Hilbert's third problem asks: do there exist two polyhedra with the same volume which are not scissors congruent? In other words, are there polyhedra P and Q of the same volume such that, no matter how P is cut into finitely many polyhedral pieces (intersecting only on their boundaries), these pieces cannot be reassembled (intersecting only on their boundaries) to yield Q? In 1901 Dehn answered this question in the negative by constructing a second scissors congruence invariant now called the "Dehn invariant," and showing that a cube and a regular tetrahedron never have equal Dehn invariants, regardless of their volumes. We can then restate Hilbert's third problem: do the volume and Dehn invariant separate the scissors congruence classes? In 1965 Sydler showed that the answer is yes; in 1968 Jessen showed that this result extends to dimension 4, and in 1982 Dupont and Sah constructed analogs of such results in spherical and hyperbolic geometries. However, the problem remains open past dimension 4. By iterating Dehn invariants Goncharov constructed a chain complex, and conjectured that the homology of this chain complex is related to certain graded portions of the algebraic K-theory of the complex numbers, with the volume appearing as a regulator. In joint work with Jonathan Campbell, we have constructed a new analysis of this chain complex which illuminates the connection between the Dehn complex and algebraic K-theory, and which opens new routes for extending Dehn's results to higher dimensions. In this talk we will discuss this construction and its connections to both algebraic and Hermitian K- theory, and discuss the new avenues of attack that this presents for the generalized Hilbert's third problem.

We prove the existence of Ulrich bundles on any Brauer--Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer-Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer-Severi variety X, at least if dim(X) = p-1 for an odd prime p. This formula implies that the rank of any Ulrich bundle on such a Brauer-Severi variety X must be a multiple of the period.

A finite subset $A$ of a group $G$ is said to have small doubling if there is a constant $K$ such that the set $A\cdot A$ consisting of products $a\cdot b$, with $a$ and $b$ in $A$ has size at most $K\cdot|A|$. Extreme examples of sets with small doubling are cosets of subgroups.
Theorems of Freiman-Ruzsa type assert that sets with small doubling are "not too far" from being subgroups. Freiman's original theorem asserts that a finite subset of the integers with small doubling is efficiently contained in a generalized arithmetic progression. A version of this result for abelian groups of bounded exponent was given by Ruzsa: a finite subset with small doubling $K$ of an abelian group $G$ of exponent $r$ is contained in a subgroup $H$ of $G$ of size bounded by $K$, $r$ and $|A|$ (but the bound he exhibited is exponential). A natural reformulation of the problem is the polynomial Freiman-Ruzsa conjecture, one of the central open problems in additive combinatorics, which aims to find polynomial bounds (in $K$) so that any subset $A$ of small doubling $K$ in an infinite-dimensional vector space over $\mathbb{F}_2$ can be covered by finitely many translates of some subspace, whose size is commensurable to the size of $A$. Improvements of this result have been subsequently obtained by many authors for arbitrary (possibly infinite and non-abelian) groups.
Motivated by work of E. Hrushovski, we will present an on-going work with D. Palacin (Freiburg) and J. Wolf (Cambridge) of Freiman-Ruzsa type under some assumptions of model-theoretic nature. No prior knowledge of mathematical logic is necessary for this talk.

We consider control problems for partial differential equations where the distributed control should take on values only from a given discrete and hence non- convex set. Such problems occur for example in parameter identification or topology optimization. Similar to their use in sparse optimization, L1-type norms can be used to formulate a convex relaxation which can be solved by semi-smooth Newton methods. We illustrate this approach using linear model problems and discuss the extension to vector-valued and nonlinear control problems.

The Navier-Stokes equations, a nonlinear system of partial differential equations, are the most important model to describe the flow of a viscous incompressible fluid. A fundamental work on existence dates back to Jean Leray (1934), but leaves open several important questions on regularity and uniqueness. In the talk we first describe some basic difficulties in the analysis of the Navier-Stokes equations and discuss the most important results from 1934 until today. Then we concentrate on the famous open problem of regularity which is known to a wider public as one of the seven millennium problems of Clay Mathematics Institute (2000): Are the weak solutions of J. Leray and E. Hopf - in the three-dimensional case - regular for all times? Finally, several new results on non-uniqueness and the construction of solutions with singularities will be presented.

Mathematics in climate research is often thought to be mainly a provider of techniques for solving the continuum mechanical equations for the flows of the atmosphere and oceans, for the motion and evolution of Earth’s ice masses, and the like. Three examples will elucidate that there is a much wider range of opportunities.
Climate modellers often employ reduced forms of “the continuum mechanical equations” to efficiently address their research questions of interest. The first example discusses how mathematical analysis can provide systematic guidelines for the regime of applicability of such reduced model equations.
Meteorologists define “climate”, in a narrow sense, as “the statistical description in terms of the mean and variability of relevant quantities over a period of time” (World Meteorological Society; see the website for a broader sense definition). Now, climate researchers are most interested in changes of the climate over time, and yet there is no unique, well-defined notion of “time dependent statistics”. In fact, there are restrictive conditions which data from time series need to satisfy for classical statistical methods to be applicable. The second example describes recent developments of analysis techniques for time series with non-trivial temporal trends.
Modern climate research has joined forces with economy and the social sciences to generate a scientific basis for informed political decisions in the face of global climate change. One major type of problems hampering progress of the related interdisciplinary research consists of often subtle language barriers. The third example describes how mathematical formalization of the notion of “vulnerability” has helped structuring related interdisciplinary research efforts.