Tuesday, February 1, 2011 - 11:00 , Location: Skiles 006 , Peter Kim , University of Utah , email@example.com , Organizer:
The immune system is a complex, multi-layered biological system, making it difficult to characterize dynamically. Perhaps, we can better understand the system’s construction by isolating critical, functional motifs. From this perspective, we will investigate two simple, yet ubiquitous motifs:state transitions and feedback regulation.Numerous immune cells exhibit transitions from inactive to activated states. We focus on the T cell response and develop a model of activation, expansion, and contraction. Our study suggests that state transitions enable T cells to detect change and respond effectively to changes in antigen levels, rather than simply the presence or absence of antigen. A key component of the system that gives rise to this change detector is initial activation of naive T cells. The activation step creates a barrier that separates the slow dynamics of naive T cells from the fast dynamics of effector T cells, allowing the T cell population to compare short-term changes in antigen levels to long-term levels. As a result, the T cell population responds to sudden shifts in antigen levels, even if the antigen were already present prior to the change. This feature provides a mechanism for T cells to react to rapidly expandingsources of antigen, such as viruses, while maintaining tolerance to constant or slowly fluctuating sources of stimulation, such as healthy tissue during growth.For our second functional motif, we investigate the potential role of negative feedback in regulating a primary T cell response. Several theories exist concerning the regulation of primary T cell responses, the most prevalent being that T cells follow developmental programs. We propose an alternative hypothesis that the response is governed by a feedback loop between conventional and adaptive regulatory T cells. By developing a mathematical model, we show that the regulated response is robust to a variety of parameters and propose that T cell responses may be governed by a simple feedback loop rather than by autonomous cellular programs.
Wednesday, January 26, 2011 - 15:00 , Location: Skiles 006 , Aaron Lauda , Columbia University , Organizer: Stavros Garoufalidis
The Jones polynomial is a link invariant that can be understood in terms of the representation theory of the quantum group associated to sl2. This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants. In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups. These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.
Monday, January 24, 2011 - 15:00 , Location: Skiles 006 , Greg Blekherman , University of California, San Diego , Organizer: John Etnyre
A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons. I will present two approaches to studying the differences between nonnegative polynomials and sums of squares. Using techniques from convex geometry we can conclude that if the degree is fixed and the number of variables grows, then asymptotically there are significantly more nonnegative polynomials than sums of squares. For the smallest cases where there exist nonnegative polynomials that are not sums of squares, I will present a complete classification of the differences between these sets based on algebraic geometry techniques.
Thursday, January 20, 2011 - 15:00 , Location: Skiles 005 , Jelena Bradic , Princeton University , Organizer: Liang Peng
High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of non-concave penalized methods for non-polynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that non-concave penalties lead to significant reduction of the "irrepresentable condition" needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the non-concave regularized estimator, is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.
Wednesday, January 12, 2011 - 14:00 , Location: Skiles 006 , Nguyen Hoai-Minh , Courant Institute of Mathematical Sciences , Organizer: Wilfrid Gangbo
A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equations. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a nite range or the full range of frequencies, and the possibility of achieving perfect cloaking. At the end of my talk, I will also discuss some results related to the wave equation in 3d.
Tuesday, April 13, 2010 - 11:00 , Location: Skiles 269 , Rafael de le Llave , Department of Mathematics, University of Texas, Austin , Organizer:
Many mechanical systems have the property that some small perturbations can accumulate over time to lead to large effects. Other perturbations just average out and cancel. It is interesting in applications to find out what systems have these properties and which perturbations average out and which ones grows. A complete answer is far from known but it is known that it is complicated and that, for example, number theory plays a role. In recent times, there has been some progress understanding some mechanisms that lead to instability. One can find landmarks that organize the long term behavior and provide an skeleton for the dynamics. Some of these landmarks provide highways along which the perturbations can accumulate.
Thursday, February 4, 2010 - 15:00 , Location: Skiles 269 , Karim Lounici , University of Cambridge , Organizer:
We consider the statistical deconvolution problem where one observes $n$ replications from the model $Y=X+\epsilon$, where $X$ is the unobserved random signal of interest and where $\epsilon$ is an independent random error with distribution $\varphi$. Under weak assumptions on the decay of the Fourier transform of $\varphi$ we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators $f_n$ for the density $f$ of $X$, where $f: \mathbb R \to \mathbb R$ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of $f$ if the Fourier transform of $\varphi$ decays exponentially, and that a corresponding result holds true for the hard thresholding wavelet estimator if $\varphi$ decays polynomially. We also analyze the case where $f$ is a 'supersmooth'/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global nonasymptotic confidence bands for the density $f$.
Thursday, January 28, 2010 - 16:00 , Location: Skiles 269 , Josephine Yu , Georgia Tech , Organizer: Matt Baker
Tropical geometry can be thought of as geometry over the tropical semiring, which is the set of real numbers together with the operations max and +. Just as ordinary linear and polynomial algebra give rise to convex geometry and algebraic geometry, tropical linear and polynomial algebra give rise to tropical convex geometry and tropical algebraic geometry. I will introduce the basic objects and problems in tropical geometry and discuss some relations with, and applications to, polyhedral geometry, computational algebra, and algebraic geometry.
Tuesday, January 26, 2010 - 11:05 , Location: Skiles 269 , Shamgar Gurevich , Institute for Advanced Study, Princeton , firstname.lastname@example.org , Organizer: Christopher Heil
In the lecture I will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. I will begin the lecture by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then I will explain how to generalize the construction to obtain a larger collection of functions that we call "The oscillator dictionary". Functions in the oscillator dictionary admit many interesting pseudo-random properties, in particular, I will explain several of these properties which arise in the context of problems of current interest in communication theory.
Thursday, January 21, 2010 - 15:00 , Location: Skiles 255 , Yekaterina Epshteyn , Carnegie Mellon University , Organizer: Sung Ha Kang
In this talk, I will first discuss several chemotaxis models includingthe classical Keller-Segel model.Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. In our work we propose a family of stable (even at times near blow up) and highly accurate numerical methods, based on interior penalty discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. This model is the basic step in the modeling of many real biological processes and it is described by a system of a convection-diffusion equation for the cell density, coupled with a reaction-diffusion equation for the chemoattractant concentration.We prove theoretical hp error estimates for the proposed discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.Numerical experiments to demonstrate the stability and accuracy of the proposed methods for chemotaxis models and comparison with other methods will be presented. Ongoing research projects will be discussed as well.