# Seminars & Events for PACM/Applied Mathematics Colloquium

##### Still running! Recent work on the neuromechanics of insect locomotion

I will describe several models for running insects, from an energy-conserving biped with passively-sprung legs to a muscle-actuated hexapod driven by a neural central pattern generator(CPG). Phase reduction and averaging theory collapses some 300 differential equations that describe this neuromechanical model to 24 one-dimensional oscillators that track motoneuron phases. The reduced model accurately captures the dynamics of unperturbed gaits and the effects of impulsive perturbations, and phase response and coupling functions provide improved understanding of reflexive feedback mechanisms.

##### Vertex-disjoint paths in tournaments

The question of linking pairs of terminals by disjoint paths is a standard and well-studied question in graph theory. The setup is: given vertices$ s1,\ldots,sk$ and $t1,\ldots,tk$, is there a set of disjoint path $P1,\ldots,Pk$ such that $Pi$ is a path from $si$ to $ti$? This question makes sense in both directed and undirected graphs, and the paths may be required to be edge- or vertex-disjoint. For undirected graphs, a polynomial-time algorithm for solving both the edge-disjoint and the vertex-disjoint version of the problem (where the number k of terminals is fixed) was first found by Robertson and Seymour, and is a part of their well-known Graph Minors project. For directed graphs, both problems are NP-complete, even when $k=2$ (by a result of Fortune, Hopcroft and Wyllie).

##### Extrapolation Models

We discuss the role of linear models for two extrapolation problems. The rst is the ex-trapolation to the limit of innite series, i.e. convergence acceleration. The second is an extension problem: Given function values on a domain $D_0$, possibly with noise, we would like to extend the function to a larger domain $D, D_0 \subset D$. In addition to smoothness at the boundary of $D_0$, the extension on $D \ D_0$ should also resemble behavioral trends of the function on $D_0$, such as growth and decay or even oscillations. In both problems we discuss the univariate and the bivariate cases, and emphasize the role of linear models with varying coefficients.

##### Information Aggregation in Complex Networks

Over the past few years there has been a rapidly growing interest in analysis, design and optimization of various types of collective behaviors in networked dynamic systems. Collective phenomena (such as flocking, schooling, rendezvous, synchronization, and formation flight) have been studied in a diverse set of disciplines, ranging from computer graphics and statistical physics to distributed computation, and from robotics and control theory to ecology, evolutionary biology, social sciences and economics. A common underlying goal in such studies is to understand the emergence of some global phenomena from local rules and interactions.

##### Feedback, Lineages and Cancer

A multispecies continuum model is developed to simulate the dynamics of cell lineages in solid tumors. The model accounts for spatiotemporally varying cell proliferation and death mediated by the heterogeneous distribution of oxygen and soluble chemical factors. Together, these regulate the rates of self-renewal and differentiation of the different cells within the lineages. As demonstrated in the talk, the feedback processes are found to play a critical role in tumor progression and the development of morphological instability.

##### Novel Phenomena and Models of Active Fluids

Fluids with suspended microstructure - complex fluids - are common actors in micro- and biofluidics applications and can have fascinating dynamical behaviors. A new area of complex fluid dynamics concerns "active fluids" which are internally driven by having dynamic microstructure such as swimming bacteria. Such motile suspensions are important to biology, and are candidate systems for tasks such as microfluidic mixing and pumping. To understand these systems, we have developed both first-principles particle and continuum kinetic models for studying the collective dynamics of hydrodynamically interacting microswimmers. The kinetic model couples together the dynamics of a Stokesian fluid with that of an evolving "active" stress field.

##### A New Formalism for Electromagnetic Scattering in Complex Geometry

We will describe some recent, elementary results in the theory of electromagnetic scattering in R3. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials, due to Lorenz, Debye and Mie, valid only in the exterior (or interior) of a sphere. In extending the Lorenz-Debye-Mie approach to arbitrary geometry, we have encountered some new mathematical questions involving differential geometry, partial differential equations and numerical analysis. This is joint work with Charlie Epstein.

##### Wavelet Frames and Applications

This talk focuses on the tight wavelet frames derived from multiresolution analysis and their applications in imaging sciences. One of the major driven forces in the area of applied and computational harmonic analysis over the last two decades is to develop and understand redundant systems that have sparse approximations of different classes of functions. Such redundant systems include wavelet frame, ridgelet, curvelet, shearlet and so on. In this talk, we will first give a brief survey on the development of the unitary extension principle and its generalizations. The unitary extension principle and its extensions give systematical constructions of wavelet frames from multiresolution analysis that can be used in various problems in imaging science. Then we will focus on applications of wavelet frames.

##### Diffusions Interacting Through Their Ranks, and the Stability of Large Equity Markets

We introduce and study ergodic multidimensional diusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.

##### Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles

I reformulate the covering and quantizer problems, well-known problems in discrete geometry, as the determination of the ground states of interacting particles in d-dimensional Euclidean space that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the "void" nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplifies the deep interplay between geometry and physics, allow one now to employ optimization techniques to analyze and solve these energy minimization problems.

##### High-dimensional reservoir neural dynamics: rules and rewards

Neural activity recorded in behaving animals is highly variable and heterogeneous, which is especially true for neurons in the prefrontal cortex (PFC), the so called 'CEO of the brain' of central importance to many cognitive functions. In this talk, I will present a reservoir-type model of randomly connected neurons to account for the diversity of neural signals in the prefrontal cortex. Specifically, I will show that such a network gives rise to mixed-selectivity of neurons that can encode task rules underlying flexible behaviors, and a broad range of time constants for short-term memory.

##### A few computational and applied math problems in density functional theory related calculations

Density functional theory (DFT) has become the most widely used quantum mechanical method in material science simulations. Due to the change of computer architectures, and the corresponding change in the scope of problems amenable by the DFT method, the algorithms used in the DFT calculations are also changing. In this talk, I will discuss a few commonly used ground state based algorithms for large scale DFT calculations. I will also discuss what it will take to speed up the DFT molecular dynamics simulations by a thousand times, and the self-consistent problems often encountered in large system simulations. Some of our recent effort in implementing our planewave DFT code in GPU will also be discussed.

##### Generalized Markov models in population genetics

Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical works rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next. We have introduced a broad class of Markov models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We have characterized the behavior of standard population-genetic quantities across this family of generalized models.

##### Brother, can you spare a compacton?

Unlike certain personal or national tragedies which may extend indefinitely, patterns observed in nature are of finite extent. Yet, as a rule, the solitary patterns predicted by almost all existing mathematical models extend indefinitely with their tails being a by product of their analytical nature. Rather then viewing such tails as a manifestation of the inherent limitation of math to model physics in detail, we adopt the opposite view: the persistence of tails in a large variety of solitary patterns points to a missing mechanism capable to constrain the pattern. Clearly, to induce a compact pattern one has to escape the curse of analyticity. Differently stated, one has to supplement the existing models with a mechanism(s) which may beget a local singularity.

##### Stirring Tails of Evolution

One of the most fundamental issues in biology is the nature of evolutionary transitions from single cell organisms to multicellular ones. Not surprisingly for microscopic life in a fluid environment, many of the processes involved are related to transport and locomotion, for efficient exchange of chemical species with the environment is one of the most basic features of life. This is particularly so in the case of flagellated eukaryotes such as green algae, whose members serve as model organisms for the study of transitions to multicellularity. In this talk I will focus on recent experimental and theoretical studies of the stochastic nonlinear dynamics of these flagella, whose coordinated beating leads to graceful locomotion but also to fluid flows that can out-compete diffusion.

##### From (basic) image denoising to surface evolution

It is relatively easy to make a connection between the implicit time-discrete approaches for the mean curvature flow and the "Rudin-Osher-Fatemi" total variation based approach for image denoising. This connection has interesting consequences, allowing to build explicit solutions for the flow of the total variation or study regularity issues, up to showing the existence of the crystalline curvature flow of convex sets or building up efficient algorithms. The talk will explain the relationship between all these problems. (Joint works with V. Caselles, M. Novaga, J. Darbon, T. Pock, D. Cremers).

##### Likelihood and algebraic maps for stochastic biochemical network models

With the development of new sequencing technologies of modern molecular biology, it is increasingly common to collect time-series data on the abundance of molecular species encoded within the genomes. This presentation shall illustrate how such data may be used to infer the parameters as well as the structure of the biochemical network under mass-action kinetics. Given certain constraints on the geometry of the stoichiometric space, we use algebraic methods as an alternative to conventional hierarchical graphical models, to carry out network structure inference by identifying reaction rate constants which are significantly different from zero.

##### Learning from Labeled and Unlabeled Data: Global vs. Multiple Approaches

In recent years there is increasing interest in learning from both labeled and unlabeled data (a.k.a. semi-supervised learning, or SSL). The key assumption in SSL, under which an abundance of unlabeled data may help, is that there is some relation between the unknown response function to be learned and the marginal density of the predictor variables. In the first part of this talk I'll present a statistical analysis of two popular graph based SSL algorithms: Laplacian regularization method and Laplacian eigenmaps. In the second part I'll present a novel multiscale approach for SSL as well as supporting theory. Some intimate connections to harmonic analysis on abstract data sets will be discussed. Joint work with Nati Srebro (TTI), Xueyuan Zhou (Chicago), Matan Gavish (WIS/Stanford) and Ronald Coifman (Yale).