures on $C^n$ with $n = 3,4,5$. The quasi-classical limit of the famous elliptic Sklyanin algebra is a particular important example of such structures. We shall discuss the Heisenberg group invariancy and unimodularity of elliptic Poisson algebras. The case of $n=5$ is of a special interest because of presence of two non-isomorphic families of elliptic algebras (Odesskii-Feigin). Their relation with Cremona transformations in $\mathbb P^4$ is described.

#### June 15, 2016, Wednesday, 17:00 Semyon Alesker (Tel-Aviv) -,

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#### June 24, 2016, Friday, 17:00: (, , , Laboratoire J.-V.Poncelet)

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#### July 1, 2016, Friday, 17:00: Fedor Bogomolov (Courant Institute and HSE) "Elliptic curves and unramified correspondences"

We define two different ( but related) notions of dominance. We will mostly consider them for curves defined over number fields of $\bar F_p$ though they can be defined for curves over any field Definition 1 For a curve $C$ of genus $g \geq 2$ we will say that $C$ is dominant over $C'$ if there is an unramified covering $\tilde C$ of $C$ with a surjection onto $C'$.

In the case of elliptic curves we have a different notion ( assuming $p\neq 2$ ) There is a involution $x\to -x$ on elliptic curve $E$ if we fix $0$ anthe quotient of this involution is $P^1$. Thus we have projection map $p: E\to P^1$ of degree $2$ with $4$ branch points $(a,b,c,d)$ corresponding to points of order $2$ on $E$. Such a map is unique modulo projective autmorphism of $P^1$. Vice versa we can associate to any quadruple of points in $P^1$ modulo projective autmorphism of $P^1$ unique elliptic curve $E$ modulo isomorphism. Moreover since the curve $E$ is an abelian group we can also define the subset $P_E(tors)\subset P^1$ which is the image of torsion points in $E$ in $P^1$.

Defintion 2 We will say that $E$ dominates $E'$ if $E'$ corresponds to a quadruple of points contained in $P_E(tors)$.

In my talk I will the relation between these two notions and nontrivial results relating them.

The talk is based on my works with Yuri Tschinkel and our more recent results with Hang Fu and Jin Qian.

#### August 12, 2016, Hiroshi Iritani (Kyoto) Mirror symmetry for toric varieties

Abstract: Abstract: Seidel representation associates to a Hamiltonian circle action on a symplectic manifold an invertible element in the quantum cohomology. This can be lifted to the action on quantum D-module and is called "shift operators". In this talk, I will explain how mirrors of toric varieties can be constructed via Seidel representation and shift operators tautologically in Gromov-Witten theory.

#### August 19, 2016, Kyusik Hong (KIAS) On factorial nodal hypersurfaces in P^4.

I will talk about the factoriality of a nodal hypersurface in P^4. For instance, the factoriality of a nodal quartic hypersurface in P^4 is strongly related to the rationality problem. In particular, I formulate a conjecture which would answer the factoriality problem of nodal hypersurfaces in P^4; the conjecture holds for some cases.

#### August 26, 2016, Andrey Soldatenkov (University of Bonn) IHS manifolds and sheaves on cubic 4-folds.

It is well known that the variety of lines on a cubic 4-fold X is an irreducible holomorphic symplectic (IHS) manifold. More recently Lehn et. al. have constructed another IHS manifold Z which is related to the variety of twisted cubics on X. We will discuss how to describe an open subset of this manifold in terms of moduli spaces of sheaves on X. We will see that the existence of symplectic form on Z is related to the structure of the derived category of X. The talk will be based on a joint work with E. Shinder.

#### September 2, 2016, Alexander Beilinson (Chicago)

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#### September 9, 2016, Boris Kruglikov (Tromsø) .

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#### September 16, 2016, Andrzej Szczepanski (Gdansk) Hantzsche-Wendt manifolds

There are a flat Riemannian manifolds of odd dimension n with holonomy group (\Z_2)^{n-1}. From Bieberbach theorems its fundamental group G is torsion free and defines a short exact sequence 0--->\Z^n----->G----->(\Z_2)^{n-1}---->, where \Z^n is a maximal abelian in G. This class of manifolds (groups) has many very interesting properties:
• - they are rational homology spheres,
• - they are homology rigid i.e M is diffeomorphic to M' if and only if cohomology rings H^{*}(M,\F_2) and H^{*}(M',\F_2) are isomorphic
• - they have not a Spin structure
I would like to present some introduction to this class of flat manifolds.

#### September 23, 2016, ݣ ͣ

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#### September 30, 2016, Dima Kaledin (MIRAS) Witt vectors, commutative and non-commutative

Witt vectors W(A) of a commutative ring A were discovered 80 years ago, but they still pop up in unexpected places and are the subject of continuous research. One question that has been solved only recently is how to generalize Witt vectors to the case when A is not commutative. I am going to review the classical theory, and then show how a very natural modification leads to the non-commutative case.

#### October 5, 2016, (Wednesday), 18:30, room 306: Fedor Bogomolov Fedor Bogomolov, Affine structures and VII_0 surfaces

Fedor Bogomolov, Affine structures and VII_0 surfaces In my talk I will describe the main steps of the proof of the following result: If $X$ is a VII_0 surface with $b_2=0$ then it's tangent bundle contains a subbundle of rank $1$ and hence it is either Hopf or Inoue surface. The proof is based on the study of surfaces with affine structures. The latter means the existense of a covering by compact balls with coordinates such that holomorphic transofrms of the coordinates between intersecting balls are affine.

#### October 7, 2016, Jihun Park (Pohang University of Science and Technology) Product theorem for K-stability

After the Yau-Tian-Donaldson conjecture concerning the existence of Kahler-Einstein metrics on Fano manifolds and stability is settled, testing K-stability of concrete Fano manifolds arises as a hot issue in algebraic and complex geometry. However direct testing of K-stability through all possible degenerations seems almost infeasible to carry out. Recent K-stability research has focused on detour methods to test K-stability of Fano varieties. In this talk, I will introduce an algebro-geometric method to show that a product of K-stable Fano manifolds is again K-stable. Under a certain conjectural condition, this method works perfectly. The conjectural condition has also been verified to hold good for 2-dimensional Fano manifolds.

#### October 14, 2016, Takanori Ayano (HSE) Jacobi Inversion Formulae for Telescopic Curves

For a hyperelliptic curve of genus g, it is well known that the symmetric products of g points on the curve are expressed in terms of their Abel-Jacobi image by the hyperelliptic sigma function (Jacobi inversion formulae). Matsutani and Previato gave a natural generalization of the formulae to the more general algebraic curves defined by y^r=f(x), which are special cases of (n,s)-curves. In this talk we extend the formulae to the telescopic curves proposed by Miura and derive new vanishing properties of the sigma function of telescopic curves. The telescopic curves contain the (n,s)-curves as special cases.

#### October 21, 2016, (Laboratoire Painleve, Lille, IUF ) 2.

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#### October 28, 2016, (Institut de Mathematiques de Jussieu) ( , , ) .

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#### Extra session: November 2, 2016, 18:30 Frederic Mangolte (Angers) Fake real planes: exotic affine algebraic models of R^2

We study topologically minimal complexifications of the Euclidean affine plane R up to isomorphism and up to birational diffeomorphism. A fake real plane is a smooth geometrically integral surface S defined over R such that:
• The real locus S(R) is diffeomorphic to R^2;
• The complex surface S_C(C) has the rational homology type of A^2_C.;
• S is not isomorphic to A^2_R as surfaces defined over R.
The analogous study in the compact case, that is the classification of complexifications of the real projective plane P(R) with the rational homology of the complex projective plane is well known: P_C is the only one. We prove that fake real planes exist by giving many examples and we tackle the question: does there exist fake planes S such that S(R) is not birationally diffeomorphic to A^2_R(R)? (Joint work with Adrien Dubouloz.)

#### November 4, 2016, 17:00, Misha Verbitsky (HSE) Mixed Hodge-Riemann relations for hyperkahler manifolds and Kuga-Satake construction for total cohomology.

Mixed Hodge-Riemann relations were discovered by Vladlen Timorin in 1998 for cohomology of the torus; the proof was purely algebraic. Geometric version was proven for an arbitrary Kahler manifold in 2005 by Dingh and Nguyen. I would explain how a stronger version of their result can be proven for a hyperkahler manifold. The argument is purely algebraic, and involves a generalized version of Kuga-Satake construction. The original Kuga-Satake is used to embed the Hodge structure of a K3 surface to the second cohomology of an appropriate torus. I would show that the whole cohomology space of a hyperkahler manifold M can be embedded to the cohomology of a torus, and this embedding is compatible with the Hodge decomposition, the Poincare pairing, and with the action of the algebra generated by all Lefschetz triples on M. Then the mixed Hodge-Riemann relations follow directly from Timorin's theorem. Using mixed Hodge-Riemann relations, we are able to generalize the Bogomolov-Beauville-Fujiki form to all dimensions. This is a joint work with Nikon Kurnosov.

#### November 11, 2016: Evgeny Shinder (Sheffield) K3 L-zero divisors in the Grothendieck ring of varieties

We discuss classes of non-isomorphic algebraic varieties X, Y such that [X] - [Y] is annihilated by a power of the affine line in the Grothendieck ring of varieties. Interesting examples include some derived equivalent Calabi-Yau threefolds (Borisov, Ito-Miura-Okawa-Ueda), and derived equivalent K3 surface pairs of degrees 8 and 2, 12 and 12, 16 and 4 (work in progress joint with A.Kuznetsov).

#### November 24, 2016: Lev Soukhanov (HSE) Models of diffusion-orthogonal polynomials of maximal degree.

A model of diffusion-orthogonal polynomials is a triple $(\Omega, L, \mu)$, where $\Omega$ is a (compact) domain in $\mathbb{R}^d$, $\mu$ is a measure on $\Omega$, $L$ is a second order differential operator, self-adjoint on $L^2(\Omega, \mu)$ preserving the subspace of polynomials of degree $\leq n$ for any $n$. In the dimension $2$ these models were classified by Bakry, Orevkov and Zani and it was observed that if the degree of the boundary of $\Omega$ is maximal possible, the model can always be obtained from the group generated by reflections. I will explain the generalization of this fact in the arbitrary dimension.

#### December 2, 2016: Andrew Staal (HSE) Irreducibility of Random Hilbert Schemes

What are the geometric properties of a typical Hilbert scheme? In this talk, we consider Hilbert schemes parametrizing closed subschemes in some projective space with a fixed Hilbert polynomial. We first explain how to make the collection of all such Hilbert schemes into a discrete probability space. Exploiting the underlying combinatorial structure, we show why a random Hilbert scheme is smooth and irreducible with probability greater than 0.5. I will explain the generalization of this fact in the arbitrary dimension.

#### December 9, 2016: (, )

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#### December 16, 2016: Sergey Fomin (University of Michigan) Noncommutative Schur functions

The problem of expanding various families of symmetric functions in the basis of Schur functions arises in many mathematical contexts such as combinatorial representation theory and Schubert calculus. I will discuss an approach to this problem that employs noncommutative analogues of Schur functions. The talk is based on joint work with Jonah Blasiak and Curtis Greene.

#### December 23, 2016: (UC Davis)

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#### December 26 and 28, 2016, room 306, 15:30: a special session by Ivan Cheltsov Stable and unstable del Pezzo surfaces.

Yau-Tian-Donaldson conjecture, recently proved by Chen, Donaldson and Sun, says that a Fano manifold is Kahler-Einstein if and only if it is K-stable. Its stronger form, still open, says that a polarized manifold (M,L) is K-stable if and only if M admits a constant scalar curvature with Kahler class in L. In these 4 lectures, I will describe K-stability of ample line bundles on smooth del Pezzo surfaces (two-dimensional Fano manifolds). I will show how to apply recent result of Dervan to prove K-stability and how to use flop-version of Ross and Thomas's obstruction to prove instability.
• Lecture 1: K-stability of polarized manifolds. Alpha-invariant of polarized manifolds. Dervan's criterion for K-stability.
• Lecture 2: K-stable ample line bundles on smooth del Pezzo surfaces.
• Lecture 3. Slope-stability of Ross and Thomas and Atiyah flops.
• Lecture 4. K-unstable ample line bundles on smooth del Pezzo surfaces.

#### December 28, 2016 (Wednesday), 17:00: Dmitry Zakharov (NYU) The theta-relations and the tautological ring

The tautological ring of the moduli space of curves M_g is a natural subring of either the cohomology or the Chow ring that is generated by a collection of natural classes. A number of vanishing results have been established about the tautological ring of M_g and its various compactifications. I will show that these results can be derived in a uniform and constructive way from Pixton's double ramification cycle relations, which were recently established by Clader and Janda. Moreover, these relations can be used to derive boundary formulas for any tautological class vanishing on the open locus. This is joint work with Clader, Grushevsky, Janda and Wang.

#### December 30, 2016 (Friday), 17:00: Dmitry Tonkonog (Cambridge) Wall-crossing for mutations of Lagrangian tori, and symplectic cohomology

Given a Lagrangian torus with an attached Lagrangian disk, a procedure called mutation produces a new Lagrangian torus out of it. Iterating this procedure allows to construct infinitely many monotone Lagrangian tori in del Pezzo surfaces, which was performed recently by Vianna. We prove the wall-crossing formula which solves the problem of enumerating holomorphic Maslov index 2 disks on these tori; the answer matches the prediction of Galkin and Usnich. Time permitting, I will mention the related Laurent phenomenon, and what it has to do with the symplectic cohomology of certain domains. This is joint work with James Pascaleff.

#### January 6, 2017: Dmitri Panov (King's College London) Real line arrangements with Hirzebruch property.

A line arrangement of 3n lines in CP2 satisfies Hirzebruch property if each line intersect others in n+1 points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in CP2 is real, confirming that there exist exactly four such arrangements.

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#### January 20, 2017: (MIT) " p, "

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#### January 27, 2017: Misha Verbitsky (HSE and ULB) Algebraic and Kahler dimension of nilmanifolds.

Let M be a complex nilmanifold, that is, a quotient of a nilpotent Lie group with left-invariant complex structure by a cocompact lattice, and h the dimension of its space of holomorphic differentials. S. Salamon has shown that \dim M >= h > 0 for any nilmanifold, with equality realized if and only if M is a torus. Algebraic dimension a(M) is transcendental dimension of the field of meromorphic functions on M. It is known that algebraic dimension is bounded from above by the usual dimension. I will show that a(M) is bounded by h (dimension of the space of holomorphic differentials) and explain when this bound is realised and how a(M) can be computed explicitly in terms of the Lie algebra. Also I would show that h bounds the Kahler dimension of M, that is, the maximal dimension of a compact Kahler manifold X such that there exists a dominant meromorphic map M -> , and explain when this bound is realized. This is a joint work with Gueo Grantcharov and Anna Fino.

#### February 3, 2017, () " "

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#### Wednesday, February 8, 2017, 17:00, (Bonn) .

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#### February 10, 2017: Mikhail Belolipetsky (IMPA) Systoles of hyperbolic manifolds

The systole of a Riemannian manifold M is the length of a shortest geodesic loop in M. I am going to discuss systoles and their higher dimensional analogues of hyperbolic n-manifolds. A special role will be played by arithmetic hyperbolic manifolds and their congruence coverings, which I am going to define in the first part of the talk.

#### February 17, 2017: Michael Finkelberg (HSE) "Kostka-Shoji polynomials".

These polynomials were introduced and studied by Shoji; they are analogues of Kostka polynomials $K_{\lambda,\mu}$ when $\lambda$ and $\mu$ are multipartitions. I will present an analogue of the Lusztig-Kato formula for Kostka-Shoji polynomials proved by Shoji last week, and their related geometric interpretation as multiplicities in the spaces of sections of certain line bundles over Lusztig's convolution diagrams for cyclic quivers, proving their positivity. This is a joint work with Andrei Ionov.

#### February 24, 2017: Fedor Bogomolov (Courant Institute and HSE) "Geometry of sets of torsion points on elliptic curves"

In this talk I introduce and discuss geometry of curves parametrizing subset of points in $P^1$ obtained as projections of torsion points of elliptic curves. For every subset of different $k$ points in $P^1$ we can define it's image in the moduli $M_{0,k}$ of $k$-tuples of points which is essentially a quotient of projective space $S^kP^1= P^k$ by the action of $PGL(2)$. Thus $M_{0,k}$ is a rational variety of dimension $k-3$. If we consider the images of points of finite order in different elliptic curves under natural projections then we obtain an( infinite) system of modular typoe curves with maps into $M_{0,k}$ I will formulate three conjectures (semi theorems) about properties of such maps which provide a possiblity of realistic universal estimate for intersections between subset of torsion points for different elliptic curves.

#### March 3, 2017: Misha Verbitsky (HSE and ULB) Kuga-Satake construction for higher cohomology.

Let M be a hyperkahler manifold of complex dimension n. Kuga-Satake construction gives an embedding of H^2(M) to H^2(torus) compatible with the Hodge structure. We construct a torus T of dimension n+k and an embedding of cohomology space H^*(M) -> H^{*+k}(T) which is compatible with the Hodge structures and the Poincare pairing. This is a joint work with Nikon Kurnosov and Andrei Soldatenkov.

(cancelled due to flight cancellation).

#### March 3, 2017: Aleksey Gorinov (HSE) A purity theorem for configuration spaces of smooth compact algebraic varieties

B. Totaro showed \cite{totaro} that the rational cohomology of configuration spaces of smooth complex projective varieties is isomorphic as an algebra to the E_2 term of the Leray spectral sequence corresponding to the open embedding of the configuration space into the Cartesian power. In this note we show that the isomorphism can be chosen to be compatible with the mixed Hodge structures. In particular, we prove that the mixed Hodge structures on the configuration spaces of smooth complex projective varieties are direct sums of pure Hodge structures.

#### March 10, 2017: Chris Brav (HSE) Relative Calabi-Yau structures

We introduce the notion of a Calabi-Yau structure on a dg functor between smooth dg categories. We discuss examples coming from topology, algebra, and algebraic geometry, explain how to glue together Calabi-Yau structures in the same way that one glues together oriented manifolds along a common boundary component, and show how this notion gives rise to symplectic/Lagrangian structures on moduli of objects in dg categories. This is joint work with Tobias Dyckerhoff from the University of Bonn.

#### March 17, 2017: ( ) , .

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#### March 22, 2017: Seidai YASUDA (Osaka University) Pseudo-tame rational functions on curves in characteristic two.

We introduce the notion of pseudo-tame morphisms of curves in characteristic two. Using this notion we prove that any curve over an algebraically closed field admits a morphism to the projective line which is tamely ramified everywhere. As a corollary, we obtain an analogue of Belyi's theorem in positive characteristic. This talk is based my joint work with Yusuke Sugiyama.

#### March 24, 2017: -- ( )

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#### March 24, 2017: P.V. Bibikov (Institute of Control Sciences RAS) Differential invariants in algebraic geometry and algebra in differential equations

The aim of the talk is to show new relationships between geometric theory of differential equations, algebraic geometry and classical invariant theory. We show how methods and constructions from the theory of differential equations can be used for studying of various algebraic problems. Also we explain how classical algebraic constructions generate new questions in the theory of differential equations. Great attention will be paid to open questions and problems.

#### 15:30, March 31, 2017: Yoshinori Gongyo (University of Tokyo) Cone theorems

We will discuss several cone theorems which appear in birational geometry.

#### 17:00, March 31, 2017: () , .

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#### April 7, 2017: Nicholas Shepherd-Barron (King's College) Fano 3-folds in positive characteristic

Fano 3-folds in positive characteristic I will discuss Kodaira vanishing and multiple projection for smooth Fano threefolds in positive characteristic.

#### Exceptional groups and del Pezzo surfaces

I shall extend the construction by Brieskorn and others that contains the simultaneous resolution of du Val singularities to the environment of principal bundles under exceptional groups over elliptic curves. This recovers the simultaneous log resolutions of simply elliptic singularities and gives a direct geometrical path from exceptional groups to del Pezzo surfaces. This is joint work with Grojnowski.

#### April 14, 2017: Sergey Galkin (HSE) The conifold point

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer to a question of Ostrover and Tyomkin: the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand. Moreover, it gives a good evidence that the same statement holds for any Fano manifold.

#### Wednesday, April 19: Liviu Ornea (Bucharest) Recent results in locally conformally Kahler geometry.

After a brief account on LCK geometry, with focus on LCK with potential and Vaisman manifolds, I shall describe several new results concerning compact LCK with potential, concerning their LCK rank and the fact that they contain Hopf surfaces.

#### April 21, 2017: Pavel Safronov (Geneva) Introduction to derived Poisson geometry with examples

Derived Poisson geometry studies higher Poisson structures on (derived) algebraic stacks. I will explain what higher Poisson structures are and how to define them on stacks following the work of Calaque, Pantev, Toen, Vaqui'e and Vezzosi. Moreover, one can define an interesting generalization of the notion of a coisotropic submanifold in this context which I will describe following joint work with Melani. In the second half of the talk I will give several examples of these constructions some of which come from Poisson-Lie groups.

#### April 28, 2017: Paul Zinn-Justin (Melbourne) Schubert calculus and quantum integrability

We formulate new combinatorial (puzzle) rules for Schubert calculus in the d-step flag variety, d<=4. More precisely, generalizing my previous work for d=1, we show how to define an integrable model that computes the structure constants of the (equivariant) cohomology (or K-theory) of the d-step flag variety in the basis of Schubert classes. Deligne's exceptional series appears naturally. We also explain the connection to Maulik-Okounkov stable classes. This is joint work with A. Knutson.

#### May 5, 2017: /Grey Violet ( ., ). $D$- .

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#### May 12, 2017: Anatol Kirillov (Kyoto) Introduction to Dilogarithm Identities, Rigged Configurations and Fomin-Kirillov algebras

The Dilogarithm function had been introduced by L. Euler more than $250$ years ago, and since that time the Dilogarithm function has been extensively studied by many mathematicians and physicists including N. Abel, E. Kummer, L. Rogers, S. Ramanujan, L. Lewin, L.D. Faddeev, D. Zagier, A. Goncharov, H. Gangl, A.Al. Zamolodchikov, among many others. Dilogarithm and its quantum analogue have found numerous deep applications in Number Theory, Hyperbolic Geometry, Knot invariants, Algebraic K-theory, Representation Theory, Mathematical Physics and Applied Mathematics. In my talk I'm planning to draw attention of the audience to some remarkable identities for the values of the Rogers dilogarithm function at some very special families of algebraic numbers. These relations admit an interesting interpretation in algebraic K-theory and Conformal Field Theory. I'm planning to talk about Rigged Configuration Bijection (RC-bijection), which originated from the analysis of the Bethe Ansatz Equations for the XXX and XXZ Heisenberg models, and has a big variety of applications to Combinatorics, Representation Theory, Discrete Integrable Systems, among other interesting applications. I'm also planning to talk about some families of quadratic algebras (the so-called "Fomin-Kirillov" type algebras) with applications to Schubert Calculus, Quantum (and Elliptic) Cohomology and K-theory of flag varieties, and beyond.
Reference

• Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, 1958.
• D. Zagier, The Dilogarithm function in Geometry and Number Theory, Number Theory and related topics, Tata Inst. Fund. Res. Stud. Math. 12 Bombay (1988), 231 - 249.
• Kirillov, A. N. Dilogarithm Identities. Progr. Theor. Phys. Suppl. 118, 61-142, 1995.

#### May 19, 2017: ()

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#### May 24, 2017, Wednesday, 17:00: Giovanni Mongardi (Bologna) Calabi Yau quotients of hyperkahlers

In this talk, i will speak about a joint work with C. Camere and A. Garbagnati. We analyze which calabi yau manifolds can be obtained as resolution of quotients of hyperkahler manifolds by non symplectic automorphism. More specifically, we will deal with the case of fourfolds modulo an involution, which gives a wide range of examples. In the natural case, we also compute the Hodge numbers of the Calabi Yau manifolds.

#### May 24, 2017, Wednesday, 18:30: Dmitri Panov (Kings College) 6- .

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#### May 26, 2017, Friday, 17:00: Giovanni Mongardi (Bologna) Hodge numbers of the O'Grady six dimensional manifold

In this talk, i will speak about a joint work with A. Rapagnetta and G. Sacca. In it, we realize O'Grady's six dimensional example of irreducible holomorphic symplectic manifold as a quotient of an IHS manifold of K3$^{[3]}$--type by a birational involution, thereby computing its Hodge numbers.

#### June 2, 2017: Agnieszka Bodzenta-Skibinska (Edinburgh) Canonical divisors revisited - categorical approach

Properties of the canonical sheaf are one of the first invariants of an algebraic variety considered in birational geometry. I will describe how canonical divisors appear in the study of derived categories of birationally equivalent varieties. I will prove that, given a birational morphism, the canonical bundle and its restriction to relative canonical divisors provide a tilting generator for one category over another. I will also discuss related quasi-hereditary algebras and a system of t-structures.

#### 15:00, Thursday, June 8, 2017: ()

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#### June 9, 2017: Charles Fougeron Lyapunov exponents for hypergeometric equations.

Lyapunov exponents and their Oseledets flag decomposition are a very useful tool for describing dynamical systems. They are presented sometimes as dynamical variation of Hodge structures. My main motivation here is to understand their link to algebraic invariants of variation of Hodge structure when it exists.

In the 90's, M. Kontsevich observed that the sum of Lyapunov exponents associated to translation surfaces are equal to the degree of some holomorphic subbundle for a variation of Hodge structure associated to its Teichmüller geodesic. It is remarkable that this relation arises from wider properties like ergodicity and some algebraic rigidity on the variation of Hodge structure which is true in a much more general setting.

Recently, a similar result was observed on higher width variation of Hodge structure (which decomposition has more flags), A. Eskin, M.Kontsevich, M. Möller and A. Zorich showed indeed a lower bound of their associated Lyapunov exponents given by the parabolic degrees of their variation of Hodge structure.

I will present this result on the example of variation of Hodge structure yielded by hypergeometric equations of arbitrary order. Starting with the computation of their degrees, and presenting some computer experiments. This will motivate questions about the equality case.

#### 17:00, June 14, 2017: Alexander Gorokhovsky .

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#### 15:30, June 16, 2017: Sergei Nechaev (Orsay) Number-theoretic aspects of 1D localization: spectral statistics of sparse random graphs, "popcorn function" with Lifshitz tails, and Dedekind $\eta$-function.

We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers. The spectral density of ensemble of linear chains (graphs) distributed exponentially $\sim f^L$ (0<f<1), where $L$ is the chain length, serves as a particular example. At $f\to 1$, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function. We suggest a continuous approximation of the popcorn function, based on the Dedekind $\eta$-function near the real axis. We provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at spectral edges. We also pay attention to the connection of the Dedekind $\eta$-function near the real axis to phyllotaxis and invariant measures of some continued fractions studied by Borwein and Borwein in 1993.

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#### June 16, 2017: (Penn State University) .

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#### Monday, 17:00, June 19, 2017: Mahan Mj. (Tata Institute) Cannon-Thurston maps and Kleinian groups (1)

Let M be a closed hyperbolic 3-manifold fibering over the circle with fiber a closed surface S. The inclusion of S into M lifts to a map between universal covers \tilde{S} and \tilde{M}. In the early 80's Cannon and Thurston showed that this inclusion extends to a continuous map between their compactifications: namely the 2-disk and the 3-ball. This gives rise to a space-filling (Peano) curve from the circle onto the 2-sphere, equivariant under the action of the fundamental group of S. This led Thurston to the following questions.

1) Is this a general phenomenon for finitely generated discrete subgroups of the isometry group of hyperbolic 3-space?

2) How does this map behave with respect to sequences of representations?

In the first lecture I shall survey an affirmative answer to Question 1. In the second, I shall give a review of work (joint in parts with C. Series and K. Ohshika) leading to a resolution of Q. 2.

#### June 23, 2017: Mahan Mj. (Tata Institute) Cannon-Thurston maps in Geometric Group Theory

Let M be a closed hyperbolic 3-manifold fibering over the circle with fiber a closed surface S. The inclusion of S into M lifts to a map between universal covers \tilde{S} and \tilde{M}. In the early 80's Cannon and Thurston showed that this inclusion extends to a continuous map between their compactifications: namely the 2-disk and the 3-ball. This can be extended to a considerably broader framework in the context of (Gromov) hyperbolic groups. I shall survey some of the developments in this broader context.

#### June 30, 2017: () " - "

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#### Jule 14, 2017: () .

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#### August 4, 2017: Sergey Galkin (HSE) Fano threefolds, K3 surfaces, Mathieu group, and Brav-Dyckerhoff relative Calabi-Yau structures

I will argue that recent works of Brav-Dyckerhoff on relative CY structures (also Katzarkov-Pandit-Spaide and others on relative spherical functors) might help to bridge two so-far distant "moonshines":

1. correspondence between some subgroups of sporadic groups (M23, M24, Co_1) and symmetries of K3, that was observed by Nikulin-Mukai (1980s), Eguchi-Ooguri-Tachikawa (2010, in form of elliptic genus), and further works such as Gaberdiel-Hochenegger-Volpato

2. correspondence between conjugacy classes of same sporadic groups and G-Fano threefolds, that was observed by myself around 2009 in an attempt to generalize and refine Dolgachev-Golyshev's picture of mirror symmetry.

#### August 10 and 11, 2017: Dmitry Kaledin (Steklov Institute) Brown representability for groupoids

This is a continuation and/or refinement of my June talk on Brown representability theorem. I will show how to prove a version of Brown representability with values in groupoids, and sketch some applications. Caution: this is heavily work in progress, use as is, no warranty, don't try this at home.

The talks are organized jointly with Colloquim of Laboratory of Mirror Symmetry.

#### August 18, 2017: Fedor Bogomolov (NYU and HSE) $PGL(2)$-invariants of collections of torsion points of elliptic curves"

In this talk I will continue to discuss geometry of sets of images of torsion points of elliptic curves in $P^1$. I am going to develop some ideas which were mentioned in my previous talks on the subject. In particular I provide an argument proving the first conjecture described in previous talk for almost all $4$ tuples of the images of torsion points.

#### September 1, 2017: Andrey Soldatenkov (University of Bonn) Kuga-Satake construction and its generalizations

Let H be a rational polarized weight 2 Hodge structure of K3 type, meaning that the (2,0)-component is one-dimensional. Classical Kuga-Satake construction attaches to it an abelian variety A an an embedding of H into the second cohomology of A, compatible with Hodge structures. I will talk about our recent work with N.Kurnosov and M.Verbitsky in which we consider the case where H is the second cohomology of a hyperkahler manifold X. We show that all cohomology of X can be embedded into the cohomology of the product of several copies of A. If time permits, I will talk about our joint work with S.Schreieder, where we consider the behaviour of Kuga-Satake abelian varieties under degeneration. It turns out that one can describe the limit mixed Hodge structure on the central fibre of degenerating family of Kuga-Satake varieties.

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#### September 13, 2017 (Wednesday), 17:00, and September 15, 2017: - ,

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#### September 22, 2017, 17:00: (, ) -

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#### September 29, 2017, 17:00: Don Zagier (Bonn) Poor man's adeles and multiple zeta values

The "poor man's adeles" of the title is the informal name of the ring $\Prod_p\bigl(\Bbb Z/p\Bbb Z_p\bigr)/\Oplus_p\bigl(\Bbb Z/p\Bbb Z_p\bigr)$ whose elements are numbers" having a well-defined value modulo almost every prime number. It turns out that examples of elements of this ring show up in many places in mathematics. In the lecture I will describe several examples of this, most notably a finite-field version of the well-known multiple zeta values invented by Euler and much studied in recent years (this part is joint work with Masanobu Kaneko), but also examples coming from areas as different as quantum invariants of homology 3-spheres and transition matrices between different bases of the space of solutions of a linear differential equation with regular singularities.

(joint colloquium of Laboratory of Algebraic Geometry and Laboratory of Mirror Symmetry).

#### October 4, 2017 (Wednesday), 18:30: Jean-Louis Colliot-Thelene (CNRS, Universite Paris-Sud Paris-Saclay) Disproving stable rationality

In the last four years, a series of papers by several authors has established that some very classical, rationally connected, complex varieties (cyclic covers of projective space with ramification locus of low degree, hypersurfaces of low degree, quadric bundles over rational varieties) are not stably birational to projective space. In the first part of the talk I shall give a general description of the method and of some of its variants, and I shall try to list the main results achieved. The second part of the talk will be devoted to a recent variant of the technique.

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#### October 6, 2017, 18:30: () - - Algebro-geometric spectral data for planar Calogero-Moser systems. ( )

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My talk (based on a joint work with Igor Burban) is devoted to the algebraic analysis of planar rational Calogero-Moser systems. This class of quantum integrable systems is known to be superintegrable. This means that the underlying Schrodinger operator with Calogero-Moser potential can be included into a large family of pairwise commuting partial differential operators such that the space of joint power series eigenfunctions is generically one-dimensional.

More algebraically, any such system is essentially determined by a certain algebro-geometric datum: the projective spectral surface (defined by the algebra of planar quasi-invariants with natural filtration) and the spectral sheaf (defined by a module known to be Cohen-Macaulay of rank one). This geometric datum has very special algebro-geometric properties, the most important of which is a very special form of the Hilbert polynomial of the module (sheaf). Moreover, the spectral variety appears to be rational but very singular (only Cohen-Macaulay, even not normal). It turns out that all rank one Cohen-Macaulay modules over the algebra of planar quasi-invariants can be explicitly described in terms of very natural moduli parameters, and this description looks in some sence very similar to to the description of the generalised Jacobian for singular rational curves. The spectral module of a planar Calogero-Moser system is actually projective, and its underlying moduli parameters are explicitely determined.

Unlike the case of curves, not every Cohen-Macaulay module is spectral. The moduli space of spectral sheaves appears to be much more subtle, but its structure indicates the existence of integrable deformations of Calogero-Moser systems. I am going to explain how the classification of CM modules, combined with tools of the algebraic inverse scattering method, leads to certain new integrable deformations of Calogero-Moser systems in the algebra of differential-difference operators.

#### October 13, 2017, 17:00: Chris Brav (HSE) Functions on moduli spaces from cyclic homology

We discuss the 'moduli of objects' M_D in a dg category D and construct a map from cyclic homology of D to functions on the moduli space M_D. When D is a smooth, oriented dg category ('Calabi-Yau'), the cyclic homology HC(D) is endowed with a shifted Lie bracket ('algebraic string bracket') and the functions on M_D are endowed with a shifted Poisson bracket. We show that the map from cyclic homology to functions entwines the brackets. Examples include the Goldmann bracket of free loops on a surface, the string bracket of Chas-Sullivan, and the Hitchen system for Higgs bundles. This is joint work very much in progress with Nick Rozenblyum.