In 1952, R.H. Bing described a wild involution of S^3, that is, a homeomorphism I:S^3--S^3 such that I^2=id and the fixed point 2-sphere of I is wildly imbedded in S^3. This talk will begin by describing the heart of Bing's construction, which involves a clever strategy of shrinking a decomposition of S^3. Recently, while Michael Freedman and I were investigating the analytic properties of Bing's involutions, we discovered a surprising new method of shrinking Bing's decomposition with properties that no previously known shrinking methods have and that defied our intuition for what is possible.

The class number formula describes the behavior of the Dedekind zeta function at s=0 and s=1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s=0 and s=1 in terms of units and class numbers. The Harris-Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader Prasanna-Venkatesh conjectures. In this talk, I will introduce this big picture and describe the proof of the Harris-Venkatesh conjecture for dihedral weight-one modular forms arising as definite theta series.

In this seminar I will introduce the Morse-Bott approach to Morse homology, which doesn’t necessitate to perturb a real valued function to be a Morse function. Why is that important? In the spirit of “transversality is the opposite of symmetry”, Morse functions almost always destroy the inner symmetries of a space, i.e. don’t take them into account. Morse-Bott functions instead are a little bit more flexible in that regard and sometimes let us preserve these symmetries. I will spend 10 minutes recalling what Morse homology is (but you will appreciate the talk more if you are familiar with that already) and then talk about different approaches to Morse-Bott homology. Time permitting I will mention a couple of applications of Morse Bott homology (in symplectic geometry for example) but do not expect anything too detailed since I’m currently learning it myself :)

I will describe a mathematical approach to the relations between 4d N=2 SUSY QFT, holomorphic symplectic varieties and vertex algebras following in particular Butson's thesis arXiv:2011.14978.

Spectra are the homotopy theorist abelian groups. In particular, one can "add" a finite collection of points in a spectrum in a coherent way. Much like the way that abelian groups can be understood locally every prime at a time and then put back together, spectra can be decomposed according to ``chromatic primes''. M. Hopkins and J. Lurie had the fascinating observation for spectra localized at a single prime; the ability to add points can be extended to the ability to ``integrate'' along maps from certain spaces. We shall discuss where this extra structure arises from and show how it sheds light on the nature of these chromatic primes. In particular, we shall discuss its relationship with categorification, K-theory, Galois extensions, and Fourier-like transforms.

Applications of Optimal Transport (OT) theory have gained popularity in several fields such as machine learning and signal processing. In this seminar, we will address this point by introducing embeddings or transforms based on OT. First, we will present the Cumulative Distribution Transform (CDT), its version for signed signals/measures, and the Linear Optimal Transport Embedding (LOT). Here, the underlying conservation of mass law will be a benefit. Since these tools are new signal representations based on OT, they have suitable properties for decoding information related to certain signal displacements. We will demonstrate this by describing a Wasserstein-type metric in the embedding space and showing applications in classifying (detecting) signals under random displacements, parameter estimation problems for certain types of generative models, and interpolation. Moreover, these techniques allow faster computation of the classical Wasserstein between pairs of probability measures. However, even though the balanced mass requirement from classical OT is crucial, it also limits the performance of these transforms/embeddings. Therefore, we will finally move to Optimal Partial Transport (OPT) theory and propose a new (linear) embedding.