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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

Colloquia for the Fall 2013 Semester

Anthony Suen
USC
Monday, September 9 KAP 414 3:30 PM - 4:30 PM
Global weak solutions of the equations of compressible

We prove the global-in-time existence of weak solutions of the three-dimensional isothermal, compressible magnetohydrodynamic (MHD) equations in the whole R^3 space. The initial density is strictly positive, essentially bounded and is close to a constant in L^2, and the initial velocity and magnetic field are both small in L^2 and bounded in L^n for some n>6. Here the initial data may be discontinuous across a hypersurface of R^3.




Monday, September 30 KAP 414 3:30 PM - 4:30 PM
Panel: Applying for jobs and grants


Jacob Bedrossian
Courant Institute
Monday, October 7 KAP 414 3:30 PM - 4:30 PM
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Joint work with Nader Masmoudi.


Alan Schumitzky
USC
Monday, October 14 KAP 414 3:30 PM - 4:30 PM
Estimating an unknown probability distribution: a convexity approach

We consider the following estimation problem. Let X1,..., XN be a sequence of N independent random vectors with common but unknown probability distribution F. The { Xi } are not observed. Let Y1,..., YN be another sequence of N independent random vectors which are observed. Assume the conditional probability P(Yi | Xi) is known for i = 1,..., N. The problem is to estimate the probability distribution F given the { Yi }. The main theoretical result is that the maximum likelihood estimate of F can be found in the space of discrete distributions with no more that N support points. An elegant proof of this will be given using results of convexity theory in a topological vector space. The practical numerical problem of actually determining the discrete maximum likelihood estimator of F is still not solved in a satisfactory manner. Examples from the area of applied pharmacokinetics will be given.


Ruth Williams
UCSD
Monday, October 21 KAP 414 3:30 PM - 4:30 PM
CAMS Distinguished Lecturer
Resource Sharing in Stochastic Networks

Stochastic models of processing networks arise in a wide variety of applications in science and engineering, e.g., in high-tech manufacturing, transportation, telecommunications, computer systems, customer service systems, and biochemical reaction networks. These "stochastic processing networks" typically have entities, such as jobs, vehicles, packets, customers or molecules, that move along paths or routes, receive processing from various resources, and that are subject to the effects of stochastic variability through such variables as arrival times, processing times and routing protocols.
Networks arising in modern applications are often heterogeneous in that different entities share (i.e., compete for) common network resources. Frequently the processing capacity of resources is limited and there are bottlenecks, resulting in congestion and delay due to entities waiting for processing.
The control and analysis of such networks present challenging mathematical problems.

This talk will explore the effects of resource sharing in stochastic networks and describe associated mathematical analysis based on elegant fluid and diffusion approximations. Illustrative examples will be drawn from biology and telecommunications.


Irena Lasiecka
U of Memphis
Monday, October 28 KAP 414 3:30 PM - 4:30 PM
Long time behavior of solutions to flow-structure interactions arising in modeling of subsonic and supersonic flows of gas.

We consider flow - structure interaction comprising of a modified wave equation coupled to a nonlinear plate equation. The model has no damping imposed neither on the structure nor the flow. The regime of the parameters considered includes supersonic flows - the latter known for depleting ellipticity from the corresponding static model. Thus, both well-posedness of finite energy solutions and long time behavior of the model
have been open questions in the literature. The results presented include:
• Existence, uniqueness and Hadamard wellposedness of finite energy solutions.
• Existence of global and finite dimensional attracting sets for the structure in the absence of any mechanical dissipation. The key "hidden regularity type" inequalities, responsible for proving existence of nonlinear semigroup, are derived by using microlocal analysis.
The existence of an attracting set is proved without imposing any form of dissipation on the model. This is achieved by exploiting "compensated compactness" related to the dispersive character of the flow equation. To our knowledge, this is the first complete exhibition and rigorous justification of this fact - previously known experimentally only. In order to resolve the difficulty, we follow the decoupling method of [1] which reduces the problem to a study of nonlinear plates with the delay terms.

This presentation is based on a joint work with Igor Chueshov, Kharkov University and Justin Webster, Oregon State University.


Michael Waterman
USC
Monday, November 11 KAP 414 3:30 PM - 4:30 PM
Sequence Comparison Using Word Counts

Recently word count statistics have received attention due to their computational efficiency. Those statistics are for entire sequences. Local alignment-free sequence comparison arises in the context of identifying similar segments of sequences that may not be alignable in the traditional sense. We propose a randomized approximation algorithm that is both accurate and efficient.


Fadil Santosa
U of Minnesota
Monday, November 18 KAP 414 3:30 PM - 4:30 PM
Resonances -- Analysis and Optimization

We consider the problem of resonances for Schroedinger's equation and Helmholtz equation. These equation provide a model for an optimal design problem in which the goal is to create an optical structure that has a resonance that has low loss. In addition we will also discuss an asymptotic method for calculating low-loss resonances which is the forward problem in this inverse design problem.


Zahar Hani
Courant Institute, NYU
Monday, November 18 KAP 414 2:00 PM - 3:00 PM
Special Time
Out-of-equilibrium dynamics for the nonlinear Schroedinger equation: From energy cascades to weak turbulence.

Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations on bounded domains. This is partly due to the fact that dispersion does not translate into decay in this setting (in contrast to the case of unbounded domains like $R^d$). In this talk, we will take the cubic nonlinear Schroedinger equation as our model, and discuss some aspects of its out-of-equilibrium dynamics, from energy cascades (i.e. migration of energy from low to high frequencies) to weak turbulence.


Guo Luo
Caltech
Monday, November 25 KAP 414 3:30 PM - 4:30 PM
Potentially Singular Solutions of the 3D Incompressible Euler Equations

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data has been one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and no-flow boundary condition on the solid wall.
The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method, on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 \times 10^{12})^{2}$ near the point of singularity, we are able to advance the solution up to $\tau_{2} = 0.003505$ and predict a singularity time of $t_{s} \approx 0.0035056$, while achieving a \emph{pointwise} relative error of $O(10^{-4})$ in the vorticity vector $\omega$ and observing a $(3 \times 10^{8})$-fold increase in the maximum vorticity $\norm{\omega}_{\infty}$. The numerical data is checked against all major blowup (non-blowup) criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the
singularity. A careful local analysis also suggests that the blowing-up solution develops a self-similar structure near the point of the singularity, as the singularity time is approached.


Katya Krupchyk
University of Helsinki
Monday, December 2 KAP 414 3:30 PM - 4:30 PM
Resolvent estimates for elliptic operators and their applications.

More than 25 years ago, Kenig, Ruiz, and Sogge established uniform $L^p$ resolvent estimates for the Laplacian in the Euclidean space. Taking their remarkable estimate as a starting point, we shall describe more recent developments concerned with the problem of controlling the resolvent of elliptic self-adjoint operators in $L^p$ spaces in the context of a compact Riemannian manifold. Here some new interesting difficulties arise, related to the distribution of eigenvalues of such operators. Applications to inverse boundary problems for rough potentials and to the absolute continuity of spectra for periodic Schr\"odinger operators will be presented as well. This talk is based on joint works with Gunther Uhlmann.


James von Brecht
UCLA
Wednesday, December 4 KAP 414 2:00 PM - 3:00 PM
Special Time
Co-Dimension One Self Assembly

The interest in and study of compact, co-dimension one minimizers has at least a century-old history: in 1904, J. J. Thomson proposed minimizing the electrostatic potential over sets of particles restricted to a sphere as part of his model of the atom. Modern physical examples of these assemblies occur in the realm of interacting nanoparticles. Many species of virus rely on the formation of a hollow sphere to enclose and deliver their genetic material, for example. Inorganic polyoxometalate (POM) macroions also form into hollow spherical structures in a similar way. I will discuss recently developed mathematical theory that characterizes when spherical assemblies define energy favorable structures, as well as applications to physical models of these assemblies.