## Seminars and Colloquia Schedule

### 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

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.

### 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

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..

### 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.