Mathematics

Euler's divergent series $\sum_{n>0} n!z^n$ which converges only for $z = 0$ becomes an interesting object when evaluated with respect to a p-adic norm (which will be introduced in the talk). Very little is known about the values of the series. For example, it is an open question whether the value at one is irrational (or even non-zero). As individual values are difficult to reach, it makes sense to try to say something about collections of values over sufficiently large sets of primes. This leads to looking at primes in arithmetic progressions, which is in turn raises a need for an explicit bound for the number of primes in an arithmetic progression under the generalized Riemann hypothesis.
During the talk, I will speak about both sides of the story: why we needed good explicit bounds for the number of primes in arithmetic progressions while working with questions about irrationality, and how we then proved such a bound.

The talk is joint work with Tapani Matala-aho, Neea Palojärvi and Louna Seppälä. (Questions about irrationality with T. M. and L. S. and primes in arithmetic progressions with N. P.)

In recent years, the problem of optimal transport has received significant attention in statistics and machine learning due to its powerful geometric properties. In this talk, we introduce the optimal transport problem and present concrete applications of this theory in statistics. In particular, we will propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of "multivariate ranks" defined using the theory of optimal transport. We demonstrate the applicability of this approach by constructing exactly distribution-free tests for testing the equality of two multivariate distributions. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We further study their local power and asymptotic (Pitman) efficiency, and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic the classical efficiency results of Hodges and Lehmann (1956) and Chernoff and Savage (1958).

Florent Jouve (Institut de Mathématiques de Bordeaux, France)

Given a Galois extension of number fields L/K, the Chebotarev Density Theorem asserts that, away from ramified primes, Frobenius automorphisms equidistribute in the set of conjugacy classes of Gal(L/K). In this talk we report on joint work with D. Fiorilli in which we study the variations of the error term in Chebotarev's Theorem as L/K runs over certain families of extensions. We shall explain some consequences of this analysis: regarding first “Linnik type problems” on the least prime ideal in a given Frobenius set, and second, the existence of unconditional “Chebyshev biases” in the context of number fields. Time permitting we will mention joint work with R. de La Bretèche and D. Fiorilli in which we go one step further and study moments of the distribution of Frobenius automorphisms.

The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold, the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist. To study the interaction between Allee effect and the biased movement strategy, we mainly consider the pattern formation and local dynamics for a class of single species population models that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exist not only unstable but also linear stable steady states. Finally, we extend results of the single equation to coupled systems for two interacting species, each with different advective terms, and competing for the same resources. We also construct several non-constant steady states and analyze their stability.

This is a work in progress talk by a local graduate student.

In an introduction to proofs course, students learn to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of teaching students to write proofs informally in the language of dependent type theory. I'll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both "under the hood" to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, students could practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.