Termine

The positive first order theory of a group G is the set of all sentences only involving equalities (no negation is allowed) that are satisfied in G. All non-abelian free groups have the same positive first order theory, moreover this theory is contained in the the positive first-order theory of every other group G. When equality holds, we say that G has trivial positive first-order theory. Of course not all groups have trivial positive first-order theory. However until now all known examples of such groups contains non-abelian free subgroups. In this talk we will explain how geometric / topological tools coming from small cancellation theory and action on R-trees can be used to produce some surprising groups. Although they have trivial positive first-order theory they satisfy some other "pathological" properties : they cannot act in a non-degenerated way on a hyperbolic space; they are simple, torsion-free groups, all of whose proper subgroups are cyclic; they have no unbounded quasi-morphisms; etc. This is joint work with Francesco Fournier-Facio and Turbo Ho.

Detailed program of the event: - 4 p.m. (sharp) in SRZ 216/217: Introductory talk for PhD students and postdocs by Dr. Francesco Deangelis - 4.30 p.m. Coffee and tea will be served in the Cluster Lounge (Orléans-Ring 12) - 5 p.m. (sharp) in SRZ 216/217: Colloquium-talk by Prof. Dr. Antoine Gloria: Title: Effective ellipticity for high contrast homogenization Abstract: Homogenization is the art of averaging coefficients of differential operators in a consistent way. It is the mathematical theory associated with composite materials. The theory is by-now well-understood for linear elliptic equations with random coefficients that are uniformly bounded and elliptic. For applications to mathematical physics and to more contemporary homogenization problems, it is however desirable to move away from uniform ellipticity. In this colloquium I will introduce a concept of effective ellipticity field and ellipticity radius, which quantifies how and at which scale an elliptic operator with (degenerate and unbounded) stationary ergodic coefficients is close to a uniformly elliptic operator. Combined with nonlinear concentration of measure, this allows to extend quantitative homogenization theory to the setting of high- (and even infinite-) contrast. - 6 p.m. Informal reception in the Cluster Lounge (Orléans-Ring 12)

A first-order sentence in the language of group theory is a mathematical statement whose variables refer only to elements of a group, and two groups are said to be elementarily equivalent if they satisfy the same first-order sentences. I will introduce these concepts using simple examples, then turn to a question posed by Tarski in the 1940s, known as the Tarski problem: are non-abelian free groups elementarily equivalent? Despite the apparent simplicity of its formulation, this problem remained open until the early 2000s, when it was solved by Sela and by Kharlampovich and Myasnikov.

Abstract: Recent progress on the K-theory of "large" categories has raised interest in the algebraic K-theory of sheaves on locally compact Hausdorff spaces, which serves as a central stepping stone to modern approaches to assembly conjectures in K-theory and L-theory. A central ingredient for the computation of the K-theory of these sheaf categories is Verdier duality: The categories of sheaves and cosheaves agree when the target is a stable category. We present a category-theoretic perspective on this computation by analyzing the notion of a continuous algebra of a lax-idempotent monad. As a result we obtain a completely formal generalization of Verdier duality to a larger class of spaces - so-called stably locally compact spaces. We will elaborate on the role of classical Stone duality, as well as sketch a proof of the computation of the algebraic K-theory of a coherent space.

Steady Ricci solitons model Type II singularity formation under the Ricci flow. By utilising a procedure first introduced by Lai, we construct new one-parameter families of steady gradient Ricci solitons with positive curvature operator admitting O(p)×O(q) symmetry in dimension p+q, for any pair of integers p,q ? 2. If time permits, we shall discuss their geometry at infinity and highlight some open questions regarding their asymptotics.

We study a very general class of first-order linear hyperbolic systems that both become weakly hyperbolic and contain singular lower-order coefficients at a single time t = 0. In "critical" weakly hyperbolic settings, it is well-known that solutions lose a finite amount of regularity at t = 0. Here, we both improve upon the analysis in the weakly hyperbolic setting, and we extend this analysis to systems containing critically singular coefficients, which may also exhibit modified asymptotics and regularity loss at t = 0. In particular, we give precise quantifications for (1) the asymptotics of solutions as t approaches 0, (2) the scattering problem of solving the system with asymptotic data at t = 0, and (3) the loss of regularity due to the degeneracies at t = 0. Finally, we discuss a wide range of applications for these results, including weakly hyperbolic wave equations (and equations of higher order), as well as equations arising from relativity and cosmology (e.g. at big bang singularities). This is joint work with Bolys Sabitbek.