# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Compressive imaging: Sampling strategies and reconstruction guarantees

In many applications such as Magnetic Resonance Imaging, images are acquired using Fourier transform measurements. Such measurements can be expensive, and it is of interest to exploit the wavelet domain sparsity of natural images to reduce the number of measurements without destroying reconstruction quality. Much work in compressed sensing has been devoted to this problem in recent years. However, a rigorous theory for sampling with compressive frequency measurements has to date only been developed for bases that, unlike wavelet bases, are incoherent to the Fourier basis. Nevertheless, it has been shown empirically that variable density sampling strategies seem to overcome this obstacle.

##### Data-driven modeling and dimensionality reduction

Dimensionality reduction is a common method for rendering tractable a host of problems arising in the physical, engineering and biological sciences. In recent years, methods from data analysis have started playing critical roles in more traditional applied mathematics problems typically analyzed with dynamical systems and PDE techniques. In this talk, three disparate examples will be considered from (i) image processing, (ii) PDE solution techniques and (iii) neuroscience.

##### Flows and Decompositions of Games: Harmonic and Potential Games

We introduce a novel flow representation for finite games in strategic form. Based on this representation, we develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. Besides its intrinsic interest, this decomposition facilitates the study of Nash and correlated equilibria as well as convergence properties of natural distributed game dynamics. We explain the basic ideas, and illustrate the implications of the decomposition for dynamic analysis, pricing schemes, efficiency loss, and network games. Based on joint work with Ozan Candogan, Ishai Menache, and Asu Ozdaglar (MIT).

##### Parabolic Molecules: Curvelets, Shearlets, and Beyond

Anisotropic representation systems such as shearlets and curvelets have had a significant impact on applied mathematics in the last decade. The main reason for their success is their superior ability to optimally resolve anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. By now, a large variety of such anisotropic systems has been introduced, among which we mention second generation curvelets, bandlimited shearlets and compactly supported shearlets, all based on a parabolic dilation operation. These systems share similar sparsity properties, which is usually proven on a case-by-case basis for each different construction.

##### TBA

##### Robust Subspace Modeling

Consider a dataset of vector-valued observations that consists of a modest number of noisy inliers, which are explained well by a low-dimensional subspace, along with a large number of outliers, which have no linear structure. We describe a convex optimization problem that can reliably fit a low-dimensional model to this type of data. When the inliers are contained in a low-dimensional subspace we provide a rigorous theory that describes when this optimization can recover the subspace exactly. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### How to Design Simple Efficient Mechanisms that are also Composable

SPECIAL JOINT SEMINAR WITH COMPUTER SCIENCE. E-commerce applications require simple, and well-designed systems, and systems that work well even if users participate in multiple mechanisms (and the value of each player overall is a complex function of their outcomes). Traditional mechanism design has considered such mechanisms only in isolation, and the mechanisms it proposes tend to be complex and impractical. In contrast, players typically participate in various mechanisms, mechanisms that are run by different principals (e.g. different sellers on eBay or different ad-exchange platforms) and coordinating them to run a single combined mechanism is infeasible or impractical. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Dynamics of a Cytokine Storm

Six volunteers experienced severe inflammatory response during the Phase I clinical trial of a monoclonal antibody that was designed to stimulate a regulatory T cell response. Soon after the trial began, each volunteer experienced a "cytokine storm", a dramatic increase in cytokine concentrations. (CLICK ON SEMINAR TITLE FOR FULL ABSTRACT.)

##### Solving the GPS Problem in Almost Linear Time

A client on the earth surface wants to know her geographical location. The Global Positioning System (GPS) was built to fulfill this task. It works as follows. Satellites send to earth their location. For simplicity, the location of a satellite is a bit b=1,-1. The satellite transmits to the earth a sequence of N>1000 complex numbers S[0],S[1],...,S[N-1] multiplied by its location b. The client receives the sequence R which is a noisy version of S distorted in two parameters that encode the distance and relative radial velocity of the satellite with respect to the client. The GPS PROBLEM is to calculate the distance and the bit b. A client can compute her location by knowing the locations of at least three satellites and distances to them.

##### Controlled Active Vision/Image Processing with Applications to Medical Image Computing

In this talk, we will describe some theory and practice of controlled active vision. The applications range from visual tracking (e.g., laser tracking in turbulence, flying in formation of UAVs, etc.), nanoparticle flow control, and sedation control in the intensive care unit. Our emphasis will be on the medical side, especially image guided therapy and surgery. PLEASE CLICK ON SEMINAR TITLE FOR FULL ABSTRACT.

##### TBA

##### Approximation Bounds for Sparse Principal Component Analysis

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component.

##### What is a degree distribution?

The most studied aspect of statistical network models is their degree structure, reflecting the propensity of nodes within a network to form connections with other nodes. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Physical Reproduction of Materials with Specified Reflectance and Scattering

Although real-world surfaces can exhibit significant variation in materials - glossy, diffuse, metallic, translucent, etc. - printers are usually used to reproduce color or gray-scale images. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Coarse-grained models of gas-particle flows

Gas-particle flows are encountered widely in chemical process industries. These flows are often unstable and persistent fluctuations in velocities and particle volume fractions occur over a wide range of length and time scales. It is now well established that two-fluid models (TFM), with constitutive relations for the stresses deduced from the kinetic theory of granular materials, are able to capture the formation of such inhomogeneous structures. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Information theoretic thresholds in a class of probabilistic models on graphs

We introduce a class of probabilistic models on graphs motivated by coding and combinatorial optimization problems (e.g., LDPC codes, k-SAT, clustering). We show that for a large class of models, the conditional entropy between the information at the nodes and at the edges concentrates around a determnistic threshold. Joint work with A. Montanari.

##### Integrability and turbulence

PLEASE NOTE SPECIAL DAY AND TIME (WEDNESDAY, 3:30 PM). An important aspect of mathematical formulations of turbulence is to understand the propagation of randomness by the equations of fluid flow.

##### Interacting Bayesian Agents on Networks (joint ORFE/PACM talk)

(NOTE SPECIAL DAY & LOCATION.) Consider a network of experts who try to predict if a certain movie will win an Oscar. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### Long Memory in Stochastic Modeling: From Malliavin Calculus, to Finance, to Climatology

(NOTE SPECIAL LOCATION.) The theory of continuousstochastic processes has provided a wealth of modeling tools for random systems with complex interactions, via the building blocks of Brownian motion, martingales, and Markov processes, for instance via stochastic differential equations. PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT.

##### PACM Distinguished Lecture Series: Linear Algebra and the Shape of Bird Beaks”

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections.