Talk

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri-Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions with a "level of distribution" 1/2. In 1989, building on an idea of Maier, Friedlander and Granville showed that such equidistribution results fail if the range of the moduli q is extended up to x/(log x)^B for any B>1.  We discuss variants of this result and applications to specific functions. This is joint work with Aditi Savalia. 

The Gordian distance u(K,K') between two knots K and K' is defined as the minimal number of crossing changes needed to relate K and K'. The unknotting number of a knot K, a classical yet hard to compute knot invariant, arises as the Gordian distance between K and the trivial knot. Several lower bounds for both invariants exist. A well-known bound for the unknotting number is given by the Rasmussen invariant, which is extracted from Khovanov homology, a bigraded chain complex associated to a knot up to chain homotopy equivalence.
In this talk, I will introduce a new lower bound for the Gordian distance, \lambda, coming from Khovanov homology. After introducing all the relevant ingredients, I will present some results about \lambda. In particular, \lambda turns out to be sharper than the Rasmussen invariant as a lower bound for the unknotting number. This is based on joint work with L. Lewark and C. Zibrowius.

Manifolds with a tangential structure $\xi: B \to BO$ can be cut and glued back twisted by a $\xi$-diffeomorphism along separating codimension one $\xi$-submanifolds. This gives rise to certain groups $SKK_n^{\xi}$ of $n$-dimensional $\xi$-manifolds, modulo cut-and-paste relations (Schneiden und Kleben). Kreck, Stolz, and Teichner have, under some mild assumptions, provided a description of the kernel of the surjection $SKK_n^\xi \to \Omega_n^\xi$ in an as-yet unpublished paper. We provide an alternative geometric proof of their result in a certain special case. We calculate the $SKK_n^\xi$ groups for certain tangential structures in terms of the corresponding bordism group by determining whether the corresponding short exact sequence splits. This is joint work with Renee Hoekzema and Luuk Stehouwer.

 

My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60's and 70's, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of all bordism rings of manifolds with commuting involutions. 

This talk will provide an overview of recent developments in Fourier restriction theory, which one could describe as the study of exponential sums over restricted frequency sets with geometric structure, typically arising in PDE or number theory. Decoupling inequalities measure the square root cancellation behavior of these exponential sums. I will highlight questions from areas outside of harmonic analysis that are approachable by decoupling/Fourier restriction theory, some with satisfying answers and some which remain open.