## Oktober – November 2018

Classical results in differential geometry such as the Lichnerowicz and Bonnet-Myers theorems or isoperimetric estimates relate the Ricci curvature of a manifold to its analytic and topological properties. Originally, those estimates rely on sharp Ricci curvature lower bounds, and during the last years they have been generalized to integral curvature bounds. This talk will consider even more general Ricci curvature assumptions implying generalizations of the classical estimates. Namely, we show that, in a certain sense, relative boundedness of the Ricci curvature suffices to prove a Lichnerowicz and Bonnet-Myers type theorem. If time allows, we will also discuss isoperimetric estimates based on Kato-type assumptions.

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).

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.

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.

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.

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.

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.

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY

Vortrag in der Reihe: Oberseminar ANALYSIS – PROBABILITY