The talk is devoted to a joint work with Anne Moreau. For arbitrary connected reductive group G we consider the motivic integral over the arc space of an arbitrary Q-Gorenstein horospherical G-variety associated with a colored fan and prove a formula for the stringy E-function of a horospherical variety X which generalizes the one for toric varieties. We remark that in contrast to toric varieties the stringy E-function of a Gorenstein horospherical variety X may be not a polynomial if some cones in the fan of X have nonempty sets of colors. Using the stringy E-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.
Usually the definition of pseudo-differential operators is
given in terms of Fourier transform. This definition is based
on a local approximation of a manifold by a linear space.
In some geometric situations our manifold has additional
geometric structures that can not be approximated by linear structures.
In these cases the Fourier transform can not be used.
I will describe alternative definition of pseudo-differential
operators that is formulated in more algebraic terms.
It uses only very soft analysis, and in particular it does
not use Fourier transform.
I hope that some generalizations of this definition would allow
to extend the theory of pseudo-differential operators to more
complicated geometric situations. I will describe an important
example of such situation that naturally arises in
representation theory of reductive groups (e.g. the group G = GL(n,R)).
The connected automorphism group of a complete algebraic variety over the field of complex numbers is known to be an algebraic group; in general, it is not linear nor an abelian variety. The talk will present examples of this situation, and results which bound the non-linearity of the connected automorphism group.
Abstract: By a Schubert variety, we mean the closure of a $B$-orbit in the flag variety $G/B$ of a complex linear algebraic group $G$ with Borel subgroup $B$. Schubert varieties are parameterized by the elements of the Weyl group $W$ of the pair $(G,T)$, where $T$ is a maximal torus in $B$. Recently, Oh, Postnkov and Yoo showed that when $G$ is of type $A$, the Schubert variety corresponding to an element $w\in W$ is smooth iff the wall crossing polynomial defined by the inversion arrangement associated to $w$ equals the Poincar\'e polynomial of the Schubert variety associated to $w$. In this talk, we will make some remarks about the Poincar\'e polynomial of a smooth Schubert variety and discuss the problem of when the inversion set of an element of an arbitrary Weyl group $W$ determines a smooth Schubert variety.
In the talk I shall review some results, obtained in collaboration with Claudio Procesi and Michele Vergne, which completely dermic the possible values of the index of a G-transversally elliptic operator with respect to a compact torus G. We shall then speculate about what should happen when the group G is a compact connected Lie group and show some simple computations in this case.
Wronski map sends a vector of polynomials to their Wronski determinant. A survey of the properties of this map from the point of view of real algebraic geometry will be given. Applications to control theory will be also described.
Many important notions, related to various fields of study, can be regarded as multidimensional generalizations of the discriminant of a univariate polynomial: the Gelfand-Kapranov-Zelevinsky discriminant of a multivariate polynomial, the sparse resultant in symbolic algebra, the discriminant of a deformation in singularity theory, etc. We discuss one more generalization, which interpolates between the aforementioned ones, inherits their nice properties, does not inherit difficulties, and will be called the discriminant of a system of algebraic equations. It is a polynomial of the coefficients of the equations that vanishes whenever the system of equations is atypical in a certain sense. Its existence follows from a similar fact in tropical geometry and finds applications e. g. in the study of topology of polynomial maps.
The first step in understanding the deformation theory of an algebraic variety often consists of a concrete description of the space T^1 of first-order deformations. For a smooth complete toric variety, we give a combinatorial description of this space in terms of the corresponding fan. In this situation, there is a canonical set of one-parameter deformations over the affine line which, when the base is restricted to a fat point, span T^1. We then give a criterion for two (possibly singular) projective toric varieties to appear as special fibers in a common one-parameter flat family, and discuss applications of this to mirror symmetry.
Some recent partial results about the estimates for the number of limit cycles of quadratic vector fields will be presented. These results are due to Libre, Fishkin, and the speaker.
The Hurwitz numbers enumerate the number of possible ways to represent a given permutation as the product of a given number of transpositions. In topological terms, it describes the number of topologically distinct meromorphic functions on a Riemann surface of given genus with prescribed critical values and prescribed behavior at poles. In the case when the surface has genus zero, a closed formula for these numbers was proposed by Hurwitz a century ago. His arguments were algebraic and based on the study of combinatorics of the permutation group. Following the joint work with S. Lando and D. Zvonkine, we propose a new recursion for Hurwitz numbers which has topological origin: it is derived form the cohomological information contained in the stratification of the Hurwitz space by the multisingularity types possessed by the functions. We expect that variations of this approach could be adopted to other families of Hurwitz numbers for which closed formulas are not known at the moment.
We define higher pentagram maps on polygons in any dimension, which extend R.Schwartz's definition of the 2D pentagram map. These maps turn out to be integrable for both closed and twisted polygons. The corresponding continuous limit of the pentagram map in dimension d is shown to be the (2,d+1)-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. In the 3D case we describe the corresponding spectral curve, first integrals, Liouville tori and the motion along them. This is a joint work with Fedor Soloviev (Univ. of Toronto).
The transition constant was introduced in our 1981 paper and denoted by N(14). It is fundamental for classification of arithmetic hyperbolic reflection groups. Using some refine results and methods of our previous papers, we give its upper bound 25. It follows that the degree of ground fields of arithmetic hyperbolic reflection groups is bounded by 25 in dimensions at least 6 and by 35 in smaller dimensions. These bounds give a hope to further enumeration of arithmetic hyperbolic reflection groups.
Two-dimensional lattice polytopes without interior lattice points are completely known. In higher dimension, it is much harder to say something about lattice polytopes that have no lattice points in their interior. However, the question gets more tractable once we assume that not only the lattice polytope itself but also dilates of it have no interior lattice points. In this talk, I will present how starting with a joint paper with Victor Batyrev this elementary looking problem in Ehrhart theory has fueled ongoing research with relations to A-discriminants, toric adjunction theory, and almost-neighborly polytopes.
A function is called Noetherian if belongs to a finitely generated differentiably closed extension of the ring of polynomials. An important example are the Pfaffian functions. Khovanskii's theory of Fewnomials provides uniform bounds on the complexity of real sets defined by Pfaffian functions. For sets defined by Noetherian functions such global bounds are impossible. However, according to a conjecture of Khovanskii, one can still bound the complexity of their germs at every point. To prove this, one needs a bound on the multiplicity of a common zero of several Noetherian functions. In the case of an isolated intersection this bound was provided by Gabrielov and Khovanskii. We consider a particular notion of a multiplicity of a non-isolated intersection suggested by Gabrielov, and provide an upper bound for it in two-dimensional case.
A classical result in complex algebraic geometry states that any smooth cubic surface in P^3(C) contains precisely 27 lines. It is natural to investigate the analogous problem in real algebraic geometry: how many real lines contains a real cubic surface in P^3(R). The answer is well known: a smooth real cubic surface in P^3(R) contains 27, 15, 7 or 3 real lines. A less known result -due to Segre- states that on real cubic surfaces there exists two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choice of orientation data. Two important facts should be noticed: (1) There exists a non-trivial lower bound 3 for the total number of real lines on a smooth real cubic surface. (2) The existence of two kinds of real lines, the definition of the two kinds being intrinsic. My talk -based on a joint paper with Andrei Teleman- has the following goals: (1) explain a general principle which leads to lower bounds in real algebraic geometry, (2) explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, and show that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (3) illustrate these two principles with the enumerative problem of counting real lines in smooth real hypersurfaces of degree 2m-3 in P^m(R) .
We obtain upper bounds for the multiplicity of an isolated solution of a system of equations $f_1=...= f_M =0$ in $M$ variables, where the set of polynomials $(f_1,..., f_M)$ is a tuple of general position in a subvariety of a given codimension which does not exceed $M$, in the space of tuples of polynomials. It is proved that for $M\to\infty$ that multiplicity grows not faster than $\sqrt{M}\exp[\omega\sqrt{M}]$, where $\omega>0$ is a certain constant. It is based on the papers arXiv:1205.1998 and arXiv:1205.1995.
The homogeneous XXX model was introduced by Heisenberg in 1920s to describe a chain of atoms. Only recently it was understood that the spectrum of this model is simple and the eigenstates are labeled by pairs of polynomials f(u), g(u) satisfying the equation f(u)g(u-1)-f(u-1)g(u)=(u+1)^n.
After algebraic functions defined by polynomial equations, the next natural class of functions of one and several real or complex variables consists of "functions defined by polynomial differential equations". However, functions from this class are radically different in almost every respect: the sine has infinitely many roots, and nobody knows how many ovals may have a polynomial (differential) equation on the plane. Askold Khovanski made several breakthroughs in the study of such functions, in particular, (together with Sasha Varchenko) they proved a general finiteness result for the so called Infinitesimal Hilbert 16th problem. I will try to survey almost three decades of research in this direction, describing the recent results of G. Binyamini, D. Novikov, L. Gavrilov, S. Benditkis, G. Dor, P. Mardesic, M. Bobienski, Yu. Ilyashenko etc.
The problem in question is when the topological factor of a compact linear Lie group is homeomorphic to a vector space. Simple examples show that the answer can be both positive and negative. More exactly, the factor of a one-dimensional line by the two-element group generated by the inversion operator is homeomorphic to a closed semiline, while the factor of a two-dimensional plain by the two-element group generated by the inversion operator is homeomorphic to a two-dimensional plain. For the finite linear group case, in 1984 M.A. Michailova proved that the factor is homeomorphic to a vector space if and only if the group is generated by pseudoreflections. In the report we will discuss the results for a group with commutative connected component and for a simple three-dimensional group. In the former case, the main method is the weight system of a representation of a compact torus, and the criterion can be formulated for representations of compact groups with weight systems of a special type an arbitrary representation can be reduced to. As for the latter case, it is proved that the factor of a connected compact group can be homeomorphic to a vector space only in nine cases, two of them being still uninvestigated, six giving the positive answer and one giving the negative one.
The problem on writing convenient explicit formulae for systems of resultants is essential in algebraic geometry and computer algebra. I will write the system of resultants as coefficients of expanded resultant.
This talk addresses the existence of Kahler-Einstein metrics on Fano
varieties. Up to now there exist explicit criteria only for special
classes of Fano varieties. For example, toric Fano varieties admit such
a metric if and only if the barycenter of the corresponding polytope
equals the origin, smooth del Pezzo surfaces are Kahler-Einstein if and only if
their automorphism group is reductive. Already for the case of
(non-toric) singular del Pezzo surfaces the situation in unclear.
In this talk the situation for varieties X with the action of an
algebraic torus T of lower dimension than X itself is discussed.
In this situation, a combinatorial classification of such T-varieties is used,
which generalizes the toric description of varieties by polytopes.
The main result is a generalization of a sufficient criterion for the
existence of Kahler-Einstein metrics on toric varieties by Batyrev and
Selivanova. As an application one obtains a simple example of a Fano
threefold for studying deformations of Kahler-Einstein metrics.
I will describe the properties of a certain discrete dynamical system in the real plane, the closure of whose trajectories looks fairly smooth, and their fractal behaviour can be seen only with a very powerful microscope. A tropical analogue of that system will also be displayed.
Abstract: I will present recent results by M. Belraouti about the geometry of the initial singularity. Let M denotes a flat spacetime which is globally hyperbolic, spatially compact, ie. admitting a time function T: M --> ]0, \infty[ whose levels sets are compact. When the level sets of T are convex, the time function is quasi-concave; it is equivalent to the requirement that the induced riemannian geometry of the level sets is negatively curved. The initial singularity is the limit for t -->0 of the level sets T^{-1}(t) considered as metric spaces. We will prove that, when M has dimension 2+1, this limit exists in the setting of equivariant Gromov-Hausdorff topology of metric spaces, and that it is independant on the quasi-concave time function T. I will also discuss the higher dimensional case.
Let $f, \omega$ be a function and a one-form meromorphic on a compact Riemann surface $R$ and $\gamma\subset R$ be a closed curve. We give necessary and sufficient conditions for the generating function of the ``moments'' $$ m_s=\int_{\gamma} f^s \omega , \ \ \ i \geq 0,$$ to be rational or to vanish identically. As an application we give conditions for the identical vanishing of the Abelian integral $$ I(t)=\int_{\gamma(t)} \omega, $$ where $\omega = P(x,y)dx + Q(x,y) dy$ is a polynomial one form and $\gamma(t)$ is a continuous family of 1-cycles on the family of hyperelliptic curves $y^2-f(x)=t.$ This is a joint work with L. Gavrilov (Toulouse).
In this talk we introduce lattice trigonometric functions of angles in lattice geometry. Using these functions we show a necessary and sufficient condition for three angles to be the angles of some lattice triangle in terms of lattice tangents. This condition is translated to the global relation on singularities of toric surfaces, establishing the criterion for a triple of singularities to be on a toric surfaces whose Euler characteristic equals three. Further we discuss the relations on singularities for toric surfaces of greater Euler characteristic.
Let X be an affine or complete toric variety and Aut(X) be the automorphism group. We will give an explit description of Aut(X)-orbits on X in terms of monoids in the divisor class group Cl(X).
In his seminal article, Michel Demazure gave a combinatorial description of the automorphism group of a complete (smooth) toric variety as a linear algebraic group. The central concept is a root system associated with a complete fan. We describe the automoprhism group of a complete algebraic variety X with torus action of complexity one. The result is based on a presentation of the Cox ring R(X) in terms of trinomials and on an interpretation of Demazure's roots as homogeneous locally nilpotent derivations of R(X). This is a joint work with Juergen Hausen, Elaine Herppich, and Alvaro Liendo.
Let X be a normal affine algebraic variety with regular action of a torus \TT. The algebra A = K[X] of regular functions on X is graded by the lattice of characters of the torus \TT. The degrees of homogeneous locally nilpotent derivations (LNDs) on A are called \TT-roots of the \TT-variety X. This definition imitates in some sense the notion of a root from Lie Theory. Let G_a be the additive group of the ground field K. It is well known that LNDs on A are in bijection with regular actions of G_a on X. Moreover, an LND on A is homogeneous if and only if the corresponding G_a-action is normalized by the torus T in the group Aut(X). In these terms, a root is a character by which T acts on G_a. Let T be a subtorus in \TT. We show that each T-root of X can be obtained by restriction of some \TT-root. This allows to get an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of subtorus action on an affine toric variety are obtained.
On projective algebraic surfaces there are well-known reciprocity laws due to A.N. Parshin for the residues of rational differential two-forms and for the two-dimensional tame symbols. I will speak about a lot of new reciprocity laws on algebraic surfaces which generalize known reciprocity laws. This talk is based on a joint work with X. Zhu.
An affine algebraic variety X is called flexible if the tangent space of X at any smooth point is spanned by the tangent vectors to the orbits of one-parameter unipotent group actions. What is equivalent for X of dimension at least 2, these actions span an infinitely transitive action on the set of smooth points of X. We prove the flexibility of affine cones over del Pezzo surfaces of degree 4 and 5. The construction is general for affine cones over projective varieties and might be used for certain Fano threefolds.
Let G be a simple simply connected algebraic group, V a simple G-module, T a maximal torus in G. We are interested in pairs (G,V) such that for every v in V the closure of Tv is normal. This question has now a complete answer for all irreducible root systems and all the dominant weights. This problem has a nice combinatorial reformulation: saturatedness of certain sets of points. We will discuss methods which help check that a given set is saturated: a classical one using unimodularity, a new development of this named almost unimodularity, and the others. For all modules without this property, we will indicate a T-orbit with non-normal closure.
Let G be a complex linear algebraic group and H be a closed subgroup of G. The construction of a homogeneous space G/H admits a natural generalisation: one can take another subgroup F of G and consider double cosets FgH. The coset FgH is an orbit of g under the action of F \times H given by (f,h) \circ g = fgh^{-1}. Define the double coset variety F\\G//H to be the underlying space of the categorical quotient of G under this action. We consider two phenomena that distinguish double coset varieties from homogeneous spaces. Firstly, by a theorem of Chevalley, G/H is a smooth quasiprojective G-variety. We give examples showing that the double coset variety may not exist. Secondly, by a theorem of Kraft and Popov, if both H and G are reductive then the homogeneous space G/H is never an affine space. It is also not the case for double coset varieties. We will discuss a general approach to determining smoothness of F\\G//H and apply it in the case where G is a classical group, H is a maximal torus of G and F is a connected reductive spherical subgroup. We list all pairs (F, G) such that H\\G//F is an affine space. It turns out that H\\G//F is an affine space if and only if H\\G//F is smooth, or, equivalently, image of the identity element of G is a regular point.
We study moduli space of planar polygonal linkages. It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. We give an explicit formula of the Morse index for the signed area of a cyclic configuration. It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization. Further, we give full classification of all possible local extrema of signed area function.
The compactified Jacobian of a rational curve with one singular point is homeomorphic to the Jacobi factor of the singularity, which depends only on the analytic type of the singularity. J. Piontkowski described the homology of the Jacobi factor a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his answer, in particular, we relate it to combinatorics of partitions, cell decompositions of the Hilbert scheme of points in $\mathbb C^2,$ and the bigraded deformation of Catalan numbers introduced by A. Garsia and M. Haiman. This is a joint work with Eugene Gorsky (arXiv:1105.1151).
Klein's resolvent problem asks whether by means of a rational substitution $y=R(z,a_1,\ldots,a_n)$ the algebraic equation $z^n+a_1z^{n-1}+\ldots+a_n=0$ can be transformed into an algebraic equation on y depending on a small number of independent parameters. In 70s Arnold has proposed to use topological methods to approach a version of this problem with polynomial substitutions, rather than rational ones. Recently this approach has been applied to the original problem, even though rational substitutions, being discontinuous, are not very well suited for applying topological methods to their study. The purpose of the talk is to use this story to give one more confirmation to the thesis, which the speaker heard from A. G. Khovanskii many many times: topological methods give some of the strongest results in questions of solvability of equations in special classes of functions.
We shall survey recent results and applications on virtual polytopes (introduced by A. Pukhlikov and A. Khovanskii).
It is a classical problem in Real Algebraic Geometry that of bounding the topology of real algebraic varieties belonging to a specified family. Typical examples are the case of a nonsingular plane curve of degree d, for which the sum of its Betti numbers cannot exceed (d-1)(d-2)+2, and Khovanskii's fewnomial bound. The well known Oleinik-Petrovskii-Thom-Milnor inequality gives a general bound in the case of a real algebraic set X defined by k equations of degree at most d in n variables: the sum of the Betti numbers of X cannot exceed O(kd)^n; in the case X is the intersection of quadrics this bound is O(2k)^n. Surprisingly enough the fact that the equations defining X have degree at most two allows to interchange the role of the two numbers k and n, i.e. the number of variables and equations, and get the bound n^O(2k) - this is a classical result due to A. Barvinok. We will show that in the case X is defined by few quadrics, this bounds can be even refined; e.g. in the case of the intersection of three quadrics in P^n we get n(n+1), whereas Barvinok's one is at least of the form n^3 (this particular case is interesting by itself because of its connection with Hilbert's 16th problem). A similar approach works also in the case X is the intersection of complex quadrics and gives a formula reminiscent of Morse Theory - relating the topology of the base locus of a linear system of quadrics to the topology of its singular set.
We study smooth algebraic varieties equipped with a symplectic form and with a Hamiltonian action of a reductive group which contain an invariant Lagrangian subvariety. Cotangent bundles of algebraic varieties with a reductive group action are natural examples. It is well-known in symplectic geometry that a symplectic manifold is locally isomorphic to the cotangent bundle of any Lagrangian submanifold. This even holds in the equivariant setting, for Hamiltonian actions of compact Lie groups. However, in the category of algebraic symplectic varieties and Hamiltonian actions of reductive algebraic groups this is no longer true. However, we prove that several important invariants of a Hamiltonian action, such as corank, defect, the dimension of a general orbit, and the image of the moment map, coincide for a given symplectic variety and for the cotangent bundle of an invariant Lagrangian subvariety. The technique used in the proofs is deformation to the normal bundle of a subvariety and the local structure theorem describing the action of a certain parabolic subgroup on an open subset. Some of these results are extended to coisotropic subvarieties, which gives a hope to apply them to a conceptual proof of Elashvili's conjecture on indices of centralizers of nilpotent elements, which is now verified using case by case considerations.
Let $G$ be a reductive complex algebraic group, $B$ its Borel subgroup and $G/B$ the flag variety. Let $W$ be the Weyl group of $G$, $w$ an element of $W$ and $C_w$ the tangent cone at the point $p = B/B$ to the Schubert subvariety $X_w$ of $G/B$. We formulate and discuss some conjectures about $C_w$ in terms of the Weyl group. We prove these conjectures for some particular classes of elements of $W$. We also describe connections with the geometry of coadjoint orbits of $B$.
According to the orbit method, for unipotent groups there exists one to one correspondence between irreducible representations and coadjoint orbits. The problem is that A.A.Kirillov’s formula for a character of irreducible representation is not exact and do not present a character as a function (or generalized function) on the group. In particular, it is not clear how to characterize the support of the character, associated with a given orbit. In the talk we observe known cases of exact calculation of characters and set conjectures for general case.
Let G be a simple Lie group. The flag varieties of G are known to be spherical, i.e. they are acted upon by the Borel subgroup B with an open dense orbit. It has been shown by several authors that flag varieties can be degenerated into toric varieties with B replaced by an algebraic torus. In the talk we will describe the intermediate degeneration. Our varieties are acted upon by the unipotent abelian group (which is nothing but a product of several copies of the additive group of the field) and there exists an open dense orbit with respect to this action. We will give an explicit description of the degenerate flag varieties in type A and describe their algebro-geometric and topological properties.
We give fewnomial bounds, bounds which depend only on the number of monomials appearing in the equations, for the number of positive solutions of real tropical polynomials systems.
This is a joint work with H. Pottmann, L. Shi and F. Nilov.
Motivated by potential applications in architecture, we study surfaces in 3-dimen\-sional Euclidean space containing several circles through each point. Complete classification of such surfaces is a challenging open problem. First, we provide some partial classification results. We give a short proof of the Takeuchi theorem stating that a surface containing 7 circles through each point must be a sphere, and we show that a surface containing 4 circles through each point must be a Darboux cyclide.
Second, we study Darboux cyclides in more detail. Darboux cyclides are algebraic surfaces of order at most 4 and are a superset of Dupin cyclides and quadrics. They contain to 6 circles through each point.
We show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides distinct from spheres.
Most part of the talk is elementary and is accessible for high school students. Several open problems are stated. An opportunity to see a lot of surfaces containing several circles through each point and to hold a Darboux cyclide in hands is provided.
The speaker is supported in part by the President of the Russian Federation grant MK-3965.2012.1, by
``Dynasty'' foundation, by the Simons--IUM fellowship, and by grant RFBR-12-01-00748-a.
Schubert polynomials, defined by Lascoux and Schuetzenberger, are an essential tool for studying Schubert varieties in a full flag variety. Recall that a permutation is called a Richardson one if it is the longest element in the Weyl group $W^L$ for some Levi subgroup $L\subset GL(V)$. We show that the principal specialization of the Schubert polynomial for a Richardson permutation can be computed by a determinantal formula involving the Carlitz-Riordan $q$-Catalan numbers. As a corollary, we obtain a formula expressing the multiplicity of the most singular point in the Schubert variety corresponding to a Richardson permutation via Catalan numbers. These multiplicities also have a nice combinatorial interpretation: they are given by the number of certain plane partitions. Time permitting, we will also discuss a K-theoretic analogue of this statement, expressing the specializations of Grothendieck polynomials via Schroeder numbers by means of the same determinantal formulas.