Seminars and Colloquia by Series

Vortex evolution and stability of fish swimming

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 28, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fangxu Jing – USC Mechanical Engineering
Vortex dynamics and solid-fluid interaction are two of the most important and most studied topics in fluid dynamics for their relevance to a wide range of applications from geophysical flows to locomotion in moving fluids. In this talk, we investigate two problems in these two areas: Part I studies the viscous evolution of point vortex equilibria; Part II studies the effects of body elasticity on the passive stability of submerged bodies.In Part I, we describe the viscous evolution of point vortex configurations that, in the absence of viscosity, are in a state of fixed or relative equilibrium. In particular, we examine four cases, three of them correspond to relative equilibria in the inviscid point vortex model and one corresponds to a fixed equilibrium. Our goal is to elucidate some of the main transient dynamical features of the flow. Using a multi-Gaussian ``core growing" type of model, we show that all four configurations immediately begin to rotate unsteadily, while the shapes of vortex configurations remain unchanged. We then examine in detail the qualitative and quantitative evolution of the structures as they evolve, and for each case show the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. Comparisons between the cases help to reveal different features of the viscous evolution for short and intermediate time ! scales of vortex structures. The dynamical evolution of passive particles in the viscously evolving flow associated with the initial fixed equilibrium is shown and interpreted in relation to the evolving streamline patterns. In Part II, we examine the effects of body geometry and elasticity on the passive stability of motion in a perfect fluid. Our main motivation is to understand the role of body elasticity on the stability of fish swimming. The fish is modeled as an articulated body made of multiple links (assumed to be identical ellipses in 2D or identical ellipsoids in 3D) interconnected by hinge joints. It can undergo shape changes by varying the relative angles between the links. Body elasticity is accounted for via the torsional springs at the joints. The unsteadiness of the flow is modeled using the added mass effect. Equations of motion for the body-fluid system are derived using Newtonian and Lagrangian approaches for both hydrodynamically decoupled and coupled models in 2D and 3D. We specifically examine the stability associated with a relative equilibrium of the equations, traditionally referred to as the ``coast motion" (proved to be unstable for a rigid elongated body model), and f! ound that body elasticity does stabilize the system. Stable regions are identified based on linear stability analysis in the parameter space spanned by aspect ratio (body geometry) and spring constants (muscle stiffness), and the findings based on the linear analysis are verified by direct numerical simulations of the nonlinear system.

Longest Cycles in Graphs with Given Independence Number and Connectivity.

Series
Combinatorics Seminar
Time
Friday, February 25, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hehui Wu – University of Illinois at Urbana-Champaign
The Chv\'atal--Erd\H{o}s Theorem states that every graph whose connectivityis at least its independence number has a spanning cycle. In 1976, Fouquet andJolivet conjectured an extension: If $G$ is an $n$-vertex $k$-connectedgraph with independence number $a$, and $a \ge k$, then $G$ has a cycle of lengthat least $\frac{k(n+a-k)}{a}$. We prove this conjecture. This is joint work with Suil O and Douglas B. West.

Tangent lines and torsion of closed space curves

Series
Geometry Topology Working Seminar
Time
Friday, February 25, 2011 - 14:00 for 2 hours
Location
Skiles 269
Speaker
Mohammad Ghomi – Ga Tech
Torsion of a curve in Euclidean 3-space is a quantity which together with the curvature completely determines the curve up to a rigid motion. In this talk we use the curve shortening flow to show that the number of zero torsion points (or vertices) v a closed space curve c and the number p of the pair of parallel tangent lines of c satisfy the following sharp inequality: v + 2p > 5.

Generating Torelli groups

Series
Geometry Topology Working Seminar
Time
Friday, February 25, 2011 - 14:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 269
Speaker
Dan Margalit – GaTech
I'll present a new, simple proof that the Torelli group is generated by (infinitely many) bounding pair maps. At the end, I'll explain an application of this approach to the hyperelliptic Torelli group. The key is to take advantage of the "complex of minimizing cycles."

Four Seemingly Unrelated Problems

Series
Algebra Seminar
Time
Friday, February 25, 2011 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Filaseta – University of South Carolina –
We begin this talk by discussing four different problems that arenumber theoretic or combinatorial in nature. Two of these problems remainopen and the other two have known solutions. We then explain how these seeminglyunrelated problems are connected to each other. To disclose a little more information,one of the problems with a known solution is the following: Is it possible to find anirrational number $q$ such that the infinite geometric sequence $1, q, q^{2}, \dots$has 4 terms in arithmetic progression?

Online Matching and the Adwords Market

Series
ACO Seminar
Time
Thursday, February 24, 2011 - 16:30 for 1 hour (actually 50 minutes)
Location
KACB 1116B
Speaker
Aranyak Mehta – Google Research
The spectacular success of search and display advertising -- to businesses and search engine companies -- and its huge growth potential has attracted the attention of researchers from many aspects of computer science. Since a core problem in this area is that of effective ad allocation, an inherently algorithmic and game-theoretic question, numerous theoreticians have worked in this area in recent years. Ad allocation involves matching ad slots to advertisers, under demand and supply constraints. In short, the better the matching, the more efficient the market. Interestingly, the seminal work on online matching, by Karp, Vazirani and Vazirani, was done over two decades ago, well before the advent of the Internet economy! In this talk, I will give an overview of several key algorithmic papers in this area, starting with its purely academic beginnings, to papers that actually address the Adwords problem. The elegant -- and deep -- theory behind these algorithms involves new combinatorial, probabilistic and linear programming techniques. Besides the classic KVV paper (STOC 1990), this talk will refer to several papers with my co-authors: Mehta, Saberi, Vazirani, Vazirani (FOCS 05, J. ACM 07), Goel, Mehta (SODA 08), Feldman, Mehta, Mirrokni, Muthukrishnan (FOCS 09), Aggarwal, Goel, Karande, Mehta (SODA 10), Karande, Mehta, Tripathi (STOC 11).

The Convex Geometry of Inverse Problems

Series
Stochastics Seminar
Time
Thursday, February 24, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skyles 005
Speaker
Ben Recht – Computer Sciences Department, University of Wisconsin
Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. The resulting inverse problems are often ill-posed because there are fewer measurements available than the ambient dimension of the model to be estimated. In practice, however, many interesting signals or models contain few degrees of freedom relative to their ambient dimension: a small number of genes may constitute the signature of a disease, very few parameters may specify the correlation structure of a time series, or a sparse collection of geometric constraints may determine a molecular configuration. Discovering, leveraging, or recognizing such low-dimensional structure plays an important role in making inverse problems well-posed. In this talk, I will propose a unified approach to transform notions of simplicity and latent low-dimensionality into convex penalty functions. This approach builds on the success of generalizing compressed sensing to matrix completion, and greatly extends the catalog of objects and structures that can be recovered from partial information. I will focus on a suite of data analysis algorithms designed to decompose general signals into sums of atoms from a simple---but not necessarily discrete---set. These algorithms are derived in a convex optimization framework that encompasses previous methods based on l1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices. I will provide sharp estimates of the number of generic measurements required for exact and robust recovery of a variety of structured models. These estimates are based on computing certain Gaussian statistics related to the latent model geometry. I will detail several example applications and describe how to scale the corresponding inference algorithms to very large data sets. (Joint work with Venkat Chandrasekaran, Pablo Parrilo, and Alan Willsky)

Where to place a hole to achieve fastest escape (What are the best sink and source in a network)

Series
School of Mathematics Colloquium
Time
Thursday, February 24, 2011 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Leonid Bunimovich – Georgia Institute of Technology
Consider any dynamical system with the phase space (set of all states) M. One gets an open dynamical system if M has a subset H (hole) such that any orbit escapes ("disappears") after hitting H. The question in the title naturally appears in dealing with some experiments in physics, in some problems in bioinformatics, in coding theory, etc. However this question was essentially ignored in the dynamical systems theory. It occurred that it has a simple and counter intuitive answer. It also brings about a new characterization of periodic orbits in chaotic dynamical systems. Besides, a duality with Dynamical Networks allows to introduce dynamical characterization of the nodes (or edges) of Networks, which complements such static characterizations as centrality, betweenness, etc. Surprisingly this approach allows to obtain new results about such classical objects as Markov chains and introduce a hierarchy in the set of their states in regard of their ability to absorb or transmit an "information". Most of the results come from a finding that one can make finite (rather than traditional large) time predictions on behavior of dynamical systems even if they do not contain any small parameter. It looks plausible that a variety of problems in dynamical systems, probability, coding, imaging ... could be attacked now. No preliminary knowledge is required. The talk will be accessible to students.

Convergence to equilibrium for a thin-film equation

Series
Math Physics Seminar
Time
Wednesday, February 23, 2011 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Almut Burchard – Department of Mathematics, University of Toronto
I will describe recent work with Marina Chugunovaand Ben Stephens on the evolution of a thin-filmequation that models a "coating flow" on a horizontalcylinder. Formally, the equation defines a gradientflow with respect to an energy that controls theH^1-norm.We show that for each given mass there exists aunique steady state, given by a droplet hanging from thebottom of the cylinder that meets the dry region withzero contact angle. The droplet minimizes the energy andattracts all strong solutions that satisfy certain energyand entropy inequalities. (Such solutions exist for arbitraryinitial values of finite energy and entropy, but it is notknown if they are unique.) The distance of any solutionfrom the steady state decays no faster than a power law.

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