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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.

Friday, October 5, 2018 - 15:05 ,
Location: Skiles 156 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk I will present a proof of a generalization of a theorem by
Siegel, about the existence of an analytic conjugation between an
analytic map, $f(z)=\Lambda z +\hat{f}(z)$, and a linear map, $\Lambda
z$, in $\mathbb{C}^n$. This proof illustrates a standar technique used
to deal with small divisors problems. I will be following the work of E.
Zehnder. This is a continuation of last week talk.

Series: Algebra Seminar

In this talk, we introduce rather exotic algebraic structures called
hyperrings and hyperfields. We first review the basic definitions and
examples of hyperrings, and illustrate how hyperfields can
be employed in algebraic geometry to
show that certain topological spaces (underlying topological spaces of
schemes, Berkovich analytification of schemes, and real schemes) are
homeomorphic to sets of rational points of schemes over hyperfields.

Series: ACO Student Seminar

Abstract:
Queueing
systems are studied in various asymptotic regimes because they are hard
to study in general. One
popular regime of study is the heavy-traffic regime, when the system is
loaded very close to its capacity. Heavy-traffic behavior of queueing
systems is traditionally studied using fluid and diffusion limits. In
this talk, I will present a recently developed
method called the 'Drift Method', which is much simpler, and is based on
studying the drift of certain test functions. In addition to exactly
characterizing the heavy-traffic behavior, the drift method can be used
to obtain lower and upper bounds for all loads.
In this talk, I will present the drift method, and its successful
application in the context of data center networks to resolve a
decade-old conjecture. I will also talk about ongoing work and some open
problems.
Bio:
Siva
Theja Maguluri is an Assistant Professor in the School of Industrial
and Systems Engineering at Georgia
Tech. Before that, he was a Research Staff Member in the Mathematical
Sciences Department at IBM T. J. Watson Research Center. He obtained his
Ph.D. from the University of Illinois at Urbana-Champaign in Electrical
and Computer Engineering where he worked on
resource allocation algorithms for cloud computing and wireless
networks. Earlier, he received an MS in ECE and an MS in Applied Math
from UIUC and a B.Tech in Electrical Engineering from IIT Madras. His
research interests include Stochastic Processes, Optimization,
Cloud Computing, Data Centers, Resource Allocation and Scheduling
Algorithms, Networks, and Game Theory. The current talk is based on a
paper that received the best publication in applied probability award,
presented by INFORMS Applied probability society.

Thursday, October 4, 2018 - 13:30 ,
Location: Skiles 006 ,
Daniel Minahan ,
Georgia Tech ,
Organizer: Trevor Gunn

We discuss the construction of spectral sequences and some of their
applications to algebraic geometry, including the classic Leray spectral
sequence. We will draw a lot of diagrams and try to avoid doing
anything involving lots of indices for a portion of the talk.

Series: Graph Theory Working Seminar

Erdős and Nešetřil conjectured in 1985 that every graph with maximum degree Δ can be strong edge colored using at most (5/4)Δ^2 colors. In this talk we discuss recent progress made in the case of Δ=4, and go through the method used to improve the upper bound to 21 colors, one away from the conjectured 20.

Series: Math Physics Seminar

We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis.