Turing patterns have long been proposed as a mechanism for spatial organization in biology, but their relevance remains controversial due to the stringent fine-tuning often required. In this talk, I will present recent efforts to engineer synthetic Turing systems in bacterial colonies, highlighting both successes and limitations. While our three-node gene circuit generates patterns, challenges remain in extending these results to broader contexts. Additionally, I will discuss our exploration of machine learning methods to address the inverse problem of pattern formation, helping the design process down the road. This work addresses the ongoing task in translating theory into robust biological applications, offering insights into both current capabilities and future directions.

Immunological health relies on a balance between the ability to mount an immune response against potential pathogens and tolerance to self. However, how we keep that balance in health and what goes wrong in disease is not well understood. Here, I will describe combination of novel experimental and computational approaches using multi-omics datasets, imaging and functional experiments to dissect the role and defects in immune cells across several disease areas in cancer and autoimmunity. We show how shared mechanisms that are disrupted across diseases, including cellular, migration, immuno-surveillance, regulation and activation, as well as the immunological features associated with better prognosis and immunomodulation.

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.