Kolloquium WS 2014/15
22.10.2014 um 17:15 Uhr in Raum 69/125
Prof. Dr. Frédéric Déglise (ENS, Lyon)
From Betti Numbers to H-Motives
I will describe the slow evolution from the notion of Betti numbers, invented by Riemann and Betti, all the developments it contained and who were revealed through the 20st century, with an emphasize on the problem of cohomology in algebraic geometry. Going from these numbers, refined as abelian groups according to a suggestion of Noether, we will go to objects of an abstract category (close to abelian groups or rather their complexes). This is the path suggested by Grothendieck, following an initial intuition due to Weil and amplified by Serre; it tries to describe the number of solutions of a given set of algebraic equations over finite fields through a subtle and still mysterious object, its motive.
I will shortly describe a work in collaboration with Denis-Charles Cisinski which sheds light on these motives following an initial program due to Voevodsky.
29.10.2014 um 17:15 Uhr in Raum 69/125
Dr. Vesna Stojanoska (Max Planck Institut, Bonn)
Homotopy and Arithmetic: A Duality Playground
Homotopy theory can be thought of as the study of geometric objects and continuous deformations between them, and then iterating the idea as the deformations themselves form geometric objects. One result of this iteration is that it replaces morphism sets with topological spaces, thus remembering a lot more information. There are many examples to show that the approach of replacing sets with spaces in a meaningful way can lead to remarkable developments. In this talk, I will explain some of my recent work in the case of implementing homotopy theory in arithmetic in a way which produces new results and relationships between some classical notions of duality in both fields.
05.11.2014 um 17:15 Uhr in Raum 69/125
Prof. Dr. Gordon Pipa (Universität Osnabrück)
Neuroinformatics in a Nutshell
Here I will use 3 projects as show cases to give you a flavor of informatics and to demonstrate principle ideas and methodologies used in our field. The talk is intended to serve as an entry point for those that are interested into mathematical and computational neuroscience and theoretical approaches to neuronal information processing. The three projects are presented briefly are: Firstly, Statistical modelling of neuronal activity. Secondly, computation beyond von Neumann with self-organized recurrent networks. And lastly, real-time flight control of quad copter drones using neuoinspired recurrent networks.
12.11.2014 um 17:15 Uhr in Raum 69/125
Prof. Dr. Matthias Bolten (Universität Wuppertal)
Multigrid Methods for Structured Matrices - Theory and Applications
Matrices often possess a lot of structure, for example in the case of Toeplitz matrices or circulant matrices. In the case of Toeplitz or circulant matrices this structure allows for a rigorous analysis of the numerical methods that are applied to these matrices, e.g. to obtain the solution of linear systems. Often these results can be transferred to similar problems, e.g. to partial differential equations discretized on structured grids.
For many problems multigrid methods are optimal solvers. By optimality we mean that the convergence rate is bounded from above independently from the system size and that the number of arithmetic operations grows linear with the system size. Originally multigrid methods have been developed especially for the solution of linear systems that arise when partial differential equations are discretized, later this approach has been extended to general algebraic multigrid methods. Based on these developments multigrid methods for different classes of matrices, including Toeplitz and circulant matrices, have been developed.
Structured problems often arise in physics or engineering when structured discretizations of the continuous problems have been used. In many cases the system matrices belong to the aforementioned matrix classes, so the applications can directly benefit from the theoretical results. If this is not the case, the results can often be used by asymptotic arguments, as well. In any case the structure can be used for an efficient - and in some cases hardware-aware - implementation.
In the talk multigrid methods for Toeplitz and circulant matrices and applications will be presented.
19.11.2014 um 17:15 Uhr in Raum 69/125
Prof. Dr. Henning Kempka (Technische Universität Chemnitz)
Function Spaces with Variable Exponents
We introduce Lebesgue spaces with variable integrability and show how this concept can be used to derive variable versions of Besov-Triebel-Lizorkin spaces. Variable exponent Lebesgue spaces were already intruduced in 1931 by Orlicz but the modern development started in the 90s with upcoming applications in physics (electrorheological fluids), financial mathematics and image processing. We show some basic results for these spaces with integrability and variable smoothness as well as some interesting properties in these scales.
26.11.2014 um 17:15 Uhr in Raum 69/125:
Prof. Dr. Jan Rataj (Universität Prag)
Kinematic Formulas for Delta-Convex Domains
Integral geometry was developed historically in two separated contexts - convex geometry and differential geometry. A unified approach was achieved by Federer (1959) who introduced sets with positive reach including both convex and smooth domains. The notion of positive reach is closely connected with semiconvex functions. Recently, this approach has been generalized by considering DC (or delta-convex) functions instead (differences of two convex functions). One of nice features of DC domains is that the roles of the interior and exterior are symmetric (in contrary to the case of semiconvex domains).
In my talk I will review some important properties of DC functions and outline how the integral geometry for DC domains can be built, i.e., how curvature measures are introduced and why the Gauss-Bonnet formula and kinematic formulas hold.
17.12.2014 um 16:15 Uhr in Raum 69/125
Prof. Dr. Heidemarie Bräsel (Universität Magdeburg)
Alles und Nichts
Alles und Nichts, Unendlich und Null – dies hat schon seit Jahrhunderten die Phantasie der Menschen angeregt. Zuerst mathematisch nicht zu fassen, sind doch die Fortschritte zum ungefähren Begreifen des Nichts und des Unendlichen nicht zu übersehen. Schon die Formulierung „unendlich klein“ scheint ein Widerspruch in sich zu sein. Braucht die Arithmetik überhaupt das Unendliche? Lassen sich nicht alle Aussagen, die von einer natürlichen Zahl n abhängen, durch vollständige Induktion beweisen, wenn sie denn stimmen?
Warum gibt es verschiedene mathematische Interpretationen der natürlichen Zahlen und damit des Unendlichen? Warum hat das Unendlich seinen Schrecken verloren? Der Vortrag wird durch die Geschichte eilen. Die Namen großer Mathematiker und ihrer Erkenntnisse zu der Thematik werden uns sehr eindrucksvoll daran erinnern, dass sie und ihre Forschungen es waren, die für unser heutiges Wissen ein festes Fundament lieferten. Dabei ging nicht immer alles glatt, manche Fragestellungen konnten ganz einfach noch nicht umfassend gelöst werden, da die mathematischen Kenntnisse noch nicht weit genug entwickelt waren. Kurzgeschichten und mathematische Gedichte würzen den Vortrag, es wird experimentiert und auch die Kunst kommt nicht zu kurz. Sind Sie neugierig geworden? Kommen Sie mit auf meine Reise zu dieser Thematik. Nur eins muss noch bemerkt werden, erwarten Sie nicht zuviel!
Der Vortrag ist weder ALLES noch NICHTS zu dieser Problematik.
07.01.2015 um 17:15 Uhr in Raum 69/125
Prof. Dr. Bjørn Ian Dundas (Universität Bergen)
The aim of algebraic topology is to solve geometric problems by assigning algebraic invariants to spaces. These invariants are often very hard to calculate, but a successful technique - both in applications like persistent homology and in theoretical work - has been to study filtrations of the space and try to assemble the invariants for difficult spaces from simpler building blocks. We take this seriously, and study the category of filtrations - ind-spaces. On one level this is equivalent to study the spaces themselves, but carries the possibility of refining the classical invariants considerably - in surprising manners. We start by treating this on a level that should be accessible with a minimum of background, and continue to refine our methods in the second part. In particular, we explore the phenomenon that lengths of homotopies often grow out of control. This is open to attack in our setting, since it is reflected in that colimits in Ind-spaces are quite different from colimits in spaces, making it possible to both restrict how big spaces can grow, but also how long systems of homotopies are allowed to tend. We also will want to consider the proper context of the Adams spectral sequence, where we’ll see that even finite fields have exotic modules in a filtered world. The literature on the dual concept, pro-spaces is much better developed for the simple reason that spaces are built from cells, and this is badly mistreated by limits. However, many of the examples of pro-spaces arise naturally as mapping spaces of ind-spaces. For instance, if you map out of an infinite dimensional space, then the standard procedure is to consider the pro-space you get by considering the mapping space of ind-spaces out of the skeletal filtration. Hence, these examples are fundamentally examples that profitably could be handled by staying in the nicer category of ind-spaces.
14.01.2015 um 17:15 Uhr in Raum 69/125
Prof. Dr. Joseph Gubeladze (San Francisco State University)
The Category of Polytopes
Convex polytopes is the main vehicle for a major part of the contemporary combinatorics, which includes algebraic combinatorics. But the universe of convex polytopes and affine maps between them also resembles the algebraic world of vector spaces. Is there a unified approach that would allow an algebraic treatment of polytopes, like groups, rings, modules? We conjecture that such a fusion of polytopes and algebra is possible, leading to a homological theory of polytopes. The first attempts have already resulted in a series of papers. In the talk I will describe the obtained results and outline the next natural step in the direction of the conjectural theory.
21.01.2015 um 17:15 Uhr in Raum 69/125
Prof. Dr. Andreas Stein (Universität Oldenburg)
Fast Sieving Methods for Solving Discrete Logarithms on Curves and Applications to Ellipic Curve Cryptography
In recent years, elliptic curves have become objects of intense investigation because of their significance to public-key cryptography. The major advantage of ECC is that the cryptographic security is believed to grow exponentially with the length of the input parameters. This implies short parameters, short digital signatures, and fast computations. We first provide a survey of elliptic curves over finite fields and their interactions with algorithmic number theory. Hereby, we discuss various interesting methods to solve the so-called elliptic curve discrete logarithm problem (ECDLP) and their mathematical background. Then, we describe novel and improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of medium-genus to high-genus (hyperelliptic) curves defined over even characteristic fields. The most prominent improvement relies on the effective adaptation of sieving techniques known from the number field sieve, the function field sieve, and the arithmetic of number fields to the curve arithmetic. Those novel results are urgently needed in order to solve concrete problem instances arising from Frey's Weil descent attack methodology for solving the elliptic curve discrete logarithm problem.
28.01.2015 um 17:15 Uhr in Raum 69/125
Prof. Dr. Uwe Leck (Universität Flensburg)
Some Solved and Open Problems in Extremal Combinatorics
Im Vortrag werden ältere und neuere Ergebnisse zu einigen gelösten und offenen Extremalproblemen für Antiketten und Ideale in Booleschen Verbänden und in einer allgemeineren Klasse von Posets vorgestellt. Unter anderem geht es auch um die folgenden Fragen:
- Wie wählt man - für gegebene m und k - m k-Mengen, so dass deren Vereinigungsabschluss möglichst klein ist?
- Ein T-Graph ist ein Graph, in dem jede Kante in mindestens einem Dreieck enthalten ist. Ist G = (V,E) ein T-Graph mit |V | = n und T die Menge der Dreiecke in G, dann gilt |E| - |T| ≤ [(n + 1)^2/8], und diese Schranke ist bestmöglich (Grüttmüller, Hartmann, Kalinowski, Leck, und Roberts, 2009). Wie sieht eine entsprechende Ungleichung aus, wenn jede Kante in einem K_4 enthalten ist?
04.02.2015 um 17:15 Uhr in Raum 69/125
Prof. Dr. Marc van Barel (Katholieke Universiteit, Leuven)
Multivariate Orthogonal Polynomials and "Good" Points for Interpolation
Orthogonal polynomials on the real line satisfy a three term recurrence relation. This relation can be written in matrix notation by using a tridiagonal matrix. Similarly, orthogonal polynomials on the unit circle satisfy a Szeg}o recurrence relation that corresponds to an (almost) unitary Hessenberg matrix. It turns out that orthogonal rational functions with prescribed poles satisfy a recurrence relation that corresponds to diagonal plus semiseparable matrices. This leads to ecient algorithms for computing the recurrence parameters for these orthogonal rational functions by solving corresponding linear algebra problems. In the rst part of this talk we will study several of these connections between orthogonal functions and matrix computations and give some numerical examples illustrating the numerical behaviour of these algorithms. In the second part of the talk we will use multivariate orthogonal polynomials as a tool to find "good" points for polynomial interpolation for several planar regions, e.g., for the square, the L-shape, the disk, . ..