Wochenplan des Fachbereichs Mathematik und Informatik

To any Adams-type homology theory one can associate a notion of a synthetic spectrum, their homotopy theory is in a precise sense a deformation of the derived category of quasi-coherent sheaves on a certain algebraic stack whose generic fibre is given by the category of spectra. In this talk, I will survey some of the applications of synthetic spectra, such as the problem of algebraicity of chromatic homotopy theory, or the classification of highly-connected manifolds of Burklund-Hahn-Senger. I will also discuss some conjectures on the universal property of synthetic spectra and possible generalizations to a general homology theory.

Spin(7) is one of the special holonomy groups on Berger's list which gives rise to Ricci flat metrics. The condition that the holonomy of an 8-manifold reduces to Spin(7) gives rise to a complicated system of non-liner PDEs. In the non-compact situation, symmetries can be used to reduce this complexity. As manifolds with special holonomy cannot be homogeneous, the most symmetric case are group actions with cohomogeneity one, i.e. where a generic orbit has codimension one. In this case the PDE system is reduced to an ODE system. I will give an overview of recent progress in the construction of complete cohomogeneity one Spin(7) holonomy metrics. All examples have an asymptotically locally conical (ALC) or asymptotically conical (AC) geometry at infinity.

Given a fixed open set $\Omega$ in the plane we focus on optimal partitions, that is, partitions of $\Omega$ into $N$ equal-area subsets that minimize the total perimeter. Hales proved in 1999 that the asymptotic optimal energy for large $N$ is given by the hexagonal honeycomb, and thus hexagonal patterns are expected. However, since $\Omega$ may not accomodate a perfectly hexagonal partition, a polycrystalline structure may emerge, where different zones of constant orientation ("grains") are separated by "grain boundaries", similarly to what happens with dislocations in materials. In this talk I will present some ongoing work on the mathematical description of such structures.

Recently a lot of progress has been made in the theoretical understanding of deep learning. One of the very promising directions is the statistical approach, which interprets deep learning as a statistical method and builds on existing techniques in mathematical statistics to derive theoretical error bounds. The talk surveys this field and describes future challenges.

I will survey the current state of our knowledge of 3-manifolds, focusing on what we have learned from Thurston, and on the relatively recent breakthroughs. I will then present a lightly more algebraic perspective on some of these.

I will first introduce scalar-valued and operator-valued Schur multipliers and their partially defined versions, and will provide a Grothendieck-type characterisation of operator-valued Schur multipliers. After a brief introduction to the matrix setting, I will talk about the positive extension problem of Schur multipliers and characterise its affirmative solution in terms of structures on an operator system associated with the domain of the multipliers. This talk will be based on joint papers with R. Levene and I. G. Todorov.

We show that the connected component of the embedding space of links in R^3 containing the split union of m Hopf links and n unknots is homotopy equivalent to a parametrised 'round' subspace, extending work of Brendle Hatcher on the unknot case. We discuss potential generalisations and applications.This is joint work with Corey Bregman.