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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.

Series: Graph Theory Seminar

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.

Series: Math Physics Seminar

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.

Series: Math Physics Seminar

TBA

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.

Series: Analysis Seminar

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.

Series: High Dimensional Seminar

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.

Series: Research Horizons Seminar

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.

Series: Geometry Topology Seminar

Series: Algebra Seminar

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.