Seminars and Colloquia by Series

Thursday, May 2, 2019 - 11:00 , Location: Skiles 006 , Rohan Ghanta , School of Mathematics , rghanta3@math.gatech.edu , Organizer: Rohan Ghanta

An electron interacting with the vibrational modes of a polar crystal is called a polaron. Polarons are the simplest Quantum Field Theory models, yet their most basic features such as the effective mass, ground-state energy and wave function cannot be evaluated explicitly. And while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, they are built entirely on unjustified, even questionable, Ansätze for the wave function. 

In this talk I shall provide the first explicit description of the ground-state wave function of a polaron in an asymptotic regime: For the Fröhlich polaron localized in a Coulomb potential and exposed to a homogeneous magnetic field of strength $B$ it will be shown that the ground-state electron density in the direction of the magnetic field converges pointwise and in a weak sense as $B\rightarrow\infty$ to the square of a hyperbolic secant function--a sharp contrast to the Gaussian wave functions suggested in the physics literature. 

Series: PDE Seminar
Tuesday, April 30, 2019 - 15:00 , Location: skiles 006 , Chenchen Mou , UCLA , muchenchen@math.ucla.edu , Organizer: Xukai Yan

 In this talk we study master equations arising from mean field game 
problems, under the crucial monotonicity conditions.
Classical solutions of such equations require very strong technical 
conditions. Moreover, unlike the master equations arising from mean 
field control problems, the mean field game master equations are 
non-local and even classical solutions typically do not satisfy the 
comparison principle, so the standard viscosity solution approach seems 
infeasible. We shall propose a notion of weak solution for such 
equations and establish its wellposedness. Our approach relies on a new 
smooth mollifier for functions of measures, which unfortunately does not 
keep the monotonicity property, and the stability result of master 
equations. The talk is based on a joint work with Jianfeng Zhang.

Series: Other Talks
Tuesday, April 30, 2019 - 13:00 , Location: Skiles 005 , Yuchen Roy He , GT Math , Organizer:

Multidimensional data is ubiquitous in the application, e.g., images and videos. I will introduce some of my previous and current works related to this topic.
1) Lattice metric space and its applications. Lattice and superlattice patterns are found in material sciences, nonlinear optics and sampling designs. We propose a lattice metric space based on modular group theory and
metric geometry, which provides a visually consistent measure of dissimilarity among lattice patterns.  We apply this framework to superlattice separation and grain defect detection.
2) We briefly introduce two current projects. First, we propose new algorithms for automatic PDE modeling, which drastically improves the efficiency and the robustness against additive noise. Second, we introduce a new model for surface reconstruction from point cloud data (PCD) and provide an ADMM type fast algorithm.

 

 

 

Tuesday, April 30, 2019 - 13:00 , Location: Skiles 006 , Rodney Anderson , NASA Jet Propulsion Laboratory, California Institute of Technology , Rodney.L.Anderson@jpl.nasa.gov , Organizer: Bhanu Kumar

(Please note the unusual day)

New and proposed missions for approaching moons, and particularly icy moons, increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades.  These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options.  The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas.  The search for approach trajectories within highly nonlinear, chaotic regimes where multi-body effects dominate becomes increasingly complex, especially when landing, orbiting, or flyby scenarios must be considered in the analysis.  In the case of icy moons, approach trajectories must also be tied into the broader tour which includes flybys of other moons. The tour endgame typically includes the last several flybys, or resonances, before the final approach to the moon, and these resonances further constrain the type of approach that may be used.

 

In this seminar, new methods for approaching moons by traversing the chaotic regions near the Lagrange point gateways will be discussed for several examples.  The emphasis will be on landing trajectories approaching Europa including a global analysis of trajectories approaching any point on the surface and analyses for specific landing scenarios across a range of different energies.  The constraints on the approach from the tour within the context of the endgame strategy will be given for a variety of different moons and scenarios.  Specific approaches using quasiperiodic or Lissajous orbits will be shown, and general landing and orbit insertion trajectories will be placed into context relative to the invariant manifolds of unstable periodic and quasiperiodic orbits. These methods will be discussed and applied for the specific example of the Europa Lander mission concept.  The Europa Lander mission concept is particularly challenging in that it requires the redesign of the approach scenario after the spacecraft has launched to accommodate landing at a wide range of potential locations on the surface.  The final location would be selected based on reconnaissance from the Europa Clipper data once Europa Lander is in route.  Taken as a whole, these methods will provide avenues to find both fundamentally new approach pathways and reduce cost to enable new missions.  

 

Sunday, April 28, 2019 - 15:05 , Location: 006 , Liza Rebrova , UCLA , Organizer:

I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.

I am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.

Friday, April 26, 2019 - 12:00 , Location: Skiles 006 , Jaewoo Jung , Georgia Institute of Technology , jjung325@gatech.edu , Organizer: Trevor Gunn

It is known that non-negative homogeneous polynomials(forms) over $\mathbb{R}$ are same as sums of squares if it is bivariate, quadratic forms, or ternary quartic by Hilbert. Once we know a form is a sum of squares, next natural question would be how many forms are needed to represent it as sums of squares. We denote the minimal number of summands in the sums of squares by rank (of the sum of squares). Ranks of some class of forms are known. For example, any bivariate forms (allowing all monomials) can be written as sum of $2$ squares.(i.e. its rank is $2$) and every nonnegative ternary quartic can be written as a sum of $3$ squares.(i.e. its rank is $3$). Our question is that "if we do not allow some monomials in a bivariate form, how its rank will be?". In the talk, we will introduce this problem in algebraic geometry flavor and provide some notions and tools to deal with.

Wednesday, April 24, 2019 - 00:05 , Location: Skiles 006 , Lutz Warnke , Georgia Tech , Organizer:

During the last 30 years there has been much interest in random graph processes, i.e., random graphs which grow by adding edges (or vertices) step-by-step in some random way. Part of the motivation stems from more realistic modeling, since many real world networks such as Facebook evolve over time. Further motivation stems from extremal combinatorics, where these processes lead to some of the best known bounds in Ramsey and Turan Theory (that go beyond textbook applications of the probabilistic method). I will review several random graph processes of interest, and (if time permits) illustrate one of the main proof techniques using a simple toy example.

Series: Other Talks
Tuesday, April 23, 2019 - 15:00 , Location: Skiles 006 , Jaemin Park , Georgia Institute of Technology , jpark776@gatech.edu , Organizer: Jaemin Park

We study whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main results are the following:

(1) On the one hand, we are able to show that for any non-negative smooth stationary vorticity  that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation. 

(2) On the other hand, if we allow vorticity to change sign, then by applying bifurcation arguments to sign-changing radial patches, we are able to show that there exists a compactly-supported, sign-changing smooth stationary vorticity that is non-radial.

We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation, as well as stationary/rotating solutions for the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao. 

Monday, April 22, 2019 - 15:30 , Location: Skiles 006 , Eli Grigsby , Boston College , Organizer: Caitlin Leverson

One can regard a (trained) feedforward neural network as a particular type of function \mathbb{R}^d \rightarrow (0,1), where \mathbb{R}^d is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value N(x) \in (0,1) of the function on a data point x is the probability that the answer to a particular yes/no question is "yes." It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a bounded set. Last year, J. Johnson proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). His argument was novel in that it provided an explicit topological obstruction to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will tell you just enough about neural networks to understand how Johnson's result follows from some very simple ideas in piecewise linear geometry. Time permitting, I will also describe some joint work in progress with K. Lindsey aimed at developing a general theory of how the architecture of a neural network constrains its topological expressiveness.

Monday, April 22, 2019 - 14:00 , Location: Skiles 006 , Adam Levine , Duke University , Organizer: Caitlin Leverson

Given an m-dimensional manifold M that is homotopy equivalent to an n-dimensional manifold N (where n<m), a spine of M is a piecewise-linear embedding of N into M (not necessarily locally flat) realizing the homotopy equivalence. When m-n=2 and m>4, Cappell and Shaneson showed that if M is simply-connected or if m is odd, then it contains a spine. In contrast, I will show that there exist smooth, compact, simply-connected 4-manifolds which are homotopy equivalent to the 2-sphere but do not contain a spine (joint work with Tye Lidman). I will also discuss some related results about PL concordance of knots in homology spheres (joint with Lidman and Jen Hom).

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