Kolloquium SS 2014
14.05.2014 um 17:15 Uhr in 69/125:
Prof. Dr. Achill Schürmann (Universität Rostock)
Exploiting Polyhedral Symmetries
Many important problems in mathematics and its applications are modeled using linear constraints respectively polyhedra. Standard modeling often yields polyhedra having many symmetries. However, standard algorithms do not take advantage of them, and even worse, they often work particularly poorly on symmetric problems. In this talk we give an overview about ongoing work on new symmetry exploiting techniques for three fundamental task in polyhedral computations: the representation conversion problem, integer linear programming, and lattice point counting. Initial proof-of-concept results show that affine symmetries can be exploited quite well in certain situations. In order to apply these new techniques on a broader scale new theoretical results are needed.
21.05.2014 um 17:15 Uhr in 69/125:
Prof. Dr. Matthias Reitzner (Universität Osnabrück)
The Gilbert Graph
Choose a set μ of random points (''vertices'') in a convex set W and connect two points x, y ∈ µ by an edge if their distance is bounded by a constant δ. The resulting graph G ( µ, δ) is a random geometric graph, called Gilbert graph, or sometimes disk graph, interval graph (for d = 1) or distance graph. The disk graph is the maybe most natural construction of a random geometric graph. We are interested in the local behaviour of the disk graph within a convex body W, when the number of points tends to infinity and δ → 0. Classical results concern the expectations of quantities of interest. More recent developments deal with the asymptotic covariance structure and limit theorems. As an application we present a connection to a question concerning empty triangles.
28.05.2014 um 17:15 Uhr in 69/125:
Dr. Andreas Weinmann (Helmholtz-Zentrum München)
Reconstructing Functions with Jumps
In this talk we present methods for the reconstruction of functions with jumps. These methods are based on minimization of Potts and related functionals . We show applications to deconvolution problems and other types of ill-posed inverse problems.
04.06.2014 um 17:15 h in 69/125:
Prof. Dr. Ragnar Buchweitz (University of Toronto)
Maximal Cohen-Macaulay Modules on Cones over Elliptic Curves
Given a nonzero polynomial P in n variables, a matrix factorization of P consists of a pair of square matrices A, B of same size with entries from the polynomial ring such that AB = P Id, where Id stands for the appropriate identity matrix. If the polynomial is homogeneous one might further require that the entries of the matrices are homogeneous as well. Such matrix factorizations play a crucial role in the so-called Landau-Ginzburg models of String Theory in Physics. A fundamental result by Orlov implies as a special case that equivalence classes of such homogeneous matrix factorizations for a cubic polynomial that defines an elliptic curve in the projective plane are in a natural, though still largely mysterious bijection with the isomorphism classes of indecomposable objects in the derived category of coherent sheaves on that elliptic curve. The structure of the latter is known since Atiyah's famous classification of such sheaves in 1957. After recalling the background just described, I will present results by my student Sasha Pavlov who uses this machinery to determine all possibilities for the degrees and sizes of the entries of such matrix factorizations and how this will enable us to find all such matrix factorizations eventually.
11.06.2014 um 17:15 Uhr in 69/125:
Dr. Ralf Hielscher (Technische Universität Chemnitz)
The Radon Transform on Manifolds
The mathematical problem behind computerized tomography is the reconstruction of a function from its mean values along all lines. Along with the growing number of imaging methods there is an increasing interest in the reconstruction of functions defined on more general manifolds, like the sphere or the rotation group, from mean values along circles, or, more general, geodesics. We present a unified framework for filtered back projection type inversion formulas and discuss it for some special settings in detail.
18.06.2014 um 17:15 h in 69/125:
Prof. Dr. Markus Kiderlen (Universität Aarhus)
Wicksell's Corpuscle Problem
Motivated by the question of how the size distribution of biological cells
can be estimated from two-dimensional microscopy images, the statistician S.D. Wicksell formulated and solved in the 1920ies the 'corpuscle problem' for spherical particles: Assume that we observe an appropriately randomized planar section of a collection of spherical particles in three-dimensional space. Can the distribution of the sphere radii be estimated from the distribution of the section circle radii?
The present talk will give an overview of results on the corpuscle problem and its variants since its first appearance in Wicksell's works. After setting up the scene, we will discuss statistical and numerical issues of the inversion of the associated Abel integral transform, which is known to be moderately ill-posed. As the assumption of ball-shaped particles is quite restrictive, variants of the corpuscle problem have been formulated. They work with more general particle shapes, such as (rotationally symmetric) ellipsoids or polygonal shapes.
In the past, Wicksell's problem has always been considered in a setting, where the section plane is chosen independently of the particles. In the last part of the talk we consider Wicksell's problem in a formulation that is more adapted to modern confocal microscopy, where each particle is sampled individually by an isotropic random plane through a reference point -- think of the nucleus of a cell.
We will consider the spherical case and show that measurements in such section profiles allow for a determination of distributions for cell radius and position of the reference point in the cell. Suggestions for a numerical inversion and moment relations will conclude these results.
The last part of this talk on the corpuscle problem in local stereology is based on joint work with Ólöf Thórisdóttir.
25.06.2014 um 17:15 Uhr in 69/125:
Prof. Dr. Hermann Thórisson (University of Iceland)
Coupling Methods in Probability Theory
Coupling means the joint construction of two or more random variables, processes, or any random objects. The aim of the construction is usually to deduce properties of the individual objects or to gain insight into distributional relations between them. In this talk we shall consider some basic coupling examples, moving from Poisson approximation, stochastic domination, weak convergence and convergence in density, to Markov chains, perfect simulation, invariant allocations, and Brownian motion.
02.07.2014 um 17:15 Uhr in 32/109:
Prof. Dr. Thomas Geisser (Nagoya University)
Class Field Theory of Varieties
Classical class field theory gives an isomorphism between the Galois group of the maximal etale abelian covering of the ring of integers of a number ring or curve over a finite field with the class group. In the 1980's, this was generalized to smooth and proper schemes (or arbitrary dimension) over a finite field or number ring by Kato and Saito.
Over finite fields, this was further generalized to smooth (but not necessarily proper) schemes by Schmidt-Spiess. We discuss the classical results ,and give a further generalization to possibly singular varieties over finite fields.
09.07.2014 um 17:15 Uhr in 69/125:
Prof. Dr. Ivan Panin (Petersburg Department of Steklov Institute of Mathematics)
The Triangulated Category of K-Motives DK^eff_(k)
Recent results of Grigory Garkusha and myself will be presented. For any perfect ﬁeld k a triangulated category of K-motives DK^eff_(k) is constructed in the style of Voevodsky’s construction of the category DM^eff_(k). To each smooth k-variety X the K-motive M_K(X) is associated in the category DK^eff_(k) and
K_n(X) = Hom(M_K(X)[n], M_K(pt)), n∈Z,
where pt = Spec(k), K(X) is Quillen’s K-theory of X and the Hom-groups are the morphisms in DM^eff_(k).
16.07.2014 um 17:15 h in 69/125:
Dr. Matthias Wendt (Universität Essen)
Homology of Linear Groups via Buildings
In the talk, I will explain some recent computations of homology of SL_2 and GL_2 over function fields of curves, using the action of the groups on their associated Bruhat-Tits buildings. In particular, the results provide residue exact sequences computing the homology of GL_2 over rational function fields, which might be a step towards a better conceptual understanding of homology of linear groups.