The Edmund Landau Minerva Center for Research in Mathematical Analysis and Related Areas
Geometry and fundamental group
Ballman, Werner (Bonn University)
The intrinsic structure of groups is reflected by the structure of spaces on which they act. In differential geometry, curvature and other geometric invariants are related to the global geometry and topology of the underlying manifolds and thus have influence on the structure of their fundamental groups. I will discuss some examples and aspects of such relations.
Nonlinear parabolic equations:
From Hamilton-Jacobi to Navier-Stokes
Ben-Artzi, Matania (Hebrew University)
An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.
Laws of Probability and Their Counterparts in Dynamics
Manfred Denker (Göttingen University)
The ergodic theorem may be seen as the stationary version of the strong law of large numbers of probability theory. The talk will focus on other probabilistic laws for independent, identically distributed random variables which can be shown for stationary sequences of observables.
The problem of weak convergence to normality was first proved by Sinai for Hölder continuous functions and the geodesic flow on compact Riemannian manifolds of negative curvature in the sixties. There were hardly any other results until the eighties when the central limit theorem problem gained much interest, followed by waiting time distributions and large deviation results a decade later.
In the talk I will discuss some of the aspects of the theory connected to the central limit theorem (log averaging), Poisson approximation and large deviation (including zeta-functions) for subshifts of finite type.
On estimating the derivatives of symmetric diffusions in stationary random
environment, with applications to ∇φ
Deuschel, Jean-Dominique (Technical University, Berlin)
We consider diffusions on $\Bbb R^d$ or random walks on $\Bbb Z^d$ in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and Hölder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇φ interface model.
Numerical analysis for transient geometric PDEs
Dziuk, Gerhard (Freiburg University)
The main geometric flow problems are gradient flows for geometric energies. Mean curvature flow is the gradient flow for area, Willmore flow appears as gradient flow of the Willmore functional, i.e., the classical bending energy. Besides their mathematical beauty these flows are of interest for a wide range of applications such phase transition problems and image processing. These geometric flows lead to highly nonlinear parabolic and degenerate partial differential equations of second and fourth order. We give a survey over the discretization by finite elements and discuss stability and convergence of the algorithms. In all geometric flow problems the approximation of curvature is an essential part. We will show that it is not necessary to have a pointwise notion of curvature but that it is sufficient to define discrete curvature as a functional on the discrete surface.
The Theta Function in Complex Analysis and Combinatorial Number
Farkas, Hershel (Hebrew University)
In this talk I shall demonstrate the versatility and utility of the theory of theta functions by showing how even a superficial knowlege of the theory allows one to derive results in conformal mapping and combinatorial number theory. The results will involve the representation of positive integers from given sets, Picard's theorem and conformal mapping of rectangles onto the unit disc.
Dynamics of Reaction and Diffusion: A Survey
Bernold Fiedler (Brown University)
We consider the qualitative dynamics of reaction-diffusion systems like ut = D_u + f(u) in 1-3 space dimensions. In one dimension and under mild regularity and growth assumptions on the scalar nonlinearity f, this PDE possesses a compact global attractor A, which attracts all bounded sets of initial conditions. Due to a variational structure, A consists of equilibria ut = 0, and of heteroclinic orbits between them, only. We indicate how an additional nodal property of Sturm type determines the heteroclinic connectivity in the global attractor A, in a purely combinatorial way. Rotating spiral wave patterns are a characteristic feature of planar excitable media systems. Examples include convective fluids, surface catalysis, heart tissue, and the Belousov-Zhabotinsky medium. We view rotating waves as relative equilibria to a (not necessarily differentiable) Euclidean group action. A skew product formulation of bifurcations from rotating waves then allows us to explain phenomena like meandering and drifting tip motion. Scroll waves are three-dimensional stacks of rotating spirals, with tips aligned along filament curves. We introduce and explore the crossover collision as the only generic possibility for scroll wave filaments to change their topological knot or linking type. Our analysis of nodal properties, spiral wave collisions, and crossover collisions is based on elementary singularity theory and Thom transversality. Some collisions will be illustrated by numerical simulations. See also: http://dynamics.mi.fu-berlin.de/
Transportation and Integer Programming
Martin Groetschel (Matheon, TU, and ZIB Berlin)
Transportation problems occuring in practice (such as: public mass transport by bus, train, or plane (including driver assignment and vehicle circulation), dial-a-ride, truck routing, scheduling of service vehicles, logistics, ...) are much more complex tasks than what is usually called "transportation problems" in the mathematical programming terminology.
In this talk I will present an overview of some of the problems of this type that have been attacked using integer programming techniques by the transportation research group at Konrad-Zuse-Zentrum. I will present solutions of very large scale instances from practice and indicate the savings that can result from the use of advanced mathematical solution technology.
Cohomology of the Bianchi groups
Fritz Grunewald (Düsseldorf University)
In my talk I will describe new results concerning the cohomology of the groups SL(2,O) where O is the ring of integers in an imaginary quadratic numberfield.
Some aspects of Schubert calculus
F. Hirzebruch (MPI, Bonn)
We focus on the Grassmann variety of 2-dimensional linear subspaces of the standard complex vector space of dimension n+2. This is a homogeneous projective algebraic variety of Xn of complex dimension 2n which can also be described as the space of lines in the projective space of dimension n+1. We are interested in the Schubert classes fr where fr is a 2r-dimensional integral cohomology class of Xn which is the Poincaré dual of the homology class of the variety of all lines in Pn+1 which intersect a given projective subspace of dimension n - r. The fr are also the Chern classes of the tautological n-dimensional vector bundle over Xn. We want to study the Schubert-Chern numbers of the fr, namely the evaluation of a monomial in the fr of dimension 2n on the fundamental cycle of Xn.
These numbers are very
interesting from a geometric point of view, but also important in number
theory, representation theory, and combinatorics, for example the Catalan
numbers occur and also the Motzkin numbers.
See Sequence A005043 in the On-Line Encyclopedia of Integer Sequences (N. A. J. Sloane).
Isoperimetry, noise sensitivity and Fourier Analysis of
Kalai, Gil (Hebrew University)
In this talk I will describe recent results and open problems concerning discrete isoperimetric inequalies and noise sensitivity and related issues concerning Fourier analysis of Boolean function.
From Göttingen to Jerusalem and back: Edmund Landau, the Berlin Tradition
of reine mathematik and the establishment of The Einstein Institute of
Mathematics at the Hebrew University of Jerusalem
Katz, Shaul (Hebrew University)
The paper explores how it came to pass that precisely at this non-Europe university, The Hebrew University of Jerusalem inaugurated in 1925 and destined to serve the national academic needs of the Jewish people, as well as the developmental requirements of the Middle East and of the Zionist enterprise in Palestine, would be Israel --precisely there, during its first years, an institute of pure mathematics was a central institute; an institute that purposely dismissed from its court, any varied branches of applied mathematics. Since the mathematician from Göttingen, Prof. Edmund Landau (1877-1938) was the founder of The Einstein Institute of Mathematics (EIM) at The Hebrew University of Jerusalem, the paper deals with Landau's early connections with the Hebrew University's project during its first years, prior to his arrival at Jerusalem, with his coming to Jerusalem to inaugurate the EIM, and his staying there during the winter semester of 1927-1928 as director of the EIM. It also deals with the arrival at Jerusalem of Prof. Adolph Abraham Halevi Fraenkel (1891-1965) from Keil and Dr. Mihály-Michael Fekete (1886-1957) from Budapest as successors of Landau and as pious perpetuators of his mathematical ideals and attitudes.
Integration over valued fields
Kazhdan, David (Hebrew University)
I am planning to talk about "Integration over non-archimedian fields".
Let Ã : Qp ! C be a non-trivial additive character. Given a polynomial f : Fn ! F we can define the Fourier transform F(Ã(f)) of the function Ã(f) as a distribution on Fn. In the first part of the talk I'll explain how to see that the existence of a proper algebraic subset X ½ Fn such that the restriction of F(Ã(f)) is given by a locally constant function. If time allows I'll try to explain how to define the notion of integration over non-locally compact fields such as Qp((t)).
What is typical: symmetric or asymmetric manifolds?
Kreck, Matthias (Heidelberg University)
For a metric space a symmetry is an isommetry on the space. Geometrically the most interesting spaces are Riemannian manifolds. Although most Riemannian manifolds which occur "in nature" admit non-trivial symmetries, one expects that this is not typical. A much stronger expectation was formulated about 30 years ago by Raymond and Schulz, namely that a compact smooth manifold picked at random is asymmetric, meaning that for all Riemannian metrics the group of isometries is trivial. I would like to report on the state of the art. Until recently the only known examples of asymmetric manifolds were certain aspherical manifolds, where one uses the fundamental group to show that these manifolds are asymmetric. I want to report about new results concerning simply connected manifolds.
Formal differential geometry on graph algebras
Lawrence-Naimark, Ruth (Hebrew University)
Graph algebras are linear spaces of objects on which there are operations of disjoint union and gluing obeying a natural set of axioms. Typical examples are the algebras of trivalent graphs and of uni-trivalent graphs appearing in the theory of Vassiliev invariants and the Kontsevich integral.
We will start with a description of some of the structures appearing in the analysis of the generalized Jones polynomial of a link and the Witten-Reshetikhin-Turaev invariant of 3-manifolds, based on a Lie algebra, via the comparison of Witten-Chern-Simons theory and combinatorially defined invariants based on the representation theory of quantum groups.
Moving to universal invariants of knots, links and 3-manifolds, we will show the basis of the development of a full formal differential geometry on graph algebras. The main impetus for this work is the study of polynomial finite type invariants of links, where stationary phase integration over coadjoint orbits is necessary in order to study contributions of flat connections other than the trivial one to the Feynman integral form of the Witten-Reshetikhin-Turaev invariants.
This is joint work with Lev Rozansky (UNC, Chapel Hill).
Finite Groups and Hyperbolic Manifolds
Alex Lubotzky (Hebrew University)
The isometry group of a closed hyperbolic n-dimensional manifold is finite. We prove that for every n>1 and every finite group G there exists an n-dimensional closed hyperbolic manifold whose isometry group is isomorphic to G. The cases n=2 and n=3 were proved by Greenberg (1974) and by Kojima (1988), resp.
Our proof is non constructive; it uses a 'probablistic aproch', i.e., counting results from the theory of 'subgroup growth'. The talk is based on a joint paper with M. Belolipetsky with the same title- to apear in Invent. Math.
Applications to Convex Geometry of the Chernoff probabilistic
Milman, Vitali (Tel-Aviv University)
Pseudoholomorpic curves in symplectic topology
Salamon, Dietmar (ETH Zürich)
The structure of p-adic distribution algebras
Schneider, Peter (Muenster University)
In recent years we see an increasing interest (motivated by number theoretical problems) in developing the representation theory of p-adic Lie groups in locally convex vector spaces over p-adic fields. The first task at hand is to construct a reasonable category of such representations which includes all the examples one is interested in but which still is manageable. As very often in representation theory one interpretes representations as modules of some kind of group algebra. The idea then is to impose module theoretic finiteness conditions. In our case these algebras are p-adic distributions algebras. There are a number of very basic difficulties one has to fight with. For example, these algebras are very big. Even worse, p-adic Lie groups do not carry p-adic valued Haar measures. This means most of the traditional methods in the harmonic analysis on real Lie groups cannot be imitated. In joint work with J. Teitelbaum we have developed an axiomatic framework - the theory of Frechet-Stein algebras - which allows the construction of a "good" module category, and we have shown that p-adic distribution algebras are Frechet-Stein.
The lost proof of Loewner's Theorem
Simon, Barry (Caltech)
A real-valued function, F, on an interval (a,b) is called matrix monotone if F(A) < F(B) whenever A and B are finite matrices of the same order with eigenvalues in (a,b) and A < B. In 1934, Loewner proved the remarkable theorem that F is matrix monotone if and only if F is real analytic with continuations to the upper and lower half planes so that Im F > 0 in the upper half plane.
This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been "lost" in that the proof is not mentioned in various books and review article presentations of the subject, and I have found no references to the proof since 1960. The proof uses continued fractions.
I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.
A survey of the highlights of Landau's work
Wefelscheid, Heinrich (Duisburg University)
Edmund Landau (1877 - 1938) wrote 255 articles and 7 books.
In this talk I
will speak about Landau's achievements on the following topics:
Prime number theorems, the prime ideal theorem of algebraic number fields, Dedekinds zeta function, the theorem of Picard-Landau-Schottky ,the representation of definite functions as sums of squares (Hilbert's 17th problem) approximation of continous functions with polynomials, lattice points, additive number theory (Waring's problem and Goldbach's problem).
The main feature of his mathematical activity lies in his ability in creating new techniques which enabled him to simplify difficult proofs and to improve existing results of others or himself.
A new characterization of the Shannon entropy
Weiss, Benjamin (Hebrew University)
Two general ideas:
Last updated: March 30th, 2005
Copyright ©2005 , The Hebrew University of Jerusalem. All Rights
Copyright ©2005 , The Hebrew University of Jerusalem. All Rights Reserved.