Cornell University

Can one recover a matrix from only matrix-vector products? If so, how many are needed? We will consider the matrix recovery problem for the class of hierarchical rank-structured matrices. This problem arises in scientific machine learning, where one wishes to recover the solution operator of a PDE from only input-output pairs of forcing terms and solutions. Peeling algorithms are the canonical method for recovering a hierarchical matrix from matrix-vector products, however their recursive nature poses a potential stability issue which may deteriorate the overall quality of the approximation. Our work resolves the open question of the stability of peeling. We introduce a robust version of peeling and prove that it achieves low error with respect to the best possible hierarchical approximation to any matrix, allowing us to analyze the performance of the algorithm on general matrices, as opposed to exactly hierarchical ones. This analysis relies on theory for low-rank approximation, as well as the surprising result that the Generalized Nystrom method is more accurate than the randomized SVD algorithm in this setting. 

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

Data scientists are often faced with the challenge of understanding a high dimensional data set organized as a table. These tables may have columns of different (sometimes, non-numeric) types, and often have many missing entries. In this talk, we discuss how to use low rank models to analyze these big messy data sets. Low rank models perform well --- indeed, suspiciously well — across a wide range of data science applications, including applications in social science, medicine, and machine learning. In this talk, we introduce the mathematics of low rank models, demonstrate a few surprising applications of low rank models in data science, and present a simple mathematical explanation for their effectiveness.

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To fly is not to fall. How does an insect fly, why does it fly so well, and how can we infer its ‘thoughts’ from its flight dynamics?  We have been seeking  mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, a  theoretical framework for computing flight leads to new interpretations and predictions of the functions of an insect’s internal machinery that orchestrate its flight. The talk will discuss recent computational and experimental studies of the balancing act of dragonflies and fruit flies:  how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case,  the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.

We study a model for lipid-bilayer membrane vesicles exhibiting phase separation, incorporating a phase field together with membrane fluidity and bending elasticity. We prove the existence of a plethora of equilibria in the large, corresponding to symmetry-breaking solutions of the Euler-Lagrange equations. We also numerically compute a special class of such solutions, namely those possessing icosahedral symmetry. We overcome several difficulties along the way. Due to inherent surface fluidity combined with finite curvature elasticity, neither the Eulerian (spatial) nor the Lagrangian (material) description of the model lends itself well to analysis. This is resolved via a singularity-free radial-map description, which effectively eliminates the grossly under-determined mid-plane deformation. We then use well known group-theoretic selection techniques combined with global bifurcation methods to obtain our results.

In 1890, G. H. Bryan demonstrated that when a ringing wine glass rotates, the shape of the vibration pattern precesses, and this effect is the basis for a family of high-precision gyroscopes. Mathematically, the precession can be described in terms of a symmetry-breaking perturbation due to gyroscopic effects of a geometrically degenerate pair of vibration modes.  Unfortunately, current attempts to miniaturize these gyroscope designs are subject to fabrication imperfections that also break the device symmetry. In this talk, we describe how these devices work and our approach to accurate and efficient simulations of both ideal device designs and designs subject to fabrication imperfections.

For a finitely presented group, the Word Problem asks for an algorithm

which declares whether or not words on the generators represent the

identity. The Dehn function is the time-complexity of a direct attack

on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast

growing (Ackermannian) Dehn functions. I will explain why,

nevertheless, there are efficient (polynomial time) solutions to the

Word Problems of these groups. The main innovation is a means of

computing efficiently with compressed forms of enormous integers.

This is joint work with Will Dison and Eduard Einstein.

 Consider a fully-connected social network of people, companies,
or countries, modeled as an undirected complete graph with real numbers on
its edges. Positive edges link friends; negative edges link enemies.
I'll discuss two simple models of how the edge weights of such networks
might evolve over time, as they seek a balanced state in which "the enemy of
my enemy is my friend." The mathematical techniques involve elementary
ideas from linear algebra, random graphs, statistical physics, and
differential equations. Some motivating examples from international
relations and social psychology will also be discussed. This is joint work
with Seth Marvel, Jon Kleinberg, and Bobby Kleinberg. 

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood examples

of infinite discrete groups. The automorphism groups of these groups, in

contrast, are some of the most complex and intriguing groups in all of

mathematics. I will give some general comments about geometric group

theory and then describe the basic geometric object, called Outer space,

associated to automorphism groups of free groups.

This Colloquium talk is the first of a series of three lectures given by

Professor Vogtmann, who is the European Mathematical Society Lecturer. In

this series of three lectures, she will discuss groups of automorphisms

of free groups, while drawing analogies with the general linear group over

the integers and surface mapping class groups. She will explain modern

techniques for studying automorphism groups of free groups, which include

a mixture of topological, algebraic and geometric methods.

The convergence to stationarity of many finite ergodic Markov

chains presents a sharp cut-off: there is a time T such that before

time T the chain is far from its equilibrium and, after time T,

equilibrium is essentially reached. We will discuss precise

definitions of the cut-off phenomenon, examples, and some partial

results and conjectures.