Mathematics

Convective storms are highly intermittent and intense, making their occurrence and strength difficult to predict. This is especially true for climate models, which have grid resolutions much coarser (e.g., 100 km) than the scale of a storm’s microphysical and dynamical processes (< 1 km). Physically-based parameterizations struggle to account for this scale mismatch, causing large model errors in rain and lightning. This talk will explore some avenues of using statistical techniques (such as generalized linear and log-Gaussian Cox process models) and machine learning methods (such as random forests and neural networks) that are trained by satellite observations of thunderstorms to see how well they can improve upon existing physical parameterizations in producing accurate rain and lightning characteristics given a set of large-scale environmental conditions.

Exposure of bacteria to cidal stresses typically select for the emergence of stress-tolerant cells refractory to killing. Stress tolerance has historically been attributed to the regulation of discrete molecular mechanisms, including though not limited to regulating pro-drug activation or pumps abrogating antibiotic accumulation. However, fractions of mycobacterial mutants lacking these molecular mechanisms still maintain the capacity to broadly tolerate stresses. We have sought to understand the nature of stress tolerance through a largely overlooked axis of mycobacterial-environmental interactions, namely microbial biomechanics. We developed Long-Term Time-Lapse Atomic Force Microscopy (LTTL-AFM) to dynamically characterize nanoscale surface mechanical properties that are otherwise unobservable using other established advanced imaging modalities. LTTL-AFM has allowed us to revisit and redefine fundamental biophysical principles underlying critical bacterial cell processes targeted by a variety of cidal stresses and for which no molecular mechanisms have previously been described. I aim to highlighting the disruptive power of LTTL-AFM to revisit dogmas of fundamental cell processes like cell growth, division, and death. Our studies aim to uncover new molecular paradigms for how mycobacteria physically adapt to stress and provide expanded avenues for the development of novel treatments of microbial infections.

An important problem in machine learning and computational statistics is to sample from an intractable target distribution, e.g. to sample or compute functionals (expectations, normalizing constants) of the target distribution. This sampling problem can be cast as the optimization of a dissimilarity functional, seen as a loss, over the space of probability measures. In particular, one can leverage the geometry of Optimal transport and consider Wasserstein gradient flows for the loss functional, that find continuous path of probability distributions decreasing this loss. Different algorithms to approximate the target distribution result from the choice of the loss, a time and space discretization; and results in practice to the simulation of interacting particle systems. Motivated in particular by two machine learning applications, namely bayesian inference and optimization of shallow neural networks, we will present recent convergence results obtained for algorithms derived from Wasserstein gradient flows.

Mathematics can help Art Historians and Art Conservators in studying and understanding art works, their manufacture process and their state of conservation. The presentation will review several instances of such collaborations, explaining the role of mathematics in each instance, and illustrating the approach with extensive documentation of the art works.

Speaker Biography

Ingrid Daubechies is a Belgian Physicist and Mathematician, one of the leaders in the area of wavelets, a part of applied harmonic analysis. Wavelets are widely used in data compression and image encoding. Indeed, a wavelet pioneered by Daubechies is the basis of the standard for digital cinema. Ingrid Daubechies has held positions at the Free University in Brussels, Princeton University, and is currently James B. Duke Professor at Duke University. She is a Member of the National Academy of Sciences and of the National Academy of Engineering and a Fellow of the American Association for the Advancement of Science. Ingrid Daubechies has received many awards including the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society.

This talk focuses on the central role played by optimal transport theory in the study of incomplete econometric models. Incomplete econometric models are designed to analyze microeconomic data within the constraints of microeconomic theoretic principles, such as maximization, equilibrium and stability. These models are called incomplete because they do not predict a single distribution for the variables observed in the data. Incompleteness arises because of multiple equilibria in game theoretic solutions, unobserved heterogeneity in choice sets, interval predictions in auctions, and unknown sample selection mechanisms. The problem of confronting the model parameters (possibly infinite dimensional) and the data can be formulated as an optimal transport problem, where the transport cost is some measure of departure from the microeconomic theoretic principles. We will discuss a selection of inference methodologies on the model parameter based on different choices of transport cost, and applications to industrial organization, consumer demand theory and network formation.