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 nonlinearization 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 ReyesSouto
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 longtime behavior of nonnegative solutions to the Cauchy problem for the Porous Medium Equation in the presence of variable density vanishing at infinity, [RV], [KRV].



Nathan GlattHoltz
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 NavierStokes 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 nonidentical 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 ShiryaevRoberts Detection Procedure in the MultiCyclic Setup
Quickest changepoint 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 stateoftheart detection procedure is the recently proposed Generalized ShiryaevRoberts (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 wellknown 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 ``minimaxish'' multicyclic 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 distantfuture stationary regime. Specifically, the proposed method is based on the integralequations approach and uses the collocation technique with the basis functions chosen so as to exploit a certain changeofmeasure 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 changeofmeasure 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, HowardKopell, and Serre in various contexts. The Whitham modulation equations take the form, to lowest order, of a firstorder 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 fluidmusclebody 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

Threedimensional Brownian motion and its applications to CVA and trading


