In recent years, Optimal Transport and its link with the Ricci curvature in Riemannian geometry attracted a considerable amount of attention. While a lot is now known in the Riemannian setting (and more generally in geodesic spaces), little is known so far in discrete spaces (such as finite graphs or finite Markov chains), with the notable exception of some notions of (discrete) Ricci curvature proposed recently by several authors.
Unfortunately there is not yet a satisfactory (universally agreed upon) resolution. In particular, the notions of transport inequalities, interpolating paths on the measure space, displacement convexity of entropy, are yet to be properly introduced, analyzed and understood in discrete spaces. The chief aim of the lecture is to motivate the problem and mention some recent developments in this direction by the speaker and his collaborators, including Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. Time permitting, relation to classical sumset inequalities will also be mentioned.
Center for Fluid Mechanics, Division of Applied Mathematics Fluids and Thermal Systems, School of Engineering Biomedical Engineering, School of Engineering Joint Seminar Series
Alteration in single-cell physical properties (e.g., size, deformability, and shape) has been identified
to be a useful indicator of changes in cellular phenotype of importance for biological research since
mechanical properties of single cells are found to be strongly associated not only with their lineage but
also with the progression of various diseases. A intrinsic physical biomarker would likely have lower
operating costs than current molecular-based biomarkers that require pre-processing steps, dyes, and/or
costly antibodies. Furthermore, disease states of interest can be expanded to those without predetermined
immunological markers as long as a correlation between cellular biophysical phenotype and clinical
outcome is confirmed. Therefore, novel techniques, allowing high-throughput single-cell deformability
measurement and target cell enrichment based on deformability, would expand the research use and
clinical adoption of this biomarker.
Differential inertial microfluidic devices are great candidates for such tasks since they can (i) continuously but differentially position bio-particles to geometrically-determined equilibrium positions in flow, and/or (ii) isolate and maintain identical populations of cells in the designated regions in the channel without need for additional external forces. Research findings showed that dynamic equilibrium positions are strongly influenced by flowing particles’ physical properties, the flow speed as well as the channel geometry. Using differences in dynamic equilibrium positions, we adapted the system to conduct passive, label-free and continuous cell enrichment based on their physical properties. In addition, vortex-generating inertial microfluidics’ ability to contain cells in pre-determined locations and to release on-demand allowed todevelop a simple molecular probe delivery system with improved single-cell transfection capability. My lab focuses on developing techniques that have potential for high-throughput target cell detection, cost-effective cell separation, and sequential gene delivery, useful for cancer research, immunology, gene therapy and regenerative medicine.
Center for Vision Research Seminar
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Modern computing power and algorithms have greatly increased interest in mixture models as an effective tool for modeling heterogeneity in data. Statistical inference for finite mixture models can be done by the method of maximum likelihood (ML) which provides, in a single package, methods for point estimation, hypothesis testing and construction of confidence sets. Despite attractive features of finite mixture models, there are several challenges for likelihood based inference. In this talk we will address two problems: empirical identifiability on the mixture parameters and multiple local maximizers in the mixture likelihood.
Scientific Computing Seminar
In this talk, we will explore the relationship between graph cut, convex relaxation and some
recent continuous max-flow approaches proposed in the literature. Especially, we will see the
continuous versus discrete relationship between them. There are two advantages with these
approaches: 1) different fast numerical algorithms have been used for these approaches; 2)
global minimization can be guaranteed for some nonconvex problems. We shall explore the connections between some of these algorithms.
Image restoration and segmentation will be used as examples for applications of these algorithms. Extended application of ROF model, global minimization of the Chan-Vese model and different global minimization approach for multiphase labeling problems will be presented. This talk is based on collaborative work with Bae, Yuan, Liu and other collaborators.
We study a version of linearized Monge-Ampere type equation naturally arising from optimal transport problems with the Riemannian distance squared cost function. The source and target measures of the transportation are assumed only to be bounded above and below. This assumption requires the linearized equation to be defined only weakly, and to be degenerate elliptic. We assume nonnegative cross-curvature and log-concavity of the Jacobian determinant of the exponential map. We obtain a Harnack type inequality and Holder estimates for the solution. This extends the result of Caffarelli and Gutierrez given for the linearized equation of the classical Monge-Ampere equation on the Euclidean space, to the round sphere and the products of round spheres and the Euclidean spaces.
*This is joint work (in progress) with Young-Heon Kim.