FB 6 Mathematik/Informatik

Institut für Mathematik


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SS 2017

26.04.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Ole Christensen (Danmarks Tekniske Universitet Lyngby)

Series Expansions in Hilbert Spaces and Exact Representations Versus Approximations

03.05.2017 um 17:15 Uhr in Raum 69/125

Dr. Tino Ullrich (Universität Bonn/derzeit Vertretung in Osnabrück)

"Hyperbolic cross approximation"

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science, new challenging problems have appeared. In this talk we survey on classical as well as contemporary methods developed in multivariate approximation theory in the last decades. We will focus on sampling recovery, nonlinear approximation and numerical integration. The respective algorithms are known to work very well in moderate space dimensions and have potential for applications in really high dimensions. It is now well understood that this theory is important both for theoretical study and for practical applications. Nevertheless, both theoretical analysis and construction of practical algorithms turn out to be very difficult problems. Motivated by recently discovered deep connections between hyperbolic cross approximation (and related sparse grids) and other areas of mathematics (such as probability, number theory, Banach space geometry) we try to put emphases on the development of ideas and methods rather than just listing known results in this area. We will also focus on very recent results in this direction.

10.05.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Francisco Santos Leal (Universidad de Cantabria)

Classification of Empty 4-Simplices

An empty simplex is a lattice $d$-polytope with no other lattice point than its $d+1$ vertices. Lattice $3$-simplices were classified some 50 years ago by White, te classification following easily from the fact that they all have width one (that is, for each empty 3-simplex $P$ there is an affine integer functional with $f(P)=[0,1]$). In this talk I will explain what is known about empty 4-simplices. In particular, I will comment on three proofs of the fact that all but finitely many (equivalence classes of) empty 4-simplices have width one or two: - the nice (but incomplete) proof by Barile, Bernardi, Borisov, and Kantor ("On empty lattice simplices in dimension 4"., Proc. Am. Math. Soc. 139, 2011, 4247-4253). - the proof by Blanco, Haase, Hofmann, and Santos (arXiv:1607.00798), based on the classification of lattice 3-polytopes without interior lattice points by Averkov et al. - the proof by Santos and Iglesias (in preparation, a poster was presented in www.mi.fu-berlin.de/en/math/groups/discgeom/Events/Lattice_Poly_Workshop/index.html). This one comes with a complete enumeration of empty 4-simplices of width three or more. (The list of them was found by Haase and Ziegler in 2000, but it was only conjectured to be complete).

17.05.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Michael Möller (Universität Siegen)

The Homotopy Method: Following the Solution Path of ℓ1 Regularized Linear Inverse Problems

24.05.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Thomas Kahle (Universität Magdeburg)

The Semi-Algebraic Geometry of Poisson Regression

Designing experiments for non-linear regression is hard because the optimal experiment to gain information about the unknown parameters depends on the unknown parameters. We pursue a geometric approach to this chicken and egg problem: In parameter space, we describe regions of optimality of certain experimental designs. It turns out that these regions are often semi-algebraic, that is, described by polynomial inequalties. Therefore the methods from computational logic and real algebraic geometry can be applied.

31.05.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Frank Vallentin (Universität zu Köln)

Some Applications of Semidefinite Optimization

In this talk I will present some applications of semidefinite optimization in discrete geometry.
How densely can one pack given objects into a given container? Such packing problems are fundamental problems in discrete geometry. Next to being classical mathematical challenges there are many applications in diverse areas such as information theory, materials science, physics, logistics, approximation theory.
Studying packing problems one is facing two basic tasks: Constructions: How to construct packings which are conjecturally optimal? Obstructions: How to prove that a given packing is indeed optimal? In the talk I want to explain computational tools based on semidefinite programming for both tasks. I will report on computational results, which are frequently the best-known.

07.06.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Christian Haesemeyer (University of Melbourne)

Algebraic K-Theory: Stable Linear Algebra over Rings. With Particular Attention to Rings of Functions on Singularities

It is a basic result in linear algebra that every idempotent matrix over a field is diagonalisable. If we replace fields by more general rings, this fails, even stably; the failure is one property of rings measured by the invariant called algebraic K-theory. In this talk we will discuss a long-running project (joint with G. Cortiñas, M. Schlichting, M. Walker and C. Weibel) to understand algebraic K-theory of rings of functions on algebraic singularities using methods from algebra, algebraic geometry, and homotopy theory, and will attempt to explain how this theoretical work allows us to answer the following question: If every idempotent matrix over a ring R, and also over the polynomial ring R[t], is stably diagonalisable, is the same true for the ring R[s,t]? Time permitting we will finish by describing our most recent results. 

 

14.06.2017 um 17:15 Uhr im Raum 69/125

Prof. Dr. Victoria Hoskins (FU Berlin)

Moduli of Vector Bundles and their Motives

Moduli problems are classification problems in algebraic geometry whose solutions are given by moduli spaces, which could be an algebraic variety or more generally a stack. I will introduce moduli problems and focus on the classical example of moduli of vector bundles over a smooth projective algebraic curve. Moduli spaces are ubiquitous in algebraic geometry and there has been a proliferation of studying their geometry, topology and (co)homological invariants in recent decades. Following Grothendieck's vision that a motive of an algebraic variety should capture many such invariants, Voevodsky introduced a category of motives which partially realises this idea. After explaining some of the properties of this category, I will explain how one can define the motive of certain algebraic stacks. Finally I will report on joint work with Simon Pepin Lehalleur, in which we study the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of the curve.

21.06.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Bernd Wollring (Universität Kassel)

Mathematikdidaktische Diagnostik – Was eignet sich als Studienelement?

Geht man davon aus, dass zunehmend Probleme zur Heterogenität, insbesondere solche, die durch die bildungspolitische Eigendynamik des programmatischen Begriffs „Inklusion“ den Unterrichtsalltag bestimmen, so fordert dies von Lehrkräften, Lernumgebungen und Lernsituationen „adaptiv“ gestalten zu können. Das erfordert ein Klären der jeweiligen Ausgangslage anhand einer „Anamnese“, u.E. insbesondere eines fachdidaktisch angelegten diagnostischen Vorgehens. Vorgestellt wird eine Auswahl diagnostischer Konzepte zur Mathematikdidaktik, geklärt wird die jeweilige Intention, der methodische Ansatz des Instruments, reflektiert wird die „Förderergiebigkeit“. Es erfolgt eine Abgrenzung zu allgemeinpädagogischen diagnostischen Ansätzen und zu diagnostischen Ansätzen, wie sie etwa bei Intelligenztests auftreten. Diskutiert wird, ob und inwieweit diagnostische Instrumente fachdidaktisch bestimmt sind und ob und in welchem Maße sie als Studienelemente in der Lehrerbildung sinnvoll erscheinen. 

05.07.2017 um 17:15 Uhr in Raum 69/125

Prof. Dr. Zakhar Kabluchko (Westfälische Wilhelms-Universität Münster)

Convex Hulls of Random Walks: Expected Number of Faces

Let $X_1,...,X_n$ be independent identically distributed random vectors in $R^d$. We are interested in the convex hull of the random walk $0,S_1,\ldots,S_n$, where $S_k = X_1+\ldots+X_k$. We determine the expected number of $k$-dimensional faces of this random polytope. In particular, we show that under minor general position assumptions on the increments $X_i$, this number does not depend on the distribution of the increments. The problem of computing the expected number of faces will be reduced to the problem of computing the number of Weyl chambers of a reflection group intersected by a linear subspace in general position. This is joint work with Vladislav Vysotsky and Dmitry Zaporozhets.