Title: Stability of Collision-Based Periodic n-Body Problems
Abstract: The Newtonian n-body problem describes the motion of n point masses interacting with each other with gravitational force. The physical laws governing this motion were codified by Newton in his Principia in 1686. Under Newton’s laws, two colliding bodies’ velocities increase without bound as they approach collision. Levi-Civita demonstrated that through a suitable change of variables, certain collisions could be regularized, allowing the orbit to be continued past the collision. Variations on Levi-Civita’s work continue to be used in the study of collision-based orbits today. In this talk, I will present a few periodic orbits featuring collisions, describe the methods used to study their stability, and present known results about a few orbits.
Date: Thursday, February 22, 2018
Time: 4:00 PM
Room: 135 TMCB
Title: Special Values of L-functions and Fitting Ideals
Abstract: The Main Conjecture in Iwasawa theory provides a deep connection between a certain p-adic zeta function and arithmetic data associated to class groups in p-power cyclotomic extensions. Proved by Mazur and Wiles, later generalized and proved by Wiles for totally real fields, the Main Conjecture can be interpreted in terms of a Fitting ideal. I will give an overview of the importance of higher Fitting ideals in the structure theory of modules, and provide a conjecture giving a relationship between higher Fitting ideals of Iwasawa modules and special values of L-functions.
Date: Thursday, February 15, 2018
Time: 4:00 PM
Room: 135 TMCB
Title: A Rank Rigidity Result for Certain Nonpositively Curved Spaces via Spherical Geometry
Abstract: To understand the geometry of nonpositively curved (NPC) spaces, it is natural to classify the various types of spaces that can occur. The Rank Rigidity Theorem for closed NPC manifolds separates the class of closed NPC manifolds into three very distinct types, and proves that nothing else can exist.
A version of Rank Rigidity has been conjectured for more general NPC spaces (CAT(0) spaces). In this talk, we discuss some progress toward the general conjecture, by reducing the problem to looking at patterns on spheres in the boundary at infinity. In particular, we can prove the conjecture for certain NPC spaces with one-dimensional boundary. In contrast to previous rank rigidity results, we do not assume any additional structure (such as a polyhedral or manifold structure) on the space.
Date: Tuesday, February 13
Time: 4:00 PM
Room: 135 TMCB
Speaker: Tuan Pham
Title: Minimal blowup data for potential Navier-Stokes singularities in the half space
Abstract: It is known that the mild solutions to the incompressible 3D Navier-Stokes Equations (NSE), either in the whole space or a smooth domain with nonslip boundary condition, exist locally in time. Provided that a blowup solution exists, then it has been shown that a so-called minimal blowup data for NSE in the whole space exists. In this talk, I will explain how the presence of the boundary might influence the existence of minimal blowup data. This is joint work with Vladimir Sverak. Our results are motivated by a theorem of Gregory Seregin on boundary regularity.
Date: Thursday, February 8, 2018
Time: 11:00 AM (NOTE SPECIAL TIME)
Room: 135 TMCB
Title: Discovering dual groups
Abstract: Compact quantum groups are a well-known way of generalizing the concept of a group in the framework of non-commutative mathematics. Less known are dual groups. Introduced by Voiculescu in the 80’s and also known as H-algebras, they are very similar to compact quantum groups but the computliplication has values this time in a free product instead of a tensor product. We will introduce the notion of dual group, compare it to compact quantum groups and give some new results that allow us to understand them
Thursday, February 2nd
4pm 135 TMCB
Title: Tales of Our Forefathers
Abstract: This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I’ll convince you they were also human beings and that, as the Chinese say, “May you live in interesting times” really is a curse.
Time: Nov 14th 4:00-5:00PM 3106 JKB
Title: Optimal Mass Transport and Curvature Bounds
Abstract: Have you ever been in a hurry at the grocery store? Have you ever wondered if you are collecting the items on your list in the most efficient way possible? This is a discrete case of the fundamental problem of optimal mass transport theory. As a consequence of its many and varied applications in both pure and applied mathematics, the problem of optimal mass transport theory has received considerable attention during the past few years. (Indeed, Cédric Villani, one of the prominent researchers in this area, was awarded the Fields Medal in 2010.) On the pure side, the theory of optimal mass transport has found important connections to nonlinear partial differential equations (such as the critically-important Monge-Ampère equations that appear within its foundations), dynamical systems, inequalities (including isoperimetric inequalities), and geometry. On the applied side, partial answers to the transport question have had significant implications in such fields as economics, statistics, computer vision, physics, fluid dynamics, engineering, biology, and even meteorology.
In this talk, we will introduce the basic notions of optimal mass transport theory as well as discuss some interactions between optimal mass transport theory and the geometry of the underlying (and other related) spaces. More particularly, we will present some results concerning the inherited geometric properties of the 2-Wasserstein metric space of probability measures (over its parent space) and state a special case of the so-called curvature-dimension condition arising in the study of optimal transport over Riemannian manifolds. We will also explore the possibility of extending L. Caffarelli’s celebrated contraction theorem to the setting of the standard sphere as well as examine a curvature bound obtained very recently by A. Kolesnikov. We will end with a generalization of a part of E. Calabi’s work concerning Hessian metrics.
The majority of this presentation should be accessible to all graduate students, especially those with coursework in differential geometry. All are welcome to attend.
Title: The Three M’s of Scientific Computing: A High-Performance Solution to the FCH Equation
Abstract: Modern computing architecture is advancing our ability to approximate solutions to partial differential equations. I will present an example of how a graphics processing unit (GPU) can make a large problem tractable. The Functionalized Cahn-Hilliard (FCH) equation in three dimensions,
ut = Δ(ε2Δ – W″(u) + ε2η)(ε2 Δu – W′(u)),
describes pore network formation in a functionalized polymer/solvent system like those used in hydrogen fuel cells. The physical process is defined by multiple time-scales: short time phase separation, long time network growth, and slow evolution to steady state. This requires very long, accurate simulations to correctly describe the physics. I will present a fast, numerically stable, and time-accurate method for solving the FCH equation. Experimentation with several different methods for computing time evolution led to a Fourier spectral method in space with an exponential time integrator giving the desired qualities. Numerical results on a GPU show a 10x speedup over a fully-parallelized eight-core CPU.
Title: Dynamic Stability of Time-Delayed Networks
Abstract: Networks, including computer networks, social networks, and biological networks, have within the past few years attracted an enormous amount of interest. Due to both their physical size and finite processing speeds the dynamics of these networks are inherently time-delayed. In this talk we consider how the global stability of a dynamical network depends on its time-delays. We find that a network’s stability is unaffected by the introduction or removal of undistributed delays and that such delays can be characterized in terms of the network’s graph structure. By extending this technique we also describe how to remove the implicit delays in a network (or general dynamical system) to gain improved estimates of the network’s (system’s) stability. This approach of “restricting” a network is illustrated by applications to certain classes of Cohen-Grossberg neural networks.
Title: q-series and quantum modular forms
Abstract: While the theory of mock modular forms has seen great advances in the last decade, questions remain. We revisit Ramanujan’s last letter to Hardy, and prove one of his remaining conjectures as a special case of a more general result. Surprisingly, Dyson’s combinatorial rank function, the Andrews-Garvan crank function, mock theta functions, and quantum modular forms, all play key roles. Along these lines, we also show that the Rogers-Fine false theta functions, functions that have not been well understood within the theory of modular forms, specialize to quantum modular forms. This is joint work with K. Ono (Emory U.) and R.C. Rhoades (Stanford U.).