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Seminar in Geometry and Topology

TuesdayDec 12, 201716:15
Seminar in Geometry and TopologyRoom 155
Speaker:Ary ShavivTitle:Tempered Manifolds and Schwartz Functions on ThemAbstract:opens in new windowin html    pdfopens in new window

Schwartz functions are classically defined as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. This was formulated on $\mathbb{R}^n$ by Laurent Schwartz, and later on Nash manifolds  (smooth semi-algebraic varieties) by Fokko du Cloux and by Rami Aizenbud and Dima Gourevitch. In a joint work with Boaz Elazar we have extended the theory of Schwartz functions to the category of (possibly singular) real algebraic varieties. The basic idea is to define Schwartz functions on a (closed) algebraic subset of $\mathbb{R}^n$ as restrictions of Schwartz functions on $\mathbb{R}^n$.

Both in the Nash and the algebraic categories there exists a very useful characterization of Schwartz functions on open subsets, in terms of Schwartz functions on the embedding space: loosely speaking, Schwartz functions on an open subset are exactly restrictions of Schwartz functions on the embedding space, which are zero "to infinite order" on the complement to this open subset. This characterization suggests a very intuitive way to attach a space of Schwartz functions to an arbitrary (not necessarily semi-algebraic) open subset of $\mathbb{R}^n$.

In this talk, I will explain this construction, and more generally the construction of the category of tempered smooth manifolds. This category is in a sense the "largest" category whose objects "look" locally like open subsets of $\mathbb{R}^n$ (for some $n$), and on which Schwartz functions may be defined. In the development of this theory some classical results of Whitney are used, mainly Whitney type partition of unity (this will also be explained in the talk). As time permits, I will show some properties of Schwartz functions, and describe some possible applications. This is a work in progress.

MondayFeb 13, 201716:15
Seminar in Geometry and TopologyRoom 155
Speaker:Mikhail KarpukhinTitle:Eigenvalue bounds on surfaces: some recent advancesAbstract:opens in new windowin html    pdfopens in new window
We will give an overview of some recent results on Laplace and Steklov eigenvalue estimates on Riemannian surfaces. In particular, we will present an upper bound on the first Laplace eigenvalue for non-orientable surfaces, extending some classical inequalities due to Yang, Li and Yau. We will also discuss the Steklov eigenvalue problem that has attracted a lot of attention in the past decade. In particular, geometric estimates on Steklov eigenvalues of arbitrary index will be presented.
MondayJan 09, 201716:15
Seminar in Geometry and TopologyRoom 155
Speaker:Gal BinyaminiTitle:Wilkie's conjecture for restricted elementary functionsAbstract:opens in new windowin html    pdfopens in new window

Let X be a set definable in some o-minimal structure. The Pila-Wilkie theorem (in its basic form) states that the number of rational points in the transcendental part of X grows sub-polynomially with the height of the points. The Wilkie conjecture stipulates that for sets definable in $R_\exp$, one can sharpen this asymptotic to polylogarithmic.
I will describe a complex-analytic approach to the proof of the Pila-Wilkie theorem for subanalytic sets. I will then discuss how this approach leads to a proof of the "restricted Wilkie conjecture", where we replace $R_\exp$ by the structure generated by the restrictions of $\exp$ and $\sin$ to the unit interval (both parts are joint work with Dmitry Novikov). If time permits I will discuss possible generalizations and applications.

TuesdayDec 27, 201616:00
Seminar in Geometry and TopologyRoom 208
Speaker:Boris ZilberTitle:On algebraic and diophantine geometry in characteristic 1Abstract:opens in new windowin html    pdfopens in new window
I will start with a motivation of what algebraic (and model-theoretic) properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how a search of similar properties lead to a well-known now Hrushovski's construction and then formulate very precise properties that such a construction produces and so the field must satisfy. The axioms have a form of diophantine and valuation-theoretic statements in positive characteristics and the consistency of those remain an open problem. A special case of the axioms has been confirmed by a theorem of F.Bogomolov.
MondayDec 26, 201616:15
Seminar in Geometry and TopologyRoom 155
Speaker:Boris KhesinTitle:Optimal transport and geodesics on diffeomorphism groupsAbstract:opens in new windowin html    pdfopens in new window
We revisit how the Euler and Burgers equations arise as geodesics on the groups of diffeomorphisms. It turns out that the Euler hydrodynamics is in a sense dual to problems of optimal mass transport. We also describe L^2 and H^1 versions of the the Wasserstein space of volume forms. It turns out that for the homogeneous H^1 metric the Wasserstein space is isometric to (a piece of) an infinite-dimensional sphere and it leads to an integrable generalization of the Hunter-Saxton equation.
TuesdayDec 20, 201616:00
Seminar in Geometry and TopologyRoom A
Speaker:Boaz Elazar Title:Schwartz functions on real algebraic varietiesAbstract:opens in new windowin html    pdfopens in new window
We define Schwartz functions and tempered functions on affine real algebraic varieties, which might be singular. We prove that some of the important classical properties of these functions, such as partition of unity, characterization on open subsets, etc., continue to hold in this case. Some of our proves are based on the works of Milman, Bierstone and Pawlucki on Whitney's extension problem and composite differentiable functions. Joint work with Ary Shaviv.
TuesdayDec 13, 201616:00
Seminar in Geometry and TopologyRoom 208
Speaker:Ilya Kossovskiy Title:On the Gevrey regularity of CR-mappingsAbstract:opens in new windowin html    pdfopens in new window

Cauchy-Riemann maps (shortly: CR-maps) occur in complex analysis as boundary values of maps holomorphic in a domain in complex space. As a rule, CR-mappings of real-analytic hypersurfaces appear to be analytic as well. However, we recently showed in a joint work with Rasul Shafikov the existence of Stokes Phenomenon in CR-geometry: there exist real-analytic hypersurfaces, which are equivalent formally, but not holomorphically. 
Despite of this, it appears that in complex dimension 2, CR-maps necessarily posses appropriate weaker regularity properties. Namely, components of such maps necessarily belong to the well known Gevrey classes. The latter statement has the following remarkable application: if two real-analytic hypersurfaces in complex two-space are equivalent formally, then they are also equivalent smoothly. 
The proof of all these facts employs the recent multi-summability theory in Dynamical Systems. It as well employs the recent CR-DS technique that we developed, which connects CR-manifolds and certain Dynamical Systems. In this talk, I will outline the technique, as well as some details of the proof.

TuesdayAug 23, 201616:00
Seminar in Geometry and TopologyRoom 155
Speaker:Misha Verbitsky Title:Homogeneous dynamic, hyperbolic geometry and cone conjectureAbstract:opens in new windowin html    pdfopens in new window
Hyperbolic manifold is a Riemannian manifold of constant negative curvature and finite volume. Let S be a set of geodesic hypersurfaces in a hyperbolic manifold of dimension >2. Using Ratner theory, we prove that either S is dense, or it is finite. This is used to study the Kahler cone of a holomorphically symplectic manifold. It turns out that the shape of the Kahler cone is encoded in the geometry of a certain polyhedron in a hyperbolic manifold. I will explain how this correspondence works, and how it is used to obtain the cone conjecture of Kawamata and Morrison. This is a joint work with Ekaterina Amerik.
TuesdayJun 07, 201616:00
Seminar in Geometry and TopologyRoom 155
Speaker:Alexey Glutsyuk Title:On periodic orbits in complex planar billiardsAbstract:opens in new windowin html    pdfopens in new window
A conjecture of Victor Ivrii (1980) says that in every billiard with smooth boundary the set of periodic orbits has measure zero. This conjecture is closely related to spectral theory. Its particular case for triangular orbits was proved by M. Rychlik (1989, in two dimensions), Ya. Vorobets (1994, in any dimension) and other mathematicians. The case of quadrilateral orbits in dimension two was treated in our joint work with Yu. Kudryashov (2012). We study the complexified version of planar Ivrii's conjecture with reflections from a collection of planar holomorphic curves. We present the classification of complex counterexamples with four reflections and partial positive results. The recent one says that a billiard on one irreducible complex algebraic curve without too complicated singularities cannot have a two-dimensional family of periodic orbits of any period. The above complex results have applications to other problems on real billiards: Tabachnikov's commuting billiard problem and Plakhov's invisibility conjecture.
TuesdayMay 31, 201616:00
Seminar in Geometry and TopologyRoom 155
Speaker:Askold KhovanskiiTitle:Topological Galois theoryAbstract:opens in new windowin html    pdfopens in new window
In the topological Galois theory we consider functions representable by quadratures as multivalued analytical functions of one complex variable. It turns out that there are some topological restrictions on the way the Riemann surface of a function representable by quadratures can be positioned over the complex plan. If a function does not satisfy these restrictions, then it cannot be represented by quadratures. This approach, besides its geometrical appeal, has the following advantage. The topological obstructions are related to the character of a multivalued function. They hold not only for functions representable by quadratures, but also for a more wide class of functions. This class is obtained adding to the functions representable by quadratures all meromorphic functions and allowing the presence of such functions in all formulae. Hence the topological results on the non representability by quadratures are stronger that those of algebraic nature.
TuesdayMay 24, 201616:15
Seminar in Geometry and TopologyRoom 155
Speaker:Yu. Ilyashenko Title:Towards the global bifurcation theory on the planeAbstract:opens in new windowin html    pdfopens in new window
The talk provides a new perspective of the global bifurcation theory on the plane. Theory of planar bifurcations consists of three parts: local, nonlocal and global ones. It is now clear that the latter one is yet to be created. Local bifurcation theory (in what follows we will talk about the plane only) is related to transfigurations of phase portraits of differential equations near their singular points. This theory is almost completed, though recently new open problems occurred. Nonlocal theory is related to bifurcations of separatrix polygons (polycycles). Though in the last 30 years there were obtained many new results, this theory is far from being completed. Recently it was discovered that nonlocal theory contains another substantial part: a global theory. New phenomena are related with appearance of the so called sparkling saddle connections. The aim of the talk is to give an outline of the new theory and discuss numerous open problems. The main new results are: existence of an open set of structurally unstable families of planar vector fields, and of families having functional invariants (joint results with Kudryashov and Schurov). Thirty years ago Arnold stated six conjectures that outlined the future development of the global bifurcation theory in the plane. All these conjectures are now disproved. Though the theory develops in quite a different direction, this development is motivated by the Arnold's conjectures.
TuesdayMay 17, 201616:15
Seminar in Geometry and TopologyRoom 155
Speaker:Boris LevitTitle:Optimal Interpolation in approximation theory, nonparametric regression and optimal designAbstract:opens in new windowin html    pdfopens in new window

For some rectangular Hardy classes of analytic functions,an optimal method of interpolation has been previously found, within the framework of Optimal Recovery. It will be shown that this method of interpolation, based on the Abel-Jacobi elliptic functions,  is also optimal, according to corresponding criteria of Nonparametric Regression and Optimal Design.

In a non-asymptotic setting, the maximal mean squared error of the optimal interpolant is evaluated explicitly, for all noise levels away from 0. In  these results, a pivotal role is played by an interference effect, in which both the stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating processes mutually subduing each other.

TuesdayMay 12, 201516:00
Seminar in Geometry and TopologyRoom 261
Speaker:Gal BinyaminiTitle:Counting solutions of differential equationsAbstract:opens in new windowin html    pdfopens in new window

We consider the following problem: given a set of algebraic conditions on an $n$-tuple of functions and their first $l$ derivatives, admitting finitely many solutions (in a differentiably closed field), can one give an upper bound for the number of solutions?

I will present estimates in terms of the degrees of the algebraic conditions, or more generally the volumes of their Newton polytopes (analogous to the Bezout and BKK theorems). The estimates are singly-exponential with respect to $n,l$ and have the natural asymptotic with respect to the degrees or Newton polytopes, sharpening previous doubly-exponential estimates due to Hrushovski and Pillay. I will also discuss some diophantine applications to counting transcendental lattice points on algebraic varieties.

TuesdayApr 28, 201516:00
Seminar in Geometry and TopologyRoom 261
Speaker:Marina Prokhorova Title:The index theorem for self-adjoint elliptic operators with local boundary conditionsAbstract:opens in new windowin html    pdfopens in new window

The spectral flow is a well-known invariant of a 1-parameter family of self-adjoint Fredholm operators. It is defined as the net number of operator’s eigenvalues passing through 0 with the change of parameter.

Let S be a compact surface with non-empty boundary. Consider the space Ell(S) of first order self-adjoint elliptic differential operators on S with local boundary conditions. The first part of the talk is devoted to the computing of the spectral flow along loops in Ell(S), and also along paths with conjugated ends.

After that we consider more general situation: a family of elements of Ell(S) parameterized by points of a compact space X. We define the topological index of such a family and show that it coincides with the analytical index of the family. Both indices take value in K^1(X). When X is a circle, this result turns into the formula for the spectral flow from the first part of the talk.

ThursdayJan 22, 201514:00
Seminar in Geometry and TopologyRoom 1
Speaker:Egor ShelukhinTitle:Autonomous Hamiltonian flows, Hofer's geometry and persistence modulesAbstract:opens in new windowin html    pdfopens in new windowNote unusual day, time and room
We describe how persistence modules, a notion originating in data sciences, can be applied to obtain new results on Hofer's geometry, related in particular to autonomous Hamiltonian flows. Joint work with Leonid Polterovich.
TuesdayDec 23, 201416:00
Seminar in Geometry and TopologyRoom 261
Speaker:A. GabrielovTitle:Classification of spherical quadrilaterals (part 2)Abstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL PLACE AND TIME
A spherical quadrilateral (membrane) is a bordered surface homeomorphic to a closed disc, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that at most three angles at the corners are not multiples of Pi. This is a very old problem, related to the properties of solutions of the Heun's equation (an ordinary differential equation with four regular singular points). The corresponding problem for the spherical triangles, related to the properties of solutions of the hypergeometric equation, has been solved by Klein, with some gaps in Klein's classification filled in by Eremenko in 2004. The general quadrilateral case remains open. This is joint work with V. Tarasov (IUPUI). The first part is by A. Eremenko (Purdue), 13:00-14:00, room 1. The second is by A. Gabrielov (Purdue), 16:00-17:15, room 261.
TuesdayDec 23, 201413:00
Seminar in Geometry and TopologyRoom 1
Speaker:A. EremenkoTitle:Classification of spherical quadrilaterals (part 1)Abstract:opens in new windowin html    pdfopens in new windowNOTE UNUSUAL PLACE AND TIME
A spherical quadrilateral (membrane) is a bordered surface homeomorphic to a closed disc, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that at most three angles at the corners are not multiples of Pi. This is a very old problem, related to the properties of solutions of the Heun's equation (an ordinary differential equation with four regular singular points). The corresponding problem for the spherical triangles, related to the properties of solutions of the hypergeometric equation, has been solved by Klein, with some gaps in Klein's classification filled in by Eremenko in 2004. The general quadrilateral case remains open. This is joint work with V. Tarasov (IUPUI). The first part is by A. Eremenko (Purdue), 13:00-14:00, room 1. The second is by A. Gabrielov (Purdue), 16:00-17:15, room 261.
ThursdayDec 18, 201414:00
Seminar in Geometry and TopologyRoom 261
Speaker:Andrei GabrielovTitle:Lipschitz contact equivalence of functions in two variablesAbstract:opens in new windowin html    pdfopens in new windowPLEASE NOTE UNUSUAL DAY AND TIME
We consider germs at the origin in $R^2$ of continuous functions definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). We construct a complete invariant of an equivalence class of such functions with respect to Lipschitz contact equivalence. A similar construction produces a complete bi-Lipschitz invariant for a germ of a real definable two-dimensional surface in $R^n$. This is joint work with L. Birbrair and A. Fernandes (University of Ceara, Fortaleza, Brazil)
TuesdayNov 18, 201416:00
Seminar in Geometry and TopologyRoom 261
Speaker:Tony Yue YuTitle:The moduli stack of non-archimedean stable mapsAbstract:opens in new windowin html    pdfopens in new window
Tropical geometry is a powerful technique to study enumerative problems in algebraic geometry. The theory of Berkovich spaces gives us a natural framework to apply tropical techniques in a much wider context. I will begin by explaining motivations from mirror symmetry. Then I will introduce a notion of Khaler structures in non-archimedean geometry. I will explain the construction of the moduli stack of non-archimedean stable maps and an analog of Gromov-Fs compactness theorem in the non-archimedean setting. They are the first steps of enumerative non-archimedean geometry. I will also discuss the tropicalization of the space of stable maps. They are based on arXiv 1401.6452 and 1407.8444. If time permits, I will discuss a related joint work with M. Porta concerning higher non-archimedean stacks and GAGA theorems.
TuesdaySep 16, 201416:00
Seminar in Geometry and TopologyRoom 261
Speaker:Alex IsaevTitle:Isolated hypersurface singularities and associated formsAbstract:opens in new windowin html    pdfopens in new window
In our recent articles joint with M. Eastwood and J. Alper, it was conjectured that all rational $GL_n$-invariant functions of forms of degree $d>2$ on complex space $C^n$ can be extracted, in a canonical way, from those of forms of degree $n(d-2)$ by means of assigning every form with nonvanishing discriminant the so-called associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the well-known Mather-Yau theorem. Settling the conjecture is part of our program to solve the reconstruction problem for quasihomogeneous isolated hypersurface singularities. This amounts to showing that a certain system of invariants arising from the Milnor algebras of such singularities is complete, and the conjecture implies completeness in the homogeneous case. In my talk I will give an overview of the recent progress If time permits, I will further discuss the map that assigns a non-degenerate form its associated form. This map is rather natural and deserves attention regardless of the conjecture. For instance, it induces a natural equivariant involution on the space of elliptic curves with non-vanishing j-invariant. Surprisingly, the existence of such an involution appears to be a new fact.