2005-2006 Departmental Colloquium

The organizers of the seminar are  Toufik Mansour and Yuval Ginosar

---

Tuesday at 16:10 on 2005 November 8	Vladimir Rovenski: Saddle submanifolds and saddle foliations The new classes of submanifolds: (k,ε)-saddle, (kε)-parabolic, (k,ε)-asymptotic and (k,ε)-convex are defined in terms of eigenvalues of their 2-nd fundamental form. These submanifolds generalize k-saddle, etc. submanifolds (i.e., ε=0) first introduced by S.Shefel and then studied by A. Borisenko, and naturally arise among submanifolds with extrinsic (sectional, q-th Ricci, q-th scalar) curvature bounded from above (≤ε^2) and small codimension. The stability of various properties of (k,ε)-saddle, etc. submanifolds for small ε is shown. The main results are that a submanifold or the intersection of two submanifolds is s-connected under some assumption. Using homotopy results the corollaries about (co)homology of compact saddle submanifolds are obtained. The possible applications are in the analysis of ``close'' to isometries group actions on manifolds of positive curvature. The dimension of asymptotic subspaces and the index of relative nullity of submanifolds with non-positive extrinsic (sectional, q-th Ricci, q-th scalar) curvature are estimated from below. The characterizations of totally geodesic submanifolds and cylinders and extremal theorem for Riemannian spaces of positive curvature are obtained. Extending the method of proof we give characterize totally umbilical submanifolds among isotropic submanifolds with small codimension. A survey of recent results on saddle foliations completes our report.
Tuesday at 16:10 on 2005 November 15	Olivier Guedon (University of Paris 6): A localization lemma of Lovasz and Simonovits, some generalizations and applications to the geometry. We will present a localization lemma of Lovasz and Simonovits and presents some applications to isoperimetry. Moreover, we will explain why this result may be seen as a consequence of Krein-Milman theorem and will explain some possible generalization. Finally, we will talk about other applications to the study of the volumes of slabs of a convex body in R^n. Joint work with Matthieu Fradelizi.
Tuesday at 16:10 on 2005 November 22	Michael Cwikel: Calderon's interpolation spaces and compact operators. In the 1960's Alberto Calderon developed his beautiful theory of interpolation spaces defined via analytic Banach space valued functions using an abstract version of the classical theorem of Riesz-Thorin. These interpolation spaces have applications in a variety of topics in analysis. Among many other things, Calderon obtained a result concerning the compactness of linear operators acting on his interpolation spaces, but subject to some special conditions on the pair of Banach spaces used to generate the interpolation spaces. During the intervening 40 years Calderon's compactness theorem has been extended to include a number of other cases where other conditions have to be imposed. Furthermore, in the 1990's the analogous compactness theorem for Lions-Peetre interpolation spaces (also introduced in the 1960s) was shown to hold without any such special conditions. We are still trying to find a compactness theorem for Calderon's spaces without any side conditions. In this talk I will describe Calderon's construction and survey some standard and some exotic properties of his spaces. I will mention some open questions, which can be stated relatively simply, without reference to the general machinery of complex interpolation. A positive solution to any one of these questions would suffice to answer Calderon's 40 year old question. Most of this talk will be intended for a general audience familiar with no more than the basic properties of Banach spaces and analytic functions. Some of the results that may be mentioned are recent joint work with Fedor Nazarov, and rather older joint work with Nigel Kalton, Natan Krugljak and Mieczyslaw Mastylo.
Tuesday at 16:10 on 2005 November 29	Anna Melnikov:  B-orbits of nilpotent order 2 in upper-triangular matrices with application to orbital varieties. Let N be the algebra of strictly upper-triangular n x n matrices and X the subset of matrices of nilpotent order 2. Let B be the (Borel) group of invertible upper-triangular matrices acting on N by conjugation. Let B(u) be the orbit of u from X with respect to this action. Let Sym(2) be the subset of involutions in the symmetric group on n elements. We define a new partial order on Sym(2) which gives the combinatorial description of the (Zariski) closure of B(u). We also construct an ideal I(u) in the symmetric algebra of N* whose variety of zeros is the closure Of B(u). We apply these results to orbital varieties of nilpotent order 2 in sl_n(Co) in order to give a complete combinatorial description of the closure of such an orbital variety in terms of Young tableaux. We also construct the ideal of definition of such an orbital variety up to taking the radical.
Tuesday at 16:10 on 2005 December 6	Uri Onn:  Real numbers, p-adic numbers and what's in between A basic construction in algebraic geometry is compactification of varieties in order to obtain global theorems. For example, the vanishing of the sum of multiplicities of zeros and poles of a rational function does not hold for the complex plain but does hold for its compactification, the Riemann sphere. An analogous construction exists for primes in number fields; however, unlike the Riemann sphere in which all points look the same, the resulting construction is much more involved. One is lead to look at completions of the number field, and finds there Archimedean (e.g. real) and non-Archimedean (e.g. p-adic) fields. We shall discuss the above construction and a possible way to relate the different completions.
Tuesday at 16:00 on 2005 December 7	Issai Kantor (Lund University):  Dobling  process For Jordan (super) slgebras and algebraic formalism unting classical untting classical and quantum mechanics. Suppose U is a Poisson algebra , i.e. a linear space with two operations: a commutative associative operation $(x,y)\rightarrow xy$ and a Lie operation $(x,y)\rightarrow \{x,y\} $, jointly satisfying the Leibnitz identity. We introduce a functor $\;\;J$ which assign to a Poisson algebra $U$ a Jordan (super)algebra $J(U)$, which is defined on the direct sum $U\oplus U^s$, where $U^s$ consists of the same elements as $U$ but taken with opposite parity. It turns out that the (super)algebra $J(U)$ remains a Jordan algebra under weaker conditions on the commutative operation $xy$ of the Poisson algebra: associativity is replaced by the Jordan property with additional fourth-degree identity connecting the associator with the Lie operation. It is natural to call such algebras Poisson-Jordan algebras. Allowing to say that Poisson algebra corresponds to the classical mechanics one can say that the notion of Poisson-Jordan algebra include also the quantum mechanics (and possibly some others). An important property of the functor $J$ is the following: the Jordan (super)algebra $J(U)$ is simple if and only if the Poisson-Jordan algebra $U$ is simple. Thus the functor $J$ enable starting from the classification of simple Jordan finite dimensional superalgebras to obtain the classification of simple Poisson-Jordan algebras, which in finite dimensional case consists only of cases corresponding to the classical and quantum mechanics.
Tuesday at 16:10 on 2005 December 13	Aryeh Juhasz: A Freiheitssatz for cyclic presentations. Let P=<X| R> be a one- relator group presentation and assume that |X| is finite. Adding a finite number of new generators and new relators which are obtained from R by shifting the indices of the letters in R, we get a new presentation P'=<Y| R'> with |Y| =|R'| which includes the original presentation as a sub presentation. This presentation is called the cyclic presentation obtained from P. The Freiheitssatz for cyclic presentations claims that the group presented by P is a subgroup of the group presented by P'. We give a sufficient condition in terms of the combinatorial structure of the word R for the Freiheitssatz to hold.
Tuesday at 16:10 on 2005 December 20	Shahar Mendelson (Technion and ANU, Canberra):  Entropy and the combinatorial dimension We will explore recent results connecting the combinatorial dimension of a set of functions (which, roughly put, measures for every t>0 the largest dimension of a "cube" of side length t that can be found in the given set), and the metric entropy of the set. We will present some geometric applications of these results.
Tuesday at 16:10 on 2005 December 27	Jerzy Dydak (University of Tennessee):  Asymptotic dimension Gromov and Dranishnikov introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. One of the main motivations behind the research in asymptotic dimension is the result of G.Yu that the Novikov Conjecture holds for groups of finite asymptotic dimension. We (joint work with Kolya Brodskiy) define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov and Dranishnikov-Keesling-Uspienskij on the dimension of Higson corona
Tuesday at 16:10 on 2006 January 3	Dan Abramovich(Brown University): Stellar subdivisions and blowings up. This lecture is about combinatorial geometry inspired by algebraic geometry. A cornerstone of the classification of algebraic surfaces is: "Every birational map of smooth projective surfaces is composed of a sequence of blowings up followed by a sequence of blowings down." What about higher dimensions? I will present and discuss an embarrassingly simple and explicit conjecture about stellar subdivisions of simplicial complexes. This conjecture would imply the strongest relevant result for three-dimensional algebraic manifolds in characteristic 0.
Tuesday at 16:10 on 2006 January 10	John W. Chinneck (Carleton University): Analyzing Infeasible Optimization Models. Infeasibility is frequent in the early stages of developing a complex optimization model. For linear programs, algorithmic tools that assist in the analysis of infeasibility are well developed, and are available in most commercial LP solvers. There are two main approaches: (1) finding an irreducible infeasible subset of constraints and (2) finding the smallest set of constraints to remove such that the remainder constitute a feasible set. This tutorial talk will focus on these, with the goal of helping modellers make the most effective use of the available tools. In addition, we will briefly explore the state of the art in infeasibility analysis for integer and nonlinear programs, and will also survey how infeasibility analysis algorithms are proving useful in unexpected ways in applications such as data mining and logic programming.
Tuesday at 16:10 on 2006 January 17	Joan Birman: Garside Groups The canonical example of a Garside group is the classical n-strand braid group, with its classical presentation due to Artin. Related examples are any Artin group of finite type, but the class includes much more exotic examples. We will review the area, and recent work, give lots of examples, and discuss why they are an interesting class.
Tuesday at 16:10 on 2006 January 24	Izu Vaisman (Haifa University): New chapters of Poisson geometry: Dirac structures, Courant algebroids, generalized complex structures. Dirac structures, Courant algebroids and generalized complex structures are new chapters of Poisson geometry, which have developed in the last years and are motivated by the appearance of these structures in the dynamics of constrained systems and in supersymmetries in string theory. In the talk we shall present the new structures and the basic examples and include some personal contributions on transitive Courant algebroids and on the theory of submanifolds.
Tuesday at 16:10 on 2006 March 21	Ron Aharoni: Topological methods in combinatorics In 1912 two fundamental theorems were proved. One was Brower's fixed point theorem in topology, and the other Frobenius' theorem, a precursor of Hall's marriage theorem. In 2000 it was realized that the second theorem can be proved using the first. The proof yields far reaching generalizations, for example to marriages in which the men are matched not to women, but to sets, the requirement being that the sets matched to different men are disjoint. We shall describe the topological proof, and various corollaries of the stronger theorem obtained using it. Work done jointly (in various papers) with Eli Berger, Penny Haxell, Maria Chudnovsky, Roy Meshulam, Andrei Kotlov and Ran Ziv.
Tuesday at 16:10 on 2006 April 4	Jonathan Kirby: Intersection theory for some differential equations One of the basic questions in mathematics is "When does a system of equations have a solution?" In the best situations such as linear algebra and algebraic geometry, the answer depends on a dimension theory and the associated intersection theory. I will describe how this also applies to some systems of differential equations arising from Lie theory, such as the exponential equation dy/dx =y. There are applications to problems in diophantine geometry, to the theory of the complex exponential function, and to model theory. In this talk I will give an overview of the ideas and the applications. No specialist knowledge will be assumed.
Tuesday at 16:10 on 2006 April 30	Juris Steprans (York University): Geometric cardinal invariants of the continuum. Suppose it is known that all sets of reals of cardinality $\kappa$ are Lebesgue null. Does it follow that all sets of $\kappa$ lines in the plane have null union? What about circles? These and related questions will be discussed.
Tuesday at 16:10 on 2006 May 16	A. Elashvili (Institute of Math. Academy of Science of Georgia.Tbilisi; Weizmann Institute. Rehovot): Lie Algebras of Simple Hypersurface Singularities. I'll talk about structural properties and numerical invariants of the finite -dimensional solvable Lie algebras naturally associated with simple hypersurface singularities. Joint work with G. Khimshiashvili
Tuesday at 16:10 on 2006 May 23	Y. Yomdin (Weizmann Institute): Analytic reparametrization of semi-algebraic sets and local complexity bounds. In many problems in Analysis and Dynamics it is important to subdivide objects under consideration into simple pieces, keeping control of high order derivatives. It is known that semi-algebraic sets A inside the unit ball allow for a C^k-triangulation, where each simplex is represented as an image, under the ``reparametrization mapping" ψ, of the standard simplex, with all the derivatives of ψ up to order k bounded by 1. The number of simplices in this triangulation is bounded through the degree of A. The main result presented in this talk is, that if we reparametrize all the set A but its small part of a size δ> 0, we can do much more: not only to ``kill" all the derivatives at once, but to bound uniformly the analytic complexity of the pieces, while their number remains of order log(1/ δ). In contrast with the C^k-case, the number of pieces in an analytic reparametrization cannot be bounded through the degree only, and the above result is, essentially, sharp. Relations to the Bernstein type inequalities for algebraic functions, as well as some new open questions concerning the complexity of semi-algebraic sets will be stressed. Initial applications in Analytic Dynamics will be discussed, in particular, explicit bounds on the local volume growth in iterations of analytic mappings.
Tuesday at 16:10 on 2006 May 30	Tomas Kroupa: Measures on MV-algebras The notion of MV-algebra originated in mathematical logic as a certain ``non-idempotent" generalization of Boolean algebra. MV-algebras stand to infinite-valued {\L}ukasiewicz logic the same as two-valued Boolean algebras stand to classical logic. From the viewpoint of ordered algebraic structures, there exists a deep relation between the category of MV-algebras and lattice-ordered Abelian groups with order unit. In the contribution we will study measures on MV-algebras which generalize classical measures on Boolean algebras. The motivation comes from game theory: measures on MV-algebras of functions (so called tribes) were introduced in the field of coalition games by D. Butnariu. We will show an integral representation theorem for measures on MV-algebras of functions resulting from the classical characterization of Choquet simplices. Moreover, some generalizations of well-known theorems from classical theory will be mentioned (Lyapunov and Hahn decomposition theorem).
Tuesday at 16:10 on 2006 June 6	Michael Polyak: Counting Lines and Other Objects How many circles are tangent to three given plane curves? How many triangles similar to a given one may be inscribed in a plane curve? How many lines intersect a link in four points? These are the types of problems which I will consider in this talk. In general, complex enumerative geometry deals with counting algebraic- geometric objects satisfying certain restrictions, e.g., counting a number of algebraic curves of a fixed degree passing through a fixed set of points and tangent to some fixed algebraic curves. I will discuss a real counterpart of such problems, where some objects may be taken smooth instead of rigid algebraic and one counts objects with signs. I will then explain a relation of these problems with the theory of finite type invariants and propose a general setting to produce such invariants using maps of configuration spaces and homology intersections.
Tuesday at 16:10 on 2006 June 13	Johann A. Makowsky: Logic and Combinatorics We study linear recurrence relations for combinatorial counting problems on graphs. There are plenty of such recurrence relations, usually proven for each case with specially tailored methods. We show how tools from logic can be used to prove the mere existence of such recurrence relations. The talk will be quite elementary with little background in logic or graph theory required.

Back to Department of Mathematics homepage