Seminars and Colloquia by Series

Long-time dynamics for the generalized Korteweg-de Vries and Benjamin-Ono equations

Series
CDSNS Colloquium
Time
Wednesday, May 27, 2020 - 09:00 for 1.5 hours (actually 80 minutes)
Location
Bluejeans: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Benoît GrébertUniversité de Nantes

We provide an accurate description of the long time dynamics for generalized Korteweg-de Vries  and Benjamin-Ono equations on the circle without external parameters and for almost any (in probability and in density) small initial datum. To obtain that result we construct for these two classes of equations and under a very weak hypothesis of non degeneracy of the nonlinearity, rational normal forms on open sets surrounding the origin in high Sobolev regularity. With this new tool we can make precise the long time dynamics of the respective flows. In particular we prove a long-time stability result in Sobolev norm: given a large constant M and a sufficiently small parameter ε, for generic initial datum u(0) of size ε, we control the Sobolev norm of the solution u(t) for time of order ε^{−M}. 

Riemann's non-differentiable function is intermittent

Series
CDSNS Colloquium
Time
Wednesday, May 20, 2020 - 12:00 for 1 hour (actually 50 minutes)
Location
https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Victor da RochaGeorgia Tech

Please Note: (UPDATED Monday 5-18) Note the nonstandard start time of 12PM.

Riemann's non-differentiable function, although introduced as a pathological example in analysis, makes an appearance in a certain limiting regime of the theory of binormal flow for vortex lines. From this physical point of view, it also bears some qualitative similarities to turbulent fluid velocity fields in the infinite Reynolds number limit. In this talk, we'll see how this function arises in the study of the vortex filaments, and how we can adapt the notion of intermittency from the study of turbulent flows to this setting. Then, we'll study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways. One in the physical space, and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0, which highlights the (weak) intermittent nature of Riemann's function.

This is a joint work with Alexandre Boritchev (Université de Lyon) and Daniel Eceizabarrena (BCAM, Bilbao).
 

Interaction energies, lattices, and designs

Series
Dissertation Defense
Time
Wednesday, May 13, 2020 - 13:30 for 1 hour (actually 50 minutes)
Location
Bluejeans: https://bluejeans.com/9024318866/
Speaker
Josiah ParkGeorgia Tech

This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging electrons to minimize their energy. Restricting our potentials to be polynomials in the squared distance between points, we show in the Chapter 1 that there exist discrete minimal energy distributions. In addition we pose a conjecture on discreteness of minimizers for another class of energies while showing these minimizers must have empty interior.


In Chapter 2, we discover that highly symmetric distributions of points minimize energies over probability measures for potentials which are completely monotonic up to some degree, guided by the work of H. Cohn and A. Kumar. We make conjectures about optima for a class of energies calculated by summing absolute values of inner products raised to a positive power. Through reformulation, these observations give rise to new mixed-volume inequalities and conjectures. Our numerical experiments also lead to discovery of a new highly symmetric complex projective design which we detail the construction for. In this chapter we also provide details on a computer assisted argument which shows optimality of the $600$-cell for such energies (via interval arithmetic).


In Chapter 3 we also investigate energies having minimizers with a small number of distinct inner products. We focus here on discrete energies, confirming that for small $p$ the repeated orthonormal basis minimizes the $\ell_p $-norm of the inner products out of all unit norm configurations. These results have analogs for simplices which we also prove. 

Finally, in Chapter 4 we show that real tight frames that generate lattices must be rational, and that the same holds for other vector systems with structured matrices of outer products. We describe a construction of lattices from distance transitive graphs which gives rise to strongly eutactic lattices. We discuss properties of this construction and also detail potential applications of lattices generated by incoherent systems of vectors.

Random Young Towers

Series
CDSNS Colloquium
Time
Wednesday, May 13, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Bluejeans event: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Yaofeng SuUniversity of Houston and Georgia Tech

Please Note: The attendee link is https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh

I will discuss random Young towers and prove an quenched Almost Sure Invariant Principle for them, which implies many quenched limits theorems, e.g., Central Limit Theorem, Functional Central Limit Theorem etc.. I will apply my result to some random perturbations of some nonuniformly expanding maps such as unimodal maps, Pomeau-Manneville maps etc..

Adaptive Tracking and Parameter Identification

Series
Applied and Computational Mathematics Seminar
Time
Monday, May 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/614972446/
Speaker
Prof. Michael Malisoff Louisiana State University

Please Note: Virtual seminar held on BlueJeans

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

Rayleigh-Taylor instability with heat transfer

Series
Dissertation Defense
Time
Saturday, May 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/603353375/4347?src=calendarLink
Speaker
Qianli HuGeorgia Tech

Please Note: Online at https://bluejeans.com/603353375/4347?src=calendarLink

In this thesis, the Rayleigh-Taylor instability effects in the setting of the Navier-Stokes equations, given some three-dimensional and incompressible fluids, are investigated. The existence and the uniqueness of the temperature variable in the the weak form is established under suitable initial and boundary conditions, and by the contraction mapping principle we investigate further the conditions for the solution to belong to some continuous class; then a positive minimum temperature result can be proved, and with the aid of the RT instability effect in the density and the velocity, the instability for the temperature is established.

A dynamic view on the probabilistic method: random graph processes

Series
School of Mathematics Colloquium
Time
Thursday, May 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://gatech.bluejeans.com/344615810
Speaker
Lutz WarnkeGeorgia Tech

 Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology).
Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way.

The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms.
Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory).

In this talk I shall survey several random graph processes of interest (in the context of the probabilistic method), and give a glimpse of their analysis.
If time permits, we shall also illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.

Every surface is a leaf

Series
Geometry Topology Student Seminar
Time
Wednesday, May 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Justin LanierGeorgia Tech

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

Constructive methods in KAM theory- from numerics to regularity

Series
CDSNS Colloquium
Time
Wednesday, May 6, 2020 - 09:00 for 1 hour (actually 50 minutes)
Location
Attendee link: https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
Speaker
Rafael de la LlaveGeorgia Tech

Please Note: This is the first installment of our CDSNS virtual colloquium, which will be held in a Bluejeans event space on Wednesdays at 9AM (EST).

We will present the "a-posteriori" approach to KAM theory.

We formulate an invariance equation and show that an approximate-enough solution which verifies some non-degeneracy conditions leads to a solution.  Note that this does not have any reference to integrable systems and that the non-degeneracy conditions are not global properties of the system, but only properties of the solution. The "automatic reducibility" allows to take advantage of the geometry to develop very efficient Newton methods and show that they converge.

This leads to very efficient numerical  algorithms (which moreover can be proved to lead to correct solutions), to validate formal expansions. From a more theoretical point of view, it can be applied to other geometric contexts (conformally symplectic, presymplectic) and other geometric objects such as whiskered tori. One can deal well with degenerate systems, singular perturbation theory and obtain simple proofs of monogenicity and Whitney regularity.

This is joint work with many people.

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