Upcoming talks

Some new results in higher order Fourier analysis

Higher order Fourier analysis is a generalization of the classical Fourier analysis in which the role of linear phases is played by polynomial phases and related objects. It originates from the work of Gowers in which he proved quantitative bounds in Szemerédi's theorem on arithmetic progressions and introduced a family of norms on functions on an abelian group, now known as the uniformity norms. These norms, denoted by U^k for k ≥ 2, are the central objects of study in this subject and a key question is the inverse problem, which is to understand the structure of functions with a large value of uniformity norm.

In this talk, I will present some novel results in this field: a general inverse theory for the U^4 norm, as well as a joint work with Žarko Ranđelović, in which we prove that the Möbius function does not correlate with polynomial phases in function fields over prime fields.

Counting connected graphs

How many connected graphs have a prescribed degree sequence?
 This classical combinatorial question turns out to admit a natural probabilistic approach.

In joint ongoing work with Sasha Bell and Remco van der Hofstad, we derive asymptotic formulas for the number of connected graphs with a given degree sequence. Our approach is an example of the probabilistic method. Concretely, we construct a random graph in which (an approximation of) the prescribed degree sequence appears with high probability inside a large connected component. This perspective allows us to translate questions about enumeration into probabilistic statements about random graphs.

Along the way, I will discuss several key probabilistic tools, including the configuration model, branching process approximations, and local weak convergence, and explain how they combine to yield asymptotic counting results.

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