Oktober – November 2018

Max Pfeffer (MPI MIS, Leipzig): Learning paths from signature tensors
Okt 17 @ 11:00 – 12:00

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their signature tensors of order three. We establish identifiability results and recovery algorithms for piecewise linear paths, polynomial paths, and generic dictionaries. A detailed analysis of the relevant condition numbers is presented. We also compute the shortest path with a given signature tensor.

Felix-Klein-Colloquium, Iskander Taimanov (Novosibirsk State University): Closed geodesics on non-simply-connected manifolds
Okt 17 @ 16:30 – 17:30

We shall give a survey on results and open problems on the existence of several closed geodesics on non-simply-connected manifolds

Etienne Sandier (Université Paris 12): Two-component Bose-Einstein condensates with rotation
Okt 23 @ 15:15 – 16:45

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY
In this joint work with Amandine Aftalion we study an energy functional in two-dimensions describing a rotating two-component Bose-Einstein condensate. The mathematical difficulty in this model is that it exhibits defects which are both 1-dimensional (curves) and 0-dimensional (vortices).

Max Fathi (CNRS & Université Paul Sabatier): Stability for the Bakry-Emery theorem
Okt 23 @ 16:45 – 18:15

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY
The Bakry-Emery theorem states that if a probability measure is in some sense more
log-concave than the standard Gaussian measure, then certain functional
inequalities (such as the Poincare inequality and the logarithmic
Sobolev inequality) hold, with better constants than for the associated
Gaussian inequalities. I will show how we can combine Stein’s method and
simple variational arguments to show that if the Bakry-Emery bound is
almost sharp for a given measure, then that measure must almost split
off a Gaussian factor, with explicit quantitative bounds. Joint work with Thomas Courtade.

Piotr Hajac (Polska Akademia Nauk): Operator algebras that one can see
Okt 24 @ 10:00 – 11:00

Operator algebras are the language of quantum mechanics just as much as differential geometry is the language of general relativity. Reconciling these two fundamental theories of physics is one of the biggest scientific dreams. It is a driving force behind efforts to geometrize operator algebras and to quantize differential geometry. One of these endeavours is noncommutatvive geometry, whose starting point is natural equivalence between commutative operator algebras (C*-algebras) and locally compact Hausdorff spaces. Thus noncommutative C*-algebras are thought of as quantum topological spaces, and are researched from this perspective. However, such C*-algebras can enjoy features impossible for commutative C*-algebras, forcing one to abandon the algebraic-topology based intuition. Nevertheless, there is a class of operator algebras for which one can develop new (“quantum”) intuition. These are graph algebras, C*-algebras determined by oriented graphs (quivers). Due to their tangible hands-on nature, graphs are extremely efficient in unraveling the structure and K-theory of graph algebras. We will exemplify this phenomenon by showing a CW-complex structure of the Vaksman-Soibelman quantum complex projective spaces, and how it explains its K-theory.

Mariusz Tobolski (Polska Akademia Nauk): Pullbacks of Leavitt path algebras from pushouts of graphs
Okt 24 @ 11:00 – 12:00

To a directed graph one can assign its Leavitt path algebra, which is a quotient of the path algebra of the extended graph by a certain ideal. An appropriate choice of morphisms in a category whose objects are directed graphs makes this assignment into a covariant functor into the category of algebras. In spite of the apparent covariant nature of the construction of Leavitt path algebras, we prove that, for a suitable class of graphs, pushouts of directed graphs give rise to pullbacks of the underlying Leavitt path algebras. This talk is based on the joint work with Piotr M. Hajac and Sarah Reznikoff.

Paul Breiding (MPI MIS, Leipzig): Pencil-based algorithms for tensor rank decomposition are not stable
Nov 1 @ 11:00 – 12:00

I will discuss the existence of an open set of n1× n2× n3 tensors of rank r on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically followed by a generalized eigendecomposition, is arbitrarily numerically forward unstable. The analysis shows that this problem is caused by the fact that the condition number of the tensor rank decomposition can be much larger for n1× n2× 2 tensors than for the n1× n2× n3 input tensor. Moreover, I present a lower bound for the limiting distribution of the condition number of random tensor rank decompositions of third-order tensors. The numerical experiments illustrate that for random tensor rank decompositions one should anticipate a loss of precision of a few digits. Joint work with Carlos Beltran and Nick Vannieuwenhoven.

Chenqi Mou (Beihang University Beijing): Chordal Graphs in Triangular Decomposition in Top-Down Style
Nov 2 @ 11:00 – 12:00

In this talk we present some underlying connections between symbolic computation and graph theory. Inspired by the two papers of Cifuentes and Parrilo in 2016 and 2017, we are interested in the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style. Viewing triangular decomposition in top-down style as multivariate generalization of Gaussian elimination, we show that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets, including all the computed triangular sets, appearing in one specific algorithm for triangular decomposition in top-down style (Wang’s method) have associated graphs which are subgraphs of the chordal graph of the input polynomial set. Potential applications of chordal graphs in symbolic computation are also discussed.

Dirk Blömker (Universität Augsburg): to be announced
Nov 6 @ 13:45 – 15:15

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Pierre Mathieu (Université d’Aix-Marseille): to be announced
Nov 6 @ 15:15 – 16:45

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Letzte Änderung: 15. Mai 2013