Seminars and Colloquia by Series

Thursday, October 11, 2018 - 13:30 , Location: Skiles 006 , Trevor Gunn , Georgia Tech , Organizer: Trevor Gunn
I will discuss some elementary theory of symmetric functions and give a brief introduction to representation theory with a focus on the symmetric groups. This talk relates to the discussion of Schubert calculus in the intersection theory reading course but can be understood independent of attending the reading course.
Thursday, October 11, 2018 - 12:00 , Location: Skiles 005 , Greg Blekherman , Georgia Tech , Organizer: Xingxing Yu
 A sum of squares of real numbers is always nonnegative. This elementary observation is quite powerful, and can be used to prove graph density inequalities in extremal combinatorics, which address so-called Turan problems. This is the essence of semidefinite method of Lov\'{a}sz and   Szegedy, and also Cauchy-Schwartz calculus of Razborov. Here multiplication and addition take place in the gluing algebra of partially  labelled graphs. This method has been successfully used on many occasions and has also been extensively studied theoretically. There are two  competing viewpoints on the power of the sums of squares method. Netzer and Thom refined a Positivstellensatz of Lovasz and Szegedy by  showing that if f> 0 is a valid graph density inequality, then for any a>0 the inequality f+a > 0 can be proved via sums of squares. On the  other hand,  Hatami and Norine showed that testing whether a graph density inequality f > 0 is valid is an undecidable problem, and also provided explicit but  complicated examples of inequalities that cannot be proved using sums of squares. I will introduce the sums of squares method, do several  examples of sums of squares proofs, and then present simple explicit inequalities that show strong limitations of the sums of squares method. This  is joint work in progress with Annie Raymond, Mohit Singh and Rekha Thomas. 
Wednesday, October 10, 2018 - 16:00 , Location: Skiles 005 , David Borthwick , Dept. of Math. and Comp. Science, Emory University , Organizer: Michael Loss
 Non-compact hyperbolic surfaces serve as a model case for quantum scattering theory with chaotic classical dynamics.  In this talk I’ll explain how scattering resonances are defined in this context and discuss our current understanding of their distribution.  The primary focus of the talk will be on some recent conjectures inspired by the physics of quantum chaotic systems.  I will introduce these and discuss the numerical evidence as well as recent theoretical progress. 
Wednesday, October 10, 2018 - 16:00 , Location: Skiles 005 , David Borthwick , Dept. of Math. and Comp. Science, Emory University , Organizer: Michael Loss
Wednesday, October 10, 2018 - 14:00 , Location: Skiles 006 , Sudipta Kolay , Georgia Tech , Organizer: Sudipta Kolay
This talk will be an introduction to the homotopy principle (h-principle). We will discuss several examples. No prior knowledge about h-principle will be assumed.
Wednesday, October 10, 2018 - 13:55 , Location: Skiles 005 , Lenka Slavikova , University of Missouri , , Organizer: Michael Lacey
In this talk I will discuss the Mikhlin-H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. I will show that this theorem does not hold in the limiting case $|1/p - 1/2|=s/n$. I will also present a sharp variant of this theorem involving a space of Lorentz-Sobolev type. Some of the results presented in this talk were obtained in collaboration with Loukas Grafakos.
Wednesday, October 10, 2018 - 12:55 , Location: Skiles 006 , Josiah Park , Georgia institute of Technology , , Organizer: Galyna Livshyts
It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.
Wednesday, October 10, 2018 - 12:20 , Location: Skiles 005 , Guillermo Goldsztein , Georgia Tech , Organizer: Trevor Gunn
In 1665, Huygens discovered that, when two pendulum clocks hanged from a same wooden beam supported by two chairs, they synchronize in anti-phase mode. Metronomes provides a second example of oscillators that synchronize. As it can be seen in many YouTube videos, metronomes synchronize in-phase when oscillating on top of the same movable surface. In this talk, we will review these phenomena, introduce a mathematical model, and analyze the the different physical effects. We show that, in a certain parameter regime, the increase of the amplitude of the oscillations leads to a bifurcation from the anti-phase synchronization being stable to the in-phase synchronization being stable. This may explain the experimental observations.
Monday, October 8, 2018 - 14:00 , Location: Skile 006 , None , None , Organizer: John Etnyre
Monday, October 8, 2018 - 14:00 , Location: Skiles 006 , Harry Richman , Univ. of Michigan , Organizer: Matt Baker
The set of (higher) Weierstrass points on a curve of genus g > 1 is an analogue of the set of N-torsion points on an elliptic curve. As N grows, the torsion points "distribute evenly" over a complex elliptic curve. This makes it natural to ask how Weierstrass points distribute, as the degree of the corresponding divisor grows. We will explore how Weierstrass points behave on tropical curves (i.e. finite metric graphs), and explain how their distribution can be described in terms of electrical networks. Knowledge of tropical curves will not be assumed, but knowledge of how to compute resistances (e.g. in series and parallel) will be useful.