The discrete fundamental groups of a metric space can be thought of as fundamental groups that `ignore' closed loops up to some specified size R. As the parameter R grows, these groups have been used to produce interesting invariants of coarse geometry. On the other hand, as R gets smaller one would expect to retrieve the usual fundamental group as a limit. In this talk I will try to briefly illustrate both these aspects.

KW states that every finite abelian extension of the rationals is contained in a cyclotomic extension. In a previous talk, this was reduced to considering cyclic extensions of the local fields Q_p of prime power order l^r. When l\neq p, general theory is sufficient, however for l=p, more specific (although not necessarily more abstruse) descriptions become necessary.
I will focus on the simple structure of Q_p's extensions to obstruct the remaining obstructions to KW (and hopefully provoke some interest in local fields in those less familiar). Time-permitting, I will talk about this theorem in the context of class field theory and/or Hilbert's 12th problem.

In the spirit of the Ax-Kochen-Ershov principle, one could conjecture that the imaginaries in equicharacteristic zero Henselian fields can be entirely classified in terms of the Haskell-Hrushovski-Macpherson geometric imaginaries, residue field imaginaries and value group imaginaries. However, the situation is more complicated than that. My goal in this talk will be to present what we believe to be an optimal conjecture and give elements of a proof.

We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
 

We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve".  In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

Fireshape is a shape optimization library based on finite elements. In this talk I will describe how Florian Wechsung and I developed Fireshape and will share my reflections on lessons learned through the process.