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CAMS Distinguished Colloquia for the Fall 2015 Semester

Stanley Osher
Monday, October 12 KAP 414 3:30 PM - 4:30 PM
Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere

It is well known that time dependent Hamilton-Jacobi-Isaacs partial differential equations (HJ PDE) play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they involve geometric motion is the level set method. The cost of these algorithms, and, in fact, all PDE numerical approximations is exponential in the space dimensions and time.

In this work we propose and test methods for solving a large class of HJ PDE without the use of grids or numerical approximations. For this wide class, which includes many linear control problems, we can obtain methods which are rapidly convergent, low memory, easily parallelizable and apparently very low complexity in dimension. We can evaluate the solution in many dimensions at between 10(-4) to 10(-8) seconds per evaluation on a laptop.

In addition, as a step needed in our procedure, we have developed a new and equally fast and efficient method to find the closest point xopt lying in the union of compact convex sets in Rn, (n large) to any point x exterior to this set.

The term "curse of dimensionality" was coined by Richard Bellman in 1957 when he considered problems in dynamic optimization.

Osher’s research interests include scientific computing, applied PDE, shock capturing methods, and image processing techniques.

Osher’s many honors and awards include membership of the National Academy of Sciences, Fellow of the American Academy of Arts and Sciences, Fellow of SIAM, Fellow of the AMS, honorary degrees from Hong Kong and ENS in Paris, the ICIAM Pioneer Prize and the SIAM Kleinman Prize. Most recently Osher received the Carl Friedrich Gauss Prize whose citation credited "his far ranging inventions that have changed our conception of physical, perceptual and mathematical concepts, giving us new tools to apprehend the world".

CAMS Distinguished Colloquia for the Spring 2015 Semester

Sylvester Gates
University of Maryland
Monday, January 26 KAP 414 3:30 PM - 4:30 PM
How Attempting To Answer A Physics Question Led Me to Graph Theory, Error-Correcting Codes, Coxeter Algebras, and Algebraic Geometry

We discuss how a still unsolved problem in the representation theory of Superstring/M-Theory has led to the discovery of previously unsuspected connections between diverse topics in mathematics.

Reception: Emmanuel Candes

Monday, April 13 Gerontology Courtyard 3:15 PM - 4:00 PM
You are cordially invited to attend

Emmanuel Candes
Stanford University, Joint with the Marshall School of Business
Monday, April 13 Gerontology Auditorium 4:00 PM - 5:00 PM
Around the Reproducibility of Scientific Research: A Knockoff Filter for Controlling the False Discovery Rate

The big data era has created a new scientific paradigm: collect data first, ask questions later. When the universe of scientific hypotheses that are being examined simultaneously is not taken account, inferences are likely to be false. The consequence is that follow up studies are likely not to be able to reproduce earlier reported findings or discoveries. This reproducibility failure bears a substantial cost and this talk is about new statistical tools to address this issue. Imagine that we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR)---the expected fraction of false discoveries among all discoveries---is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. We introduce the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, and the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. This work is joint with Rina Foygel Barber.

Grace Wahba
University of Wisconsin
Monday, May 4 KAP 414 3:30 PM - 4:30 PM
Learning Genetic Risk Models Using Distance Covariance

We extend an approach suggested by Li, Zhong and Zhu (2012) to use distance covariance (DCOV) as a variable selection method by providing the DCOV Variable Selection Theorem, which gives a principled stopping rule for a greedy variable selection algorithm. We apply the resulting DCOV Variable Selection Method in two genetic based classification problems with small sample size and large vectors of gene expression data.

The first problem involves the well known SBRCT (Small Blue Round Cell Tumor) childhood Leukemia data, which involves gene expression data from four different types of Leukemia, and it is well known that these data are easy to classify.

The second involves Ovarian Cancer data from The Cancer Genome Atlas, and involves Ovarian Cancer patients that are either sensitive or resistant to a platinum based cancer chemotherapy. The Ovarian Cancer data presents a difficult classification problem.

CAMS Distinguished Colloquia for the Spring 2014 Semester

Luis Caffarelli
UT Austin
Monday, March 3 KAP 414 3:30 PM - 4:30 PM
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.

CAMS Distinguished Colloquia for the Fall 2013 Semester

Ruth Williams
Monday, October 21 KAP 414 3:30 PM - 4:30 PM
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.

CAMS Distinguished Colloquia for the Spring 2013 Semester

Ronald Graham

Monday, March 4 KAP 414 2:00 PM - 3:00 PM
Special Time
Juggling Mathematics and Magic

The mystery of magic and the art of juggling have surprising links to interesting ideas from mathematics. In this talk, I will illustrate some of these connections.

Fan Chung Graham
UC San Diego
Monday, March 4 KAP 414 3:30 PM - 4:30 PM
Can you hear the shape of a network? New directions in spectral graph theory

We will discuss some recent developments in several new directions of spectral graph theory, including random walks for directed graphs, ranking algorithms, graph gauge theory, network games, graph limits and graphlets, for example.

CAMS Distinguished Colloquia for the Fall 2012 Semester

Bradley Efron
Stanford University
Monday, October 15 GFS 106 3:45 PM - 4:45 PM
Special Time & Location
Bayes and Empirical Bayes Information (Learning from the experience of others)

Bayesian methods require a catalog of prior experience for the interpretation of statistical evidence. In the absence of prior information, empirical Bayes methods rely instead on a catalog of cases similar to the problem of interest. The crime rate in one small city, for example, may be estimated by modifying its observed rate with evidence from other cities.
I will give some examples that show how powerful the empirical Bayes approach can be in practice, both for estimation and testing. The use of "other" cases then raises the question of just which others are relevant, and how their information bears on the case of interest.

CAMS Distinguished Colloquia for the Spring 2012 Semester

Edward Witten
Institute for Advanced Study
Wednesday, March 28 SAL 101 3:45 PM - 4:45 PM
Special Time & Location
Khovanov Homology And Gauge Theory

In this talk, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory.

CAMS Distinguished Colloquia for the Fall 2011 Semester

Nancy Kopell
Boston University
Monday, December 5 GER Auditorium 3:30 PM - 4:30 PM
Special Location
Connecting The Dots: Propofol, Parkinson’s Disease and Brain Rhythms

Rhythms of the nervous system are produced in all cognitive states, and have been shown to be highly associated with a myriad of cognitive tasks. Thus, changes in these rhythms, however they come about, are likely to change the ability to do such tasks. This talk focuses on the beta (12-30 Hz) and alpha (9-11) rhythms, and pathological states due to anesthesia and PD; it is about three related studies, the latter two emerging from the first one. The first concerns an early stage of anesthesia, in which, paradoxically, the subject gets more excited and disoriented. With low propofol, the brain rhythms show an increase in beta oscillations, which in normal awake state is associated with brain functions including motor preparation and higher-order processing.
The second concerns the beta oscillations associated with abnormal motor control in Parkinson’s disease. The relationship between the two phenomena can be seen from the underlying physiology using modeling as well as experiments. Finally, the first story led to looking at higher doses of propofol at which consciousness is lost, and uses experimental data to get new ideas about the physiological basis for the loss of consciousness. Again, the focus on relevant physiology of the rhythms is what led, though modeling, to the new insights. Applications to other states of consciousness, such as coma, may be discussed.

CAMS Distinguished Colloquia for the Spring 2011 Semester

Dennis Sullivan
SUNY and CUNY Graduate Center
Monday, March 7 KAP 414 3:30 PM - 4:30 PM
Correlated finite energy models of Navier Stokes time evolution

If one has an AT (Algebraic Topology) model of a system of fields and operations in Riemannian geometry, there is a natural way to construct derived models at each scale of resolution. In addition there are transition mappings between these derived models at different scales.The process of constructing derived models is based on the key idea of AT: chain homotopy equivalences between chain complexes.
If a nonlinear PDE among the original system of fields and operations can be reformulated in the derived models, one can obtain a system of finite energy or finite scale models which are correlated by structure mappings.
Incompressible Navier Stokes evolution in 3D can be described by the differential algebra of differential forms, the Hodge star operator and the projections of the Hodge decomposition. These objects are naturally interpreted in AT.
The lecture will discuss this AT approach to deriving computational fluid models.

CAMS Distinguished Colloquia for the Fall 2010 Semester

Charles Fefferman
Princeton University, CAMS Distinguished Lecturer
Friday, December 3 KAP 414 3:30 PM - 4:30 PM
The Muskat Problem

The problem concerns the evolution of the interfaces between two or more fluids in a porous medium. The talk presents new phenomena arising when at least three fluids are present. (Joint work with several coauthors)

CAMS Distinguished Colloquia for the Spring 2010 Semester

James Glimm
CAMS Distinguished Lecturer
Monday, March 22 Andrus Gerontology Center 4:00 PM - 5:00 PM
Mathematical and Numerical Principles for Turbulent Mixing

Turbulent mixing is an important aspect of a number of practical problems, often combined with combustion or some other reaction. Due to the importance of this problem, considerable effort has been invested in verification (mathematical correctness of numerical solutions) and validation (correctness and applicability of the equations to be solved). A standard test problem of this class is Rayleigh-Taylor mixing, the problem of a heavy fluid over a light one, mixing under the acceleration force of gravity.

CAMS Distinguished Colloquia for the Fall 2009 Semester

Margaret Wright
Courant Institute, NYU (the CAMS Distinguished Lecture, preceded by a reception at 3:00)
Friday, October 9 Andrus Gerontology Center 3:30 PM - 4:30 PM
Optimization without derivatives: consensus and controversies

Non-derivative methods for optimization have had a sometimes rocky relationship for more than 50 years with applied mathematicians who specialize in optimization.
Although practitioners have never wavered in their fondness for non-derivative methods, their mathematical foundations were mostly lacking until the late 1980s. Since then, significant progress has been made concerning theoretical underpinnings, but several perplexing mysteries remain.
In addition, there has been continuing and lively controversy about which methods are ``most effective''
on real-world applications, with disagreements about both the selection of test problems and the choice of criteria for assessing computational results. This talk will briefly survey the current state of the art, trying along the way to highlight a few of the interesting open questions.

CAMS Distinguished Colloquia for the Spring 2009 Semester

Terence Tao
UCLA, Joint with the Whiteman Lectures
Thursday, February 19 Gerentology Auditorium 3:30 PM - 4:30 PM
Compressed Sensing

Suppose one wants to recover an unknown signal x in Rn from a given vector Ax=b in Rm of linear measurements of the signal x. If the number of measurements m is less than the degrees of freedom n of the signal, then the problem is underdetermined and the solution x is not unique. However, if we also know that x is sparse or compressible with respect to some basis, then it is a remarkable fact that (given some assumptions on the measurement matrix A) we can reconstruct x from the measurements b with high accuracy, and in some cases with perfect accuracy. Furthermore, the algorithm for performing the reconstruction is computationally
feasible. This observation underlies the newly developing field of compressed sensing. In this talk we will discuss some of the mathematical foundations of this.

George Papanicolaou
Stanford, CAMS Distinguished Lecturer
Friday, April 17 Gerentology Auditorium 3:30 PM - 4:30 PM
Imaging with Noise

It is somewhat surprising at first that it is possible to locate a network of sensors from cross correlations of noise signals that they record. This is assuming that the speed of propagation in the ambient environment is known and that the noise sources are sufficiently diverse. If the sensor locations are known and the propagation speed is not known then it can be estimated from cross correlation information. Although a basic understanding of these possibilities had been available for some time, it is the success of recent applications in seismology that have revealed the great potential of correlation methods, passive sensors and the constructive use of ambient noise in imaging. I will introduce these ideas in an interdisciplinary, mathematical way and show that a great deal can be done with them.
Things become more complicated, and a mathematically more interesting, when the ambient medium is also strongly scattering. I will end with a review of what is known so far in this case, and what might be expected.

CAMS Distinguished Colloquia for the Fall 2008 Semester

Persi Diaconis
Friday, October 24 KAP 249 3:30 PM - 4:30 PM
Adding Numbers and Shuffling Cards

The usual process of "carries" when adding numbers turns out to have interesting mathematics hidden in it. It begins with an "amazing" matrix discovered by Holte, which has close connections to the usual way of mixing cards by riffle shuffling. The connections give new results for addition and for shuffling. This is joint work with Jason Fulman.