Scientific

In this talk I will explain how to obtain a local to global principle for expected values over free ℤ-modules of finite rank. We use the same philosophy as Ekedhal’s Sieve for densities, later extended and improved by Poonen and Stoll in their local to global principle for densities. This strategy can also be extended to higher moments and to holomorphy rings of any global function field.

These results were obtained in collaboration with A. Hsiao, J. Ma, G. Micheli, S. Tinani, V. Weger, Y.Q. Wen.

Consider a poker game where you have to mix the deck of cards between two turns. How (many times) should you shuffle it to prevent any cheating? In this talk we will introduce the theory of mixing times, and explain how representation theory can be used to study card mixing and diffusions on other objects.

I will try to present an overview of some results of unbalanced optimal transport for positive measures with different total masses, showing the crucial role of the so-called cone representation and of the corresponding homogeneous marginals. The cone perspective naturally arises in the convex-relaxation approach to optimal transport; in the more specific case of the Hellinger-Kantorovich (aka Fisher-Rao) metric, it provides a natural tool for representing solutions of the dual dynamical formulation via Hamilton-Jacobi equations, and it is very useful for studying the geodesic convexity of entropy type functionals. (In collaboration with M. Liero, A. Mielke, G. Sodini)

The questions of how healthy colonic crypts maintain their size under the rapid cell turnover in intestinal epithelium, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit-amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady-state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies.

In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer \(a\) different from \(-1\) or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which \(a\) is a primitive root but instead elements of an infinite subset of \(\mathbb{N}\) for which \(a\) is a generalized primitive root. In particular, we will take this infinite subset to be either \(\mathbb{N}\) itself or integers with few prime factors.