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 Colloquia for the Fall 2014 Semester Monday, September 8 KAP 414 3:30 PM - 4:30 PM Career Advice Panel Panelists: Francis Bonahon, Eric Friedlander, Jason Fulman, Cymra Haskell, Paul SobajeModerator: Kenneth AlexanderAll graduate students and postdocs are encouraged to come and ask questions about positioning themselves for their future careers. Anna Mazzucato Penn State University Monday, September 15 KAP 414 3:30 PM - 4:30 PM Optimal mixing by incompressible flows I will discuss mixing of passive scalars by incompressible flows and measures of optimal mixing. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with prescribed energy or enstrophy budget. These examples are related to loss of regularity for solutions of transport equations. Marco Sammartino University of Palermo, visiting USC Monday, October 6 KAP 414 3:30 PM - 4:30 PM Navier-Stokes Equations in the Zero Viscosity Limit: Boundary Layers, Separation and Blow Ups The appearance of a boundary layer (BL) is a ubiquitous phenomenon in applied mathematics: a BL occurs when the presence of a small parameter causes a sharp transition between the perturbed and the unperturbed regime. The concept of BL was introduced by Ludwig Prandtl to give an explanation to D'Alembert's paradox; Prandtl's 1904 paper would prove to be one of the most important fluid dynamics paper ever written. However, despite more than a century of investigations, many problems raised by Prandtl's BL theory still remain unsolved. Among them we mention the lack of a fully satisfactory mathematical theory of Prandtl's equations and the problem of the convergence, in the zero viscosity limit, of the Navier-Stokes solutions to the Euler solutions.In this talk after giving a review of some of the results that have been recently obtained in this area we shall consider an incompressible flow interacting with a boundary without assuming that the initial datum satisfies the no-slip condition at the boundary. A typical case when this situation occurs is the impulsively started disk. Other instances widely studied in the literature are when a vortical configuration, which is a steady solution of the Euler equations (like the thick core vortex or the vortex array), is assumed to interact instantaneously with a solid boundary.Focusing our analysis on the Navier-Stokes equations on a half-space, we shall construct the initialboundary layer corrector in the form of a Prandtl solution with incompatible data. This corrector is the first term of an asymptotic series that we shall prove to approximate, in the zero viscosity limit and for a short time, the Navier-Stokes solutions. Assuming analytic regularity in the tangential direction, we shall prove that this time does not depend on the viscosity. Nets Katz Caltech Monday, October 13 KAP 414 3:30 PM - 4:30 PM On the three dimensional Kakeya problem We discuss new ideas for obtaining lower bounds on the Hausdorff dimension of Kakeya sets. We discuss joint work in progress with Josh Zahl.Sometimes less is more. Charles Doering University of Michigan Monday, October 20 KAP 414 3:30 PM - 4:30 PM Wall to wall optimal transport How much stuff can be transported by an incompressible flow containing a specified amount of kinetic energy or enstrophy? We study this problem for steady 2D flows focusing on passive tracer transport between two parallel impermeablewalls, employing the calculus of variations to find divergence-free velocity field with a given intensity budget that maximize transport between the walls. The maximizing velocity fields, i.e. the optimal flows, consist of arrays of (convection-like) cells. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Peclet number Pe, the dimensionless gauge of the strength of the flow. For both energy and enstrophy constraints we find that as Pe increases, the maximum transport is achieved by cells of decreasing aspect ratio. For each of the two flow intensity constraints, we also consider buoyancy-driven flows the same constraint to see how the scalings for transport reported in the literature compare with the absolute upper bounds. This work provides new insight into both steady optimal transport and turbulent transport, an increasingly lively area of research in geophysical, astrophysical, and engineering fluid dynamics. This is joint work with Pedram Hassanzadeh(Berkeley/Harvard) and Gregory P. Chini (University of New Hampshire) published in *Journal of Fluid Mechanics **751*, 627-662 (2014). Tristan Buckmaster Courant Institute Monday, October 27 KAP 414 3:30 PM - 4:30 PM Onsager's Conjecture In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time. Michael Wolf University of Zurich Monday, November 3 KAP 414 3:30 PM - 4:30 PM Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals. David Levermore University of Maryland Monday, November 10 KAP 414 3:30 PM - 4:30 PM Scattering Theory for the Boltzmann Equation and the Arrow of Time (joint work with Claude Bardos, Irene Gamba, and Francois Golse) We develop a scattering theory for a class of eternal solutions of the Boltzmann equation posed over all space. In three spatial dimensions each of these solutions has thirteen conserved quantities. The Boltzmann entropy has a unique minimizer with the same thirteen conserved values. This minimizer is a local Maxwellian that is also a global solution of the Boltzmann equation --- a so-called global Maxwellian. We show that each of our eternal solutions has a streaming asymptotic state as time goes to minus or plus infinity. However it does not converge to the associated global Maxwellian as time goes to infinity unless it is that global Maxwellian. The Boltzmann entropy decreases as time increases, but does not decrease to its minimum as time goes to infinity. Said another way, the final step in the traditional argument for the heat death of the universe is not valid. Inwon Kim UCLA Monday, November 17 KAP 414 3:30 PM - 4:30 PM Congested crowd motion and Quasi-static evolution In this talk we investigate the relationship between a quasi-static evolution and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. We will also discuss preliminary results on general initial data. Ngoc Mai Tran University of Texas Friday, November 21 KAP 414 3:00 PM - 4:00 PM Special Time Special Colloquium: Random permutations and random partitions I will talk about various problems related to random permutations and random partitions. In particular, I discuss size-biased permutations, which have applications to statistical sampling. Then I will talk about random partitions obtained from projections of polytopes. These are related to random polytopes and zeros of random tropical polynomials. Joseph Neeman University of Texas Friday, November 21 KAP 414 4:30 PM - 5:30 PM Special Time Special Colloquium: Gaussian noise stability Given two correlated Gaussian vectors, X and Y, the noise stability of a set A is the probability that both X and Y fall in A. In 1985, C. Borell proved that half-spaces maximize the noise stability among all sets of a given Gaussian measure. We will give a new, and simpler, proof of this fact, along with some extensions and applications. Specifically, we will discuss hitting times for the Ornstein-Uhlenbeck process, and a noisy Gaussian analogue of the "double bubble" problem. Colloquia for the Summer 2014 Semester Thanasis Fokas Cambridge University Thursday, May 8 KAP 414 3:30 PM - 4:30 PM Boundary Value Problems and Medical Imaging In the late 60s a new area emerged in mathematical physics known as "Integrable Systems". Ideas and techniques of "Integrability" have had a significant impact in several areas of mathematics, science and engineering, from the proof of the Schottky problem in algebraic geometry, to optical communications. In this lecture, two such implications will be reviewed: (a) A novel method for analysing boundary value problems, which unifies the fundamental contributions to the analytical solution of PDEs of Fourier, Cauchy and Green, and also constructs a non-linearization of some of these results. This method has led to the emergence of new numerical techniques for solving linear elliptic PDEs in polygonal domains. (b) A new approach for solving the inverse problems arising in certain important medical imaging techniques, including Single Photon Emission Computerised Tomography (SPECT). Colloquia for the Spring 2014 Semester Philip Isett MIT Monday, January 27 KAP 414 3:30 PM - 4:30 PM Recent progress towards Onsager’s Conjecture Motivated by the theory of hydrodynamic turbulence, L. Onsager conjectured in 1949 that solutions to the incompressible Euler equations with Holder regularity less than 1/3 may fail to conserve energy. C. DeLellis and L. Székelyhidi, Jr. have pioneered an approach to constructing such irregular flows based on an iteration scheme known as convex integration. This approach involves correcting “approximate solutions" by adding rapid oscillations which are designed to reduce the error term in solving the equation. In this talk, I will discuss an improved convex integration framework, which yields solutions with Holder regularity as much as 1/5-. Guillermo Reyes-Souto UC Irvine Monday, February 3 KAP 414 3:30 PM - 4:30 PM Degenerate Diffusion in Heterogeneous Media In this talk, I will present some recent results on the long-time behavior of non-negative solutions to the Cauchy problem for the Porous Medium Equation in the presence of variable density vanishing at infinity, [RV], [KRV]. Nathan Glatt-Holtz Virginia Tech Monday, February 10 KAP 414 3:30 PM - 4:30 PM Inviscid Limits for the Stochastic Navier Stokes Equations and Related Systems One of the original motivations for the development of stochastic partial differential equations traces it's origins to the study of turbulence. In particular, invariant measures provide a canonical mathematical object connecting the basic equations of fluid dynamics to the statistical properties of turbulent flows. In this talk we discuss some recent results concerning inviscid limits in this class of measures for the stochastic Navier-Stokes equations and other related systems arising in geophysical and numerical settings. Arthur Toga Institute for Neuro Imaging, USC Monday, February 24 KAP 414 3:30 PM - 4:30 PM The Informatics of Brain Mapping The complexity of neurodegenerative and psychiatric diseases often requires the collection of numerous data types from multiple modalities. These can be genetic, imaging, clinical and biosample data. In combination, they can provide biomarkers critical to chart the progression of the disease and to measure the efficacy of therapeutic intervention. The difficulties lie in how can these diverse data from different subjects, collected across multiple laboratories on a wide range of instruments using non-identical protocols be aggregated and mined to discover meaningful patterns. Mapping the human brain, and the brains of other species, has long been hampered by the fact that there is substantial variance in both the structure and function of this organ among individuals within a species. Previous brain atlases have relied on information from, at best, a few samples to draw conclusions. These limitations and the lack of quantification for the variance in brain structure and function have limited the pace and accuracy of research in the field of neuroscience. There are numerous probabilistic atlases that describe specific subpopulations, measure their variability and characterize the structural differences between them. Utilizing data from structural, functional, diffusion MRI, along with gwas studies and clinical measures we have built atlases with defined coordinate systems creating a framework for mapping and relating diverse data across studies. This talk describes the development and application of theoretical framework and computational tools for the construction of probabilistic atlases of large numbers of individuals in a population. These approaches are useful in understanding multidimensional data and their relationships over time. A specific and important example of mapping multimodal data is the study of Alzheimer’s. The dynamic changes that occur in brain structure and function throughout life make the study of degenerative disorders of the aged difficult. The Alzheimer’s Disease Neuroimaging Initiative (ADNI) is a large national consortia established to collect, longitudinally, distributed and well described cohorts of age matched normals, mci's and Alzheimer’s patients. It results from the abnormal accumulation of misfolded amyloid and tau proteins in neurons and the extracellular space, ultimately leading to cell death and progressive cognitive decline. The consequences of this insult can be seen using a variety of imaging and other data analyzed from the ADNI database.Essential elements in performing this type of population based research are the informatics infrastructure to assemble, describe, disseminate and mine data collections along with computational resources necessary for large scale processing of big data such as whole genome sequence data and imaging data. This talk also describes the methods we have employed to address these challenges. Luis Caffarelli UT Austin Monday, March 3 KAP 414 3:30 PM - 4:30 PM CAMS Distinguished Lecturer Surfaces and fronts in periodic media In this lecture I will review work that concerns the behavior of surfaces and fronts in a periodic media that is highly oscillatory: minimal surfaces, whose area is weighted by a periodic factor, capillary drops sitting in a composite surface, the effective speed of flame propagation in periodic media. Gautam Iyer Carnegie Mellon Monday, March 10 KAP 414 3:30 PM - 4:30 PM Stirring and Mixing I will talk about various mixing'' questions that have attracted interest recently. For instance, Can you stir your coffee to keep it hot for longer'', or How well can you stir cream into your coffee, and at what cost?''.Mathematically these questions translate into studying a negative Sobolev norm of a passively advected scalar. The study of such questions also involves very interesting connection Bressan's (still open!) rearrangement cost conjecture. I will spend most of the talk surveying recent results, and conclude with brief description of joint work with A. Kiselev, Xiaoqian Xu and myself. Aleksey Polunchenko Binghamton University Monday, March 24 KAP 414 3:30 PM - 4:30 PM Efficient Performance Evaluation of the Generalized Shiryaev--Roberts Detection Procedure in the Multi-Cyclic Setup Quickest change-point detection is a branch of statistics concerned with the design and analysis of reliable statistical machinery for rapid anomaly detection in live'' monitored data. The subject's current state-of-the-art detection procedure is the recently proposed Generalized Shiryaev--Roberts (GSR) procedure (it was proposed in 2008, but the paper was published only in 2011). Notwithstanding its young age'', the GSR procedure has already been shown to have very strong optimality properties not exhibited by such well-known mainstream procedures as the Cumulative Sum inspection scheme'' and the Exponentially Weighted Moving Average (EWMA) chart. To foster and facilitate further research on the GSR procedure we propose a numerical method to evaluate the performance of the GSR procedure in a minimax-ish'' multi-cyclic setup where the procedure of choice is applied repetitively (cyclically) and the change is assumed to take place at an unknown time moment in a distant-future stationary regime. Specifically, the proposed method is based on the integral-equations approach and uses the collocation technique with the basis functions chosen so as to exploit a certain change-of-measure identity and the GSR detection statistic's unique martingale property. As a result, the method's accuracy and robustness improve, as does its efficiency since using the change-of-measure ploy the Average Run Length (ARL) to false alarm and the Stationary Average Detection Delay (STADD) are computed simultaneously. We show that the method's rate of convergence is quadratic and supply a tight upperbound on its error. We conclude with a case study and confirm experimentally that the proposed method's accuracy and rate of convergence are robust with respect to three factors: a) partition fineness (coarse vs. fine), b) change magnitude (faint vs. contrast), and c) the level of the ARL to false alarm (low vs. high). Since the method is designed not restricted to a particular data distribution or to a specific value of the GSR detection statistic's headstart, this work may help gain greater insight into the characteristics of the GSR procedure and aid a practitioner to design the GSR procedure as needed while fully utilizing its potential.This is joint work with Grigory Sokolov (Department of Mathematics, U. of Southern California) and Wenyu Du (Department of Mathematical Sciences, SUNY Binghamton). Kevin Zumbrun Indiana University Monday, March 31 KAP 414 3:30 PM - 4:30 PM Nonlinear modulation of spatially periodic waves Periodic waves are important features of solutions of nonlinear evolution systems in such varied contexts as optics, hydrodynamics, and reaction diffusion.A formal description of their behavior under perturbation is given by WKB expansion in terms of modulations in phase and local waveform, as pioneered by Whitham, Howard-Kopell, and Serre in various contexts. The Whitham modulation equations take the form, to lowest order, of a first-order system of conservation laws, whose characteristic speeds play a role in the nonlinear setting analogous to that of group velocity in the linear case, giving the rate of propagation of localized wave packets.In this talk we discuss recent results giving rigorous verification of this formal Whitham description using a combination of Bloch transform techniques, and techniques originating from shock wave stability and the theory of conservation laws for efficiently extracting nonlinear modulations in phase. Notably, this approach allows the treatment of situations for which the Whitham equations have multiple characteristic speeds, whereas previous techniques based on renormalization methods were limited to the case of a single characteristic speed. Indeed, the techniques introduced apply also in situations far from a periodic background, to which the Whitham equations no longer directly apply. Lisa Fauci Tulane Wednesday, April 16 150 SSL 3:30 PM - 4:30 PM Special Location Explorations in biofluids: a tale of two tails In the past decade, the study of the fluid dynamics of swimming organisms has flourished. With the possibility of using fabricated robotic micro swimmers for drug delivery, or harnessing the power of natural microorganisms to transport loads, the need for a full description of flow properties is evident. At a larger scale, the swimming of a simple vertebrate, the lamprey, can shed light on the coupling of neural signals to muscle mechanics and passive body dynamics in animal locomotion.We will present recent progress in the development of a multiscale computational model of the lamprey that examines the emergent swimming behavior of the coupled fluid-muscle-body system. At the micro scale, we will examine the function of a flagellum of a dinoflagellate, a type of phytoplankton. We hope to demonstrate that, even when the body kinematics at zero Reynolds number are specified, there are still interesting fluid dynamic questions that have yet to be answered. Alexander Lipton Bank of America Monday, April 21 KAP 414 3:30 PM - 4:30 PM Three-dimensional Brownian motion and its applications to CVA and trading
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