Curves Defined By Parametric Equations Homework Stu Schwartzman

Abstracts of talks
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Herbert Abels
The topological generating rank of Lie groups
The toplogical generating rank $d_{top}(G)$ of a topological group $G$ is the minimal number of elements of $G$ needed to generate a dense subgroup of $G$. I will present joint work with Gena Noskov where we compute the topological generating rank for connected Lie groups. Several special cases have been known before. E.g. for a linear semisimple group G the classical result $d_{top}(G) = 2$ has been improved to ”$d_{top}(G)$ is slightly bigger than one”, based on joint work with E.B.Vinberg. For the general case the Frattini subgroup is a basic tool.
Dmitri Alekseevsky
Vinberg's theory of homogeneous convex cones: developments and applications
Slides  Video
The talk is a review of developments and applications of selected basic ideas, results and notions, presented by E.B. Vinberg in his papers about classification of homogeneous convex cones. We confine ourselves to only three topics:
a) Theory of left symmetric algebras (VinbergKoszul algebras) and Hessian manifolds.
b) Information geometry , i.e. the geometry of manifolds of probability measures , developed mostly by N.N. Chentsov and SI Amary and based on ideas by R.A.Fisher, C.R.Rao, C. Shannon and S. Kullback.
c) Supergravity. Application of the theory of matrix $T$algebras by Vinberg for description of so called special geometries (very special real geometry, special Kähler geometry and special quaternionic Kähler geometry (affine and projective)) which arise as matter multiplets in Supersymmetry and Supergravity in spacetime dimension $d = 6,5,4,3$.
Ivan Arzhantsev
Torus actions and trinomials
Video
In this talk we discuss algebraic torus actions of complexity one and relate them via Cox rings to affine varieties defined by trinomials. Our aim is to survey recent results on affine trinomial varieties and their automorphism groups.
We study the flexibility property for affine varieties, find all rational trinomial varieties, characterize rigid factorial trinomial hypersurfaces, describe the automorphism group of a rigid trinomial hypersurface, solve some Diophantine equations and characterize existence of certain polynomial curves on trinomial hypersurfaces.
An explicit description of primitive homogeneous locally nilpotent derivations on trimomial affine algebras allows to describe the automorphism group of a complete rational variety with a torus action of complexity one generalizing Demazure's description of the automorphism group of a complete toric variety.
Ana Balibanu
The partial compactification of the universal centralizer
Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in $\text{Lie}(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^*G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. We show that the symplectic structure extends to a logsymplectic Poisson structure on the partial compactification, through a Hamiltonian reduction of the logarithmic cotangent bundle of the wonderful compactification.
Michel Brion
Algebraic group actions on normal varieties.
Video
Let G be a connected algebraic kgroup acting on a normal kvariety X, where k is a field. We show that X is covered by open Gstable quasiprojective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a Glinearized vector bundle on an abelian variety, quotient of G. This generalizes a result of Sumihiro on actions of affine algebraic groups, and yields a refinement of a result of Weil on birational actions.
Victor Buchstaber
Toric topology of complex Grassmann manifolds
Slides  Video
The complex Grassmann manifold G(n,k) of all kdimensional complex linear subspaces in the complex vector space C^n plays the fundamental role in algebraic topology, algebraic and complex geometry, and other areas of mathematics. The manifolds G(n,1) and G(n,n1) can be identified with the complex projective space CP(n1). The coordinatewise action of the compact torus T^n on C^n induces its canonical action on the manifolds G(n,k). The orbit space CP(n1)/T^n can be identified with the (n1)dimensional simplex. The description of the combinatorial structure and algebraic topology of the orbit space G(n,k)/T^n, where k is not 1 or (n1), is a wellknown topical problem, which is far from being solved. The talk is devoted to the results in this direction which were recently obtained by methods of toric topology jointly with Svjetlana Terzić (University of Montenegro, Podgorica).
Corrado De Concini
On some modules of covariants for a reflection group.
This is joint work with Paolo Papi. Let $W$ be an irreducible finite reflection group, $\mathfrak{h}$ its (complexified) reflection module. $\mathcal{H} = C[\mathfrak{h}]/I$, where $I$ is the ideal generated by polynomial invariants of positive degree. $A = (\Lambda (\mathfrak{h})\otimes \mathcal{H})^W$ is an exterior algebra and we completely determine the $A$−module structure of $N := hom_W (\mathfrak{h},\Lambda (\mathfrak{h})\otimes \mathcal{H})$.
When $\mathfrak{h}$ is the Cartan subalgebra of a simple Lie algebra $\mathfrak{g}$, it is well known and easy that $A$ is canonically isomorphic to $(\Lambda(\mathfrak{g}))^{\mathfrak{g}}$ and we verify that $N = hom_{\mathfrak{g}}(\mathfrak{g}, \Lambda(\mathfrak{g})$ as an $A$−module.
Finally if $V$ is an irreducible g−module whose zero weight space we denote by $V_0$, we construct a degree preserving map $$hom_{\mathfrak{g}}(V,\Lambda(\mathfrak{g}))\to hom_W (V_0,\Lambda (\mathfrak{h})\otimes \mathcal{H})$$ which we conjecture to be injective. This conjecture implies a well known conjecture by Reeder.
Boris Feigin
Shifted toroidal algebras and corresponding vertex operator algebras
Video
In my talk I introduce several constructions of affine shifted Yangians. Such Yangians are vertexoperator algebras which are constructed by "gluening" procedure .We explain the quantumgroup meaning of "gluening" and present some conjectural connections with the topology of 4dimensional manifiolds.
Evgeny Feigin
Weighted PBW degenerations and Vinberg's polytopes
Video
We study algebraic, combinatorial and geometric aspects of weighted PBWtype degenerations of flag varieties in type A. These degenerations are labeled by degree functions lying in an explicitly defined polyhedral cone. Varying the degree function in the cone, we recover the classical flag variety, its abelian PBW degeneration and toric degeneration corresponding to Vinberg's polytopes. We also identify the cone of degree functions with a maximal cone in the tropical flag variety. The talk is based on the joint work with Xin Fang, Ghislain Fourier and Igor Makhlin.
Anna Felikson
Quiver mutations, reflection groups and curves on punctured disc
Slides
Mutations of quivers were introduced by Fomin and Zelevinsky in 2002 in the context of cluster algebras. For some classes of quivers, mutations can be realised using geometric or combinatorial models. We will discuss a construction of a geometric model for all acyclic quivers. The construction is based on the geometry of reflection groups acting in quadratic spaces. As an application, we show an easy and explicit way to characterise real Schur roots (i.e. dimension vectors of indecomposable rigid representations of Q over the path algebra kQ), which proves a recent conjecture of K.H. Lee and K. Lee for a large class of acyclic quivers. This work is joint with Pavel Tumarkin.
Iordan Ganev
Slides
BeilinsonBernstein localization via wonderful asymptotics
We explain how a doubled version of the BeilinsonBernstein localization functor can be understood using the geometry of the wonderful compactification of a group, as well as the associated Vinberg semigroup. Specifically, bimodules for the Lie algebra give rise to monodromic Dmodules on the horocycle space, and to filtered Dmodules on the group that respect a certain matrix coefficients filtration. These two categories of Dmodules are related via an associated graded construction in a way compatible with localization, Verdier specialization, and additional structures. This is joint work with David BenZvi and David Nadler.
Simon Gindikin
The horospherical Cauchy transform on Riemannian symmetric spaces and curved Radon's inversion formula
Slides  Video
Following to Gelfand's conception of integral geometry the Plancherel formula on symmetric spaces is equivalent to the inversion of the horospherical transform. We modify the horospherical transform and show that there is its inversion coinciding with the inversion formula for its flat model. The last one can be obtain using the classical Fourier transform.
The basic tool is a "curving" of flat inversion formulas starting of usual Radon inversion formula on the plane. Our final Plancherel formula on Riemannian symmetric spaces differs from HarishChandra's formula.
Iva Halacheva
The periplectic Lie superalgebra from a diagrammatic perspective
One approach to the study of the representation theory of the general linear Lie algebra is to look at the endomorphism algebras of tensor powers of the vector representation and of bigger tensor products with more general representations. The classical and higher SchurWeyl dualities relate these algebras to the symmetric group and the degenerate affine Hecke algebra respectively, and have been subsequently generalized to other types, such as the symplectic and special orthogonal Lie algebras. We study the periplectic Lie superalgebra p(n) and define diagrammatically the affine VW supercategory sVW, which generalizes the Brauer supercategory and maps to the category of p(n)representations. This functor allows us to obtain a basis for the morphism spaces of sVW as well as study the representation theory of p(n).
Valentina Kiritchenko
Schubert calculus on NewtonOkounkov polytopes
Video
Theory of NewtonOkounkov convex bodies can be used to extend toric geometry to varieties with more general reductive group actions. In particular, any NewtonOkounkov polytope of a complete flag variety can be turned into a convex geometric model for Schubert calculus. Namely, we can represent Schubert cycles by linear combinations of faces of the polytopes so that the intersection product of cycles corresponds to the settheoretic intersection of faces (whenever the latter are transverse). I will talk about particular realizations of this approach in types $A$, $B$ and $C$.
Friedrich Knop
Reductive group actions over nonclosed fields of characteristic zero
Video
We present a structure theory for actions of reductive groups which, to some extent, is parallel to the classical theory of reductive groups by Satake and BorelTits. In particular, properties of reductive groups actions on varieties with enough rational points are controlled by a restricted root system and an anisotropic kernel. The theory is most effective for varieties on which a minimal parabolic has an open orbit (called kspherical). Here we construct wonderful completions. For local fields they turn out to be genuine compactification of the rational points. This is joint work with Bernhard Krötz.
Gleb Koshevoy
Geometric RobinsonSchenstedKnuth correspondence and canonical bases
Slides  Video
We define a geometric RSK correspondence for a KacMoody group and any reduced decompositon of element of its Weyl group. This correspondence is a biration map of tori of dimension equal to the length of a reduced decomposition. For classical Lie groups and the element of the Weyl group with the longest length, the tropicalization of this map turns out to be the crystal isomorphism between the Lusztig crystal on the canonical basis and the Kashiwara crystal on the dual canonical basis. The geometric RSKcorespondence provide us with a transformation of the corresponding superpotentials for geometric crystals.
Alex Lubotzky
First order rigidity of highrank arithmetic groups
Video
The family of high rank arithmetic groups is a class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2, SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasiisometric rigidity.
We will talk about a new type of rigidity: "first order rigidity". Namely if D is such a nonuniform characteristic zero arithmetic group and E a finitely generated group which is elementary equivalent to it ( i.e., the same first order theory in the sense of model theory) then E is isomorphic to D.
This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups ( many of which are lowrank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them.
Alexander Mednykh
Hyperbolic knots, links and polyhedra
Video
Knots, links or, more generally, knotted graphs admitting a hyperbolic structure on their complement are called hyperbolic. In 1975, Robert Riley discovered the existence of a family of hyperbolic knots. Today, we know that most knots and links are hyperbolic. This statement is now a special case of the Thurston Hyperbolization Theorem. Starting with noncomplete hyperbolic structure on the complement, one can produce a complete hyperbolic conemanifold whose singular set is a given knot, link or polyhedron with prescribed cone angles. This is a way to investigate geometrical properties of the knots, links and polyhedra from the unified point of view. In this lecture, we show that cone angles and lengths of singular geodesics of conemanifolds under consideration are related by quite simple and beautiful Sine, Cosine or Tangent rules. These identities allow to find integral formulas for the volume and ChernSimons invariants of the manifolds in many important cases.
Joanna Meinel
The affine nilTemperleyLieb algebra and its action on particle configurations
Video
The affine nilTemperleyLieb algebra can be characterised by its faithful action on fermionic particle configurations on a circle. This action allows for a nice visualisation, and furthermore, one can use it to derive several properties of the affine nilTemperleyLieb algebra, including an explicit description of its centre and a classification of its simple modules. This talk is based on joint work with G. Benkart and the speaker's PhD thesis.
Sergey M. Natanzon
Higher spin Riemann and Klein surfaces
Klein surfaces are generalization of Riemann surfaces to cases of nonorientable surfaces or surfaces with boundary. They are equivalent to real algebraic curves. An mspin structure on Riemann or Klein surface is a complex line bundle $l$ such that $l^{\otimes m}$ is the cotangent bundle. We nd all mspin structures on surfaces of Riemann and Klein, we describe they topological invariants and moduli spaces. The proofs are based on the theory of Fuchsian groups. The talk is based on joint works with Anna Pratoussevitch.
Grigori Olshanski
Combinatorics of characters
Video
The asymptotic representation theory reveals a surprising analogy between the symmetric and classical groups. I will describe results related to this phenomenon.
Alexander Panov
Towards a supercharacter theory for parabolic subgroups
A supercharacter theory is a system of characters of a finite group that afford a partition of the group obeying certain properties. The main example is the system of irreducible characters and the partition into conjugacy classes. For some groups, classification of irreducible characters turns out to be an extremely difficult problem. In this case, the main goal is to construct a supercharacter theory that is as close as possible to the theory of irreducible characters. In the talk, we present the supercharacter theories for certain unipotent and solvable groups, and for the parabolic subgroups in GL(n).
Dmitri Panyushev
Takiff algebras with polynomial rings of symmetric invariants
Video
(This is a joint work O.S.Yakimova) We prove that under mild restrictions on the Lie algebra $q=Lie(Q)$ having the polynomial ring of symmetric invariants, the $m$th Takiff algebra of $q$ also has a polynomial ring of symmetric invariants.
Alexander Premet
MishchenkoFomenko subalgebras for centralizers and affine Walgebras.
Let L be the centralizer of a nilpotent element in a finite dimensional semisimple Lie algebra. In my talk, based on a joint work with Arakawa, I will explain how to lift MishchenkoFomenko subalgebras of S(L) to commutative subalgebras of U(L). The construction relies in a crucial way on affine Walgebras at the critical level.
Claudio Procesi
Embedding into matrices
Slides  Video
I will discuss some history and some recent results and conjectures on the following problem.
Give necessary and sufficient conditions that an associative ring $R$ may be embedded in a ring $M_n(A)$ of $n\times n$ matrices, for some $n$, over some commutative ring $A$.
Ossip Schwarzman
Free algebras of the Hilbert modular forms
Video
The talk is based on the joint work with E.Stuken(HSE). Let $K=\mathbb{Q}\sqrt{d}$ (for $d$ a squarefree integer) be a real quadratic field, $A$ is the ring of all algebraic integers in $K$. Consider the Hilbert modular group $ \Gamma_{d}={\mathrm{PSL}}(2,A)$ acting as a discrete group of automorphisms on the product $H \times H$ of two upper half planes. Let $\tau$ be the trasposition of the half planes and $\widehat\Gamma$ be the group generated by $\Gamma_{d}$ and $\tau$.
Denote by $A(\widehat\Gamma)$ the algebra of $\widehat\Gamma$ automorphic forms on $H \times H$. The main goal of the report is the following
Theorem. If the algebra $A(\widehat\Gamma)$ is free then $d \in (2,3,5,6,13,21)$.
I would like to discuss this result as a part of more general problem: find the all discrete lattices acting on homogenous hermitian Cartan domains of type four with free algebras of automorphic forms.
Remarkable contributions to the subject was recently made by E.Vinberg
Vera Serganova
Representation theory and combinatorics related to symmetric superspaces
Video
I start with presenting certain general results and conjectures about spherical supervarieties.
Then I will concentrate on the examples of symmetric superspaces G/K associated with Jordan superalgebras via TKK construction, compute the algebra of invariant differential operators in the ring of regular functions on G/K. Our results show that in all cases the spectra of invariant differential operators can be described by certain supersymmetric polynomials which are specializations of Macdonald polynomials. (joint work with S. Sahi and H. Salmasian).
Oleg Sheinman
Lax operator algebras, integrable systems and matrix divisors
Video
I would like to report on a cycle of works which came into existence due to an illuminating discussion with E.B.Vinberg in 2013.
Lax operator algebras emerged in the joint work by Krichever and the author (2007) where it was observed that the space of Lax operators with the spectral parameter on a Riemann surface invented by Krichever in 2001, possesses a structure of a Lie algebra. However, Lax operator algebras where constructed only for classical Lie algebras, and for several years it was not clear how to define them in terms of root systems. It was a challenging problem because of obvious relation of Lax operator algebras to fundamental questions of the theory of integrable systems and holomorphic vector bundles on Riemann surfaces. In particular, the last relation is based on the Tyurin parameters of holomorphic vector bundles, which in turn go back to matrix divisors invented by A.Weil in the work (1938) considered now as a starting point of the theory of holomorphic vector bundles. For the holomorphic Gbundles, where G is an arbitrary complex reductive group, matrix divisors are responsible for the "algebraic group part" of the theory.
In course of the above mentioned discussion E.B.Vinberg associated local conditions defining Lax operator algebras with Zgradings of reductive Lie algebras. This has led to understanding that the three structures listed in the title are given by the same kind of data consisting of a Riemann surface with marked points, a complex reductive Lie algebra, a tuple of its Zgradings associated with some of the marked points, and, in the case of integrable systems, of a positive divisor supported at the remainder of the points. The theory of finitedimensional integrable systems (like Hitchin systems, gyroscopes, etc.) including their hierarchies and Hamiltonian theory, as well as the theory of matrix divisors of holomorphic Gbundles, are constructed in this general setup now.
In the talk, I am going to define the main objects in terms of the above mentioned data, and formulate main relations between them.
Seth ShelleyAbrahamson
Counting irreducible representations of rational Cherednik algebras of given support
Given a finite complex reflection group W with reflection representation V, one can consider the associated rational Cherednik algebra H_c(W, V) and its representation category O_c(W, V), depending on a parameter c. The irreducible representations in O_c(W, V) are in natural bijection with the irreducible representations of W, and each representation M in O_c(W, V) has an associated support, a closed subvariety of V. Via the KZ functor, the irreducible representations in O_c(W, V) of full support are in bijection with the irreducible representations of the Hecke algebra H_q(W). In the case that W is a finite Coxeter group, I will explain how to count irreducible representations in O_c(W, V) of arbitrary given support by introducing a functor KZ_L, generalizing the KZ functor and depending on a finitedimensional representation L of a rational Cherednik algebra attached to a parabolic subgroup of W. This is joint work with Ivan Losev.
Gregory Soifer
Affine groups acting proper and affine crystallographic groups. Mathematical developments arising from Hilbert 18th problem.
Slides
The study of affine crystallographic groups has a long history which goes back to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in \text{Aff}$(\mathbb{R}^n)$, of crystallographic groups $\G $ acting isometrically on $\mathbb{R}^n$. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group $\G$ acting isometrically on the $n$dimensional Euclidean space $\mathbb R^n$ contains a subgroup of finite index consisting of translations. In particular, such a group $\Gamma$ is virtually abelian, i.e. $\Gamma$ contains an abelian subgroup of finite index. In 1964 Auslander proposed the following conjecture
The Auslander Conjecture. Every crystallographic subgroup $\Gamma$ of \text{Aff}$(\mathbb{R}^n)$ is virtually solvable, i.e. contains a solvable subgroup of finite index. In 1977 J. Milnor stated the following question:
Question. Does there exist a complete affinely flat manifold $M$ such that $\pi_1(M) $ contains a free group ?
We will explain ideas and methods, recent and old results related to the above problems.
Vladimir Stukopin.
Yangian Double of Queer Lie Superalgebra
Video
The Yangian Double of the Queer Lie superalgebra $Q_n$ is described in terms of current Drinfeld generators. We define the PBW basis for Yangian Double and compute Hopf pairing for elements of this basis. We derive the multiplicative formula for the universal Rmatrix of the Yangian Double $DY(Q_n)$ from these Hopf pairing formulas. We also describe a general construction of the Twisted Drinfeld Yangians.
Ronan Terpereau
Automorphism groups of P1bundles over rational surfaces.
In this talk I will explain 1) how to construct a very simple moduli space (=a projective space) for the nondecomposable P1bundles with no "jumping fibres" over Hirzebruch surfaces, and 2) how to deduce the classification of all the P1bundles over rational surfaces whose automorphism group G is maximal in the sense that every Gequivariant birational map to another P1bundle is necessarily a Gisomorphism. (This is a joint work with Jérémy Blanc and Andrea Fanelli.)
Jenia Tevelev
Derived categories of moduli of stable curves
Video
I will report on a joint project with AnaMaria Castravet on birational geometry and derived category of moduli spaces of pointed rational curves and related GIT quotients. The motivating question, due to Orlov, is whether these derived categories have a full exceptional collection permuted by automorphisms of the moduli space. A similar question for the Kgroup of a toric variety was raised by Merkurjev and Panin.
Dmitry Timashev
Orbits in real loci of spherical varieties
Video
The talk is based on a joint work in progress with S. CupitFoutou. Given a spherical variety X for a complex reductive group G defined over real numbers, we address the problem of describing orbits of the real Lie group G(R) in the real locus X(R). (There may be several real orbits even if X is Ghomogeneous.) We concentrate on two cases: (1) X is a symmetric space; (2) G is split over R and X is Ghomogeneous. The answer is similar in both cases: the G(R)orbits are classified by the orbits of a finite reflection group W_X (the "little Weyl group") acting in a fancy way on the set of orbits of T(R) in Z(R), where T is a maximal torus in G and Z is a "BrionLunaVust slice" in X. The latter orbit set can be described combinatorially. We use different tools: Galois cohomology in (1) and Knop's theory of polarized cotangent bundle in (2), and we expect that the second approach can be extended to the nonsplit case.
Nikolai Vavilov
Reverse decomposition of unipotents
The talk surveys an unexpected advance in the structure theory of reductive groups over commutative rings. Decomposition of unipotents, developed by Alexei Stepanov, the author, and others over the last decades, can be viewed as an effective version of the normality of the elementary subgroup of a reductive group in its group of points. In its simplest form, it gives explicit polynomial formulae expressing a conjugate of an elementary unipotent as a product of elementary generators. This year, Raimund Preusser observed that for classical groups essentially the same calculations work also the other way, and give effective versions of description of normal subgroups. Quite surprisingly, in many cases we can get explicit polynomial bounds on the length of expressions of the elementary generators, in terms of the elementary conjugates of an arbitrary group element. In the talk I describe variants of this method for exceptional groups, as well as some applications of this idea, for instance, in description of various classes of subgroups.
Tyakal N. Venkataramana
Hypergeometric Monodromy Groups of Orthogonal Type
Video
A result of Levelt completely describes the monodromy representation associated to the "ClausenThomae hypergeometric functions". Using this and certain results of Beukers and Heckman, we show that if the real Zariski closure of the monodromy group is O(p,q) with p and q at least two, then there are many such monodromy groups which are arithmetic. The roof exploits the existence of many unipotent elements in the monodromy group.
Oksana Yakimova
Quantisation and nilpotent limits of MishchenkoFomenko subalgebras
Video
Based on a joint project with Alexander Molev.
Let A_\mu be the shift of argument, also known as MishchenkoFomenko, subalgebra associated with a linear function \mu. Relying on the explicit description of the FeiginFrenkel centre, we prove that the symmetrisation map solves Vinberg's quantisation problem in all classical types, assuming that \mu is regular, and for any \mu in types A and C. Note that in type A the results were known before. The symmetrisation map commutes with taking limits thus allowing one to quantise various limits of MishchenkoFomenko subalgebras.
Poster presentations
Alexey Petukhov
Topics in coadjoint representations and infinitedimensional Lie algebras
Slides
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