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Wintersemester 2016/17
26.10.2016 um 17:15 Uhr in Raum 69/125
Prof. Dr. Christine Bachoc (Institut de Mathématiques de Bordeaux)
Sets Avoiding the Distance 1 in IRn
What is the supremum density of a measurable set in IRn avoiding distance 1?
If the distance is the Euclidean distance, the answer is known only in the trivial case n = 1.
This problem is closely related to that of the determination of the chromatic number of Euclidean space, a surprisingly difficult problem even in dimension 2, which is open since it was posed by Nelson and Hadwiger in 1950.
We will review recent results in this area, then we will discuss the case of a distance defined by a polytope.
When the polytope tiles the space by translations, we conjecture that the answer is 2−n and that the problem should be much easier that in the Euclidean setting.
We will present a proof of thos conjecture in dimension 2 and discuss a few other cases of Voronoi polytopes associated to root lattices.
02.11.2016 um 17:15 Uhr im Raum 69/125
Prof. Dr. Bertrand Toën (Université de Toulouse)
Algebraic Geometry, Categories and Trace Formulas
In this talk I will present an approach to non-commutative geometry based on the following principle: a non-commutative variety simply is a category. This principle seems naive at first sight, but it has been very fruitful during the last 20 years, thanks to the works of many authors such as: Kapranov, Bondal-Orlov, Rosenberg, Van den Bergh, Artin-Zhang, Kontsevich-Soibelman, Keller etc.
The purpose of this lecture is to explain how we can (or can not) "do geometry" with categories with a particular focus on cohomological and numerical aspects (Euler characteristic, Lefschetz's type trace formula etc). I will illustrate this by exploring its interactions with singularity theories, both in the classical complex analytic situation and in more arithmetic settings. In the last part of the lecture I will explain how this approach to non-commutative geometry can be used in order to make progress on the, still conjectural, Bloch's conductor formula.
16.11.2016 um 17:15 Uhr im Raum 69/125
Prof. Dr. Thomas Wannerer (Universität Jena)
Integral Geometry of Exceptional Spheres
The kinematic formulas for spheres go back to Blaschke, Chern, and Santalo and are in some sense entirely classical. For spheres of dimension 6 and 7 however, new kinematic formulas have been discovered recently. In this talk we will report on ongoing joint work with Gil Solanes on kinematic formulas in these exceptional spheres. We will put our results within the framework of Alesker's theory of valuations and we will explain how properties of the octonions shape the integral geometry in these special dimensions. Moreover, we will discuss consequences for the integral geometry of general isotropic spaces.
23.11.2016 um 17:15 Uhr in Raum 69/125
Prof. Dr. Henning Krause (Universität Bielefeld)
Die Bedeutung des Kommutativen für das Nichtkommutative
In her ICM address (Zürich 1932), Emmy Noether stresses the importance of the noncommutative for the commutative. This explains the title of my talk, which is a report about an analogue of Serre duality for modular representations of finite groups. I will show how replacing sheaf cohomology by group cohomology allows the use of powerful commutative methods.
30.11.2016 um 17:15 Uhr im Raum 69/125
Prof. Dr. Philipp Rostalski (Universität Lübeck)
Semidefinite Relaxations in Convex Algebraic Geometry - Classical Results and New Developments
More than a decade ago Jean-Bernard Lasserre proposed a hierarchy of semidefinite relaxations for global polynomial optimization problems which emerged as a powerful tool and a corner stone of convex algebraic geometry. Its applications cover various areas ranging from combinatorial optimization to the characterization of real varieties. In this talk we will revisit the basic idea behind this algorithm, highlight important implementation aspects and show some recent applications, tools and open problems. The spectrum of examples will range from computationally efficient representations of real algebraic varieties to optimization problems involving non-commutating variables.
07.12.2016 um 17:15 Uhr im Raum 69/125
Prof. Dr. Anja Sturm (Universität Göttingen)
On Theory and Applications of Stochastic Interacting Particle Systems
Stochastic interacting particle systems describe the time evolution of particle systems that are distributed on a discrete space (such as the integer lattice). The random changes to these systems are local, and one of the main points of interest is to understand the global and long-time behavior that arises from a particular set of local interaction rules.
In this talk, we will introduce some classical interacting particle systems including the contact and the voter model. We will relate them to non-spatial counterparts, in particular Galton-Watson branching processes and Cannings models, and discuss properties such as long-time survival and the existence of (nontrivial) invariant laws.
We will then consider these properties for a particular one dimensional model called the cooperative branching coalescent. Here, particles perform independent random walks, only pairs of particles occupying neighboring sites can produce new particles (cooperative branching) and particles that land on an occupied site merge with the particle present on that site (coalescence).
We show that this system undergoes a phase transition as the branching rate is increased:
For small branching rates the so called upper invariant law is trivial and the process started with finitely many particles ends up with a single particle with probability 1. Both statements are not true for high branching rates. These results were obtained in joint work with Jan Swart (UTIA Prag).
14.12.2016 um 17:15 Uhr im Raum 69/125
Prof. Dr. Daniel Frohn (Universität Bielefeld)
Das Skalarprodukt in der analytischen Geometrie - Begriffliches Arbeiten und Aufbau von Grundvorstellungen
Die Analytische Geometrie in der gymnasialen Oberstufe wird zuweilen als eher langweilige Ansammlung verschiedener Verfahren zur Lage- und Abstandsbestimmung von Punkten, Geraden und Ebenen erlebt. Hinzu kommt, dass bei der Beschränkung auf lineare Objekte kaum authentische Anwendungsbezüge hergestellt werden können.
Im Vortrag soll am Beispiel des Skalarproduktes dargestellt werden, wie begriffliches Arbeiten und der Aufbau von Grundvorstellungen in den Mittelpunkt des Unterrichts gestellt werden können, um auf diese Weise Lernprozesse in der analytischen Geometrie intensiver zu fördern. Hierfür erweist sich die Verbindung von arithmetischer und geometrischer Sichtweise als grundlegende Leitidee.
11.01.2017 um 17:15 Uhr im Raum 69/125
Prof. Dr. Annie Cuyt (University of Antwerp)
Identification Problems in Multivariate Sparse Interpolation
We consider the interpolation of a d-variate exponential sum
f(x1 , . . . , xd ) = ∑nj=1(cj exp(fj,1x1 + · · · + fj,dxd)), d ≥ 1.
For d = 1 the problem statement was already solved in the 18th century by de Prony and a whole collection of numerical algorithms is now available. For d > 1 the separation of the individual f j,i , i = 1, . . . , d from a minimal number of samples remains an active topic of research.
We indicate how to identify the f j,i from (d+1)n samples along projections even if theselead to collisions. The method can also be used when d = 1 to identify the proper f j in case of aliasing. Moreover, the new technique can be combined with existing Prony-likealgorithms.
18.01.2017 um 17:15 Uhr im Raum 69/125
Prof. Dr. Christian Haase (FU Berlin)
Finiteness Theorems for Lattice Polytopes
A lattice polytope is the convex hull of finitely many points all
whose coordinates are integers. These fundamental geometric objects
appear in a variety of mathematical fields like combinatorics or
algebraic and symplectic geometry, with applications ranging from
optimization to statistics to mathematical physics in string theory.
Considerable effort has gone into several classification projects for
classes of lattice polytopes. All these classifications are based on
theorems, stating that the class under consideration has only finitely
many elements, or at least that certain parameters are bounded in the
class.
In this talk, I will give an overview of my favorite finiteness
theorems, starting in the 19th century with Pick and Minkowski and
ending with recent developments to be published in 2017+.
25.01.2017 um 17:15 Uhr im Raum 69/125
Prof. Dr. Mohamed Barakat (Universität Siegen)
Constructive Derived Equivalences
Equivalences between derived categories play an increasingly central role in mathematics as they connect various remote areas of algebra and geometry. In this talk I will demonstrate how to turn various abstract constructions (including many examples of spectral sequences) which are inherently nonconstructive in their respective Abelian categories into constructive ones by traveling through the mathematical ``worm holes'' provided by derived equivalences.
01.02.2017 um 17:15 Uhr im Raum 69/125
Prof. Dr. Arie Koster (RWTH Aachen)
Robust Optimization Challenges in Communication Networks
Since many years, communication networks have been a fruitful area to apply and advance mathematical optimization techniques. In this highly competitive market, cost-efficient designs of networks play a critical role. Due to uncertainties about future demands, networks are usually heavily overprovisioned. The methodology of robust optimization allows to reduce this overhead without losing quality of service. In this talk, we discuss the mathematical progress behind three applications of robust optimization: network design under demand uncertainty, robust virtual network embedding, and robust spectrum allocation.
