Tyrosine kinase inhibitors such as imatinib (IM), have significantly improved treatment of chronic myelogenous leukemia (CML). Yet, most patients are not cured for undetermined reasons. In this talk we will describe our recent work on modeling the autologous immune response to CML. We will also discuss our previous results on cancer vaccines, drug resistance, and the dynamics of hematopoietic stem cells.
I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.
Studying the demographic histories of humans or other species and understanding their effects on contemporary genetic variability is one of the central tasks of population genetics. In recent years, a number of methods have been developed to infer demographic histories, especially historical population size changes, from genomic sequence data. Coalescent Hidden Markov Models have proven to be particularly useful for this type of inference. Due to the Markovian structure of Coalescent Hidden Markov Models, an essential building block is the joint distribution of local genealogies, or statistics of these genealogies, in populations of variable size. This joint distribution of local genealogies has received little attention in the literature, especially under variable population size. In this talk, we present a novel method to compute the joint distribution of the total length of the genealogical trees at two linked loci for samples of arbitrary size. We show that the joint distribution can be obtained by solving a system of hyperbolic PDEs and present a numerical algorithm that can be used to efficiently and accurately solve the system and compute this distribution. Our flexible method can be straightforwardly extended to other statistics and structured populations. This is a joint work with Matthias Steinrucken (UChicago).
Anyone who has ever gotten stuck in traffic knows how the superiority of a GPS-based map or app over a traditional print map comes not necessarily just from the former’s access to more information, but more so its access to dynamical information, for eg. the flow of traffic. Similarly, in the last decade, a consensus view is emerging that we should not view proteins, and materials in general, as static entities but instead account for their ever-fluctuating dynamic nature. In this talk, we will describe how we are trying to amalgamate traditional statistical mechanics with recent developments in predictive artificial intelligence (AI) and deep learning to construct and use “dynamical maps” for molecular systems. These low-dimensional “dynamical maps” go beyond traditional static molecular maps (also called potential or free energy landscapes) by incorporating information also about dynamic quantifiables. We will then illustrate their usefulness with the fundamentally important problem of drug unbinding from proteins. A very important feature of drug efficacy is the drug’s residence time in the target protein. Structural details of the unbinding process are in general hard to capture in experiments, while the relevant timescales are far beyond the most powerful supercomputers. Here we will show how by constructing appropriate dynamical maps we are able to elucidate with unprecedented spatio-temporal resolution and statistical reliability the entire unbinding process of a variety of ligand-protein systems, shedding light on the role of protein conformations and of water molecules as molecular determinants of unbinding.
The design and optimization of the next generation of materials and applications strongly hinge on our understanding of the processing-microstructure-performance relations; and these, in turn, result from the collective behavior of materials’ features at multiple length and time scales. Although the modeling and simulation techniques are now well developed at each individual scale (quantum, atomistic, mesoscale and continuum), there remain long-recognized grand challenges that limit the quantitative and predictive capability of multiscale modeling and simulation tools. In this talk we will discuss three of these challenges and provide solution strategies in the context of specific applications. These comprise (i) the homogenization of the mechanical response of materials in the absence of a complete separation of length and/or time scales, for the simulation of metamaterials with exotic dynamic properties; (ii) the collective behavior of materials’ defects, for the understanding of the kinematics of large inelastic deformations; and (iii) the upscaling of non-equilibrium material behavior for the modeling of anomalous diffusion processes.
In the first part of the talk we will investigate a Keller-Segel model with quorum sensing and a fractional diffusion operator. This model describes the collective cell movement due to chemical sensing with flux limitation for high cell densities and with anomalous media represented by a nonlinear, degenerate fractional diffusion operator. The purpose here is to introduce and prove the existence of a properly defined entropy solution. In the second part of the talk we will analyze an equation that is gradient flow of a functional related to Hardy-Littlewood-Sobolev inequality in whole Euclidean space R^d, d \geq 3. Under the hypothesis of integrable initial data with finite second moment and energy, we show local-in-time existence for any mass of ``free-energy solutions", namely weak solutions with some free energy estimates. We exhibit that the qualitative behavior of solutions is decided by a critical value. The motivation for this part is to generalize Keller-Segel model to higher dimensions.
This is a joint work with K. H. Karlsen and E. A. Carlen.
Quadrature by Expansion, or `QBX', is a systematic, high-order approach to
singular quadrature that applies to layer potential integrals with general
kernels on curves and surfaces. The efficient and accurate evaluation of
layer potentials, in turn, is a key building block in the construction of
solvers for elliptic PDEs based on integral equation methods.
I will present a new fast algorithm incorporating QBX that evaluates layer
potentials on and near surfaces in two and three dimensions with user-specified
accuracy, along with supporting theoretical and empirical results on complexity
and accuracy. A series of examples on unstructured geometry across a variety of
applications in two and three dimensions demonstrates the applicability of
In this talk, we will review the state of the art for the synchronization problem of the Kuramoto model at the kinetic and particle level. Additionally, we will introduce new developments and variational techniques for the dynamics of this model and some of its variants and generalization.
Signal fragmentation is a method for transmitting a low frequency signal over a collection of small antennas through a modal expansion (similar to one level of a wavelet expansion), in which the mode has compact support in time. We analyze the spectral leakage and optimality of signal fragmentation. For a special choice of mode, the spectral leakage can be eliminated for sinusoidal signals and minimized for bandlimited or AM signals. We derive the optimal mode for either support size or for energy efficiency. The derivation of these results uses the Poisson summation formula and the Shannon Interpolation Formula.
Info-metrics is the science of modeling, reasoning, and drawing inferences under conditions of
noisy and insufficient information. It is at the intersection of information theory, statistical
inference, and decision-making under uncertainty. It plays an important role in helping make
informed decisions even when there is inadequate or incomplete information because it provides
a framework to process available information with minimal reliance on assumptions that cannot
be validated; it gets the data to confess with minimal torture under careful supervision from the
judge and jury.
My talk will be based on my new book ‘Foundations of Info-Metrics: Modeling, Inference, and
Imperfect Information,’ http://info-metrics.org/ that develops and examines the theoretical
underpinning of info-metrics and provides extensive interdisciplinary applications. I will present
the basic framework and theory via graphical illustrations and will then discuss several
The interface between basic ecology and applied mathematics is robust, and results from this interface are often critical to effective conservation. In my CSCAMM seminar, I will focus on one part of this interface whereby ecological observations and datasets have created new opportunities for a variety of mathematical tools and approaches. For instance, datasets derived from efforts to track the movements of wild animals (e.g., using GPS-satellite collars) have presented new opportunities to investigate the roles of learning and memory on animal performance. Likewise, to understand better how animals interact with their landscapes, we need to know more about how they perceive and react to landscape features, creating new opportunities for modeling work that goes beyond classical, diffusion-based formulations of spatial spread. Together such mathematical applications are revealing relationships among individual movements, landscape dynamics, and ecological processes, strengthening the interdisciplinary bridge linking mathematics, ecology, and conservation.