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Series: Math Physics Seminar

Consider a relativistic electron interacting with a nucleus of nuclear charge Z and coupled to its self-generated electromagnetic field. The resulting system of equations describing the time evolution of this electron and its corresponding vector potential are known as the Maxwell-Dirac-Coulomb (MDC) equations. We study the time local well-posedness of the MDC equations, and, under reasonable restrictions on the nuclear charge Z, we prove the existence of a unique local in time solution to these equations.

Series: Combinatorics Seminar

The (type A) Hecke algebra H_n(q) is an n!-dimensional q-analog of the symmetric group. A related trace space of certain functions on H_n(q) has dimension equal to the number of integer partitions of n. If we could evaluate all functions belonging to some basis of the trace space on all elements of some basis of H_n(q), then by linearity we could evaluate em all traces on all elements of H_n(q). Unfortunately there is no simple published formula which accomplishes this. We will consider a basis of H_n(q) which is related to structures called wiring diagrams, and a combinatorial rule for evaluating one trace basis on all elements of this wiring diagram basis. This result, the first of its kind, is joint work with Justin Lambright and Ryan Kaliszewski.

Friday, March 16, 2018 - 14:00 ,
Location: Skiles 006 ,
Jen Hom ,
Georgia Tech ,
Organizer: Jennifer Hom

In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module.

Series: ACO Student Seminar

A low-diameter decomposition (LDD) of an undirected graph G is a partitioning of G into components of bounded diameter, such that only a small fraction of original edges are between the components. This decomposition has played instrumental role
in the design of low-stretch spanning tree, spanners, distributed algorithms etc.
A natural question is whether such a decomposition can be efficiently maintained/updated as G undergoes insertions/deletions of edges. We make the first step towards answering this question by designing a fully-dynamic graph algorithm that maintains an
LDD in sub-linear update time.
It is known that any undirected graph G admits a spanning tree T with nearly logarithmic average stretch, which can be computed in nearly linear-time. This tree decomposition underlies many recent progress in static algorithms for combinatorial and scientific
flows. Using our dynamic LDD algorithm, we present the first non-trivial algorithm that dynamically maintains a low-stretch spanning tree in \tilde{O}(t^2) amortized update time, while achieving (t + \sqrt{n^{1+o(1)}/t}) stretch, for every 1 \leq t \leq n.
Joint work with Sebastian Krinninger.

Friday, March 16, 2018 - 10:00 ,
Location: Skiles 006 ,
Kisun Lee ,
Georgia Tech ,
klee669@gatech.edu ,
Organizer: Kisun Lee

Expanding the topic we discussed on last week, we consider the way to certify roots for system of equations with D-finite functions. In order to do this, we will first introduce the notion of D-finite functions, and observe the property of them. We also suggest two different ways to certify this, that is, alpha-theory and the Krawczyk method. We use the concept of majorant series for D-finite functions to apply above two methods for certification. After considering concepts about alpha-theory and the Krawczyk method, we finish the talk with suggesting some open problems about these.

Series: Dissertation Defense

I will discuss two topics in Dynamical Systems. A uniformly hyperbolic dynamical system preserving Borel probability measure μ is called fair dice like or FDL if there exists a finite Markov partition ξ of its phase space M such that for any integers m and j(i), 1 ≤ j(i) ≤ q one has μ ( C(ξ, j(0)) ∩ T^(-1) C(ξ, j(1)) ∩ ... ∩ T^(-m+1)C(ξ, j(m-1)) ) = q^(-m) where q is the number of elements in the partition ξ and C(ξ, j) is element number j of ξ. I discuss several results about such systems concerning finite time prediction regarding the first hitting probabilities of the members of ξ. Then I will discuss a natural modification to all billiard models which is called the Physical Billiard. For some classes of billiard, this modification completely changes their dynamics. I will discuss a particular example derived from the Ehrenfests' Wind-Tree model. The Physical Wind-Tree model displays interesting new dynamical behavior that is at least as rich as some of the most well studied examples that have come before.

Wednesday, March 14, 2018 - 14:00 ,
Location: Skiles 006 ,
Surena Hozoori ,
GaTech ,
Organizer: Anubhav Mukherjee

Assuming
some "compatibility" conditions between a Riemannian metric and a
contact structure on a 3-manifold, it is natural to ask whether
we can use methods in global geometry to get results in contact topology. There is a notion
of compatibility in this context which relates convexity concepts in
those geometries and is well studied concerning geometry questions, but
is not exploited for topological questions. I will talk about "contact
sphere theorem" due to Etnyre-Massot-Komendarczyk,
which might be the most interesting result for contact topologists.

Series: Analysis Seminar

We consider totally irregular measures $\mu$ in $\mathbb{R}^{n+1}$, that is, $$\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n} >0 \;\; \& \;\; \liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}=0$$for $\mu$ almost every $x$. We will show that if $T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu(y)$ is an operator whose kernel $K(\cdot,\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\"older continuous coefficients, then $T_\mu$ is not bounded in $L^2(\mu)$.This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform and is part of the program to clarify the connection between rectifiability of sets/measures on $\mathbb{R}^{n+1}$ and boundedness of singular integrals there. Based on joint work with Mihalis Mourgoglou and Xavier Tolsa.

Series: Research Horizons Seminar

There
is so much that the GT library can do for you, from providing research
materials to assistance with data visualization to patent guidance.
However, rather than trying to guess what you want from us, this year we
asked!
Based on the response to a short ranking survey I sent out last month, this session will cover: 1. How to find grants, fellowships, and travel money with the sponsorship database, Pivot. There are opportunities for postdocs and non US citizens too!2. How to use MathSciNet. We will cover navigating its classification index to actually getting the article
you want. 3. How to find and download articles from our systems, Google Scholar, and from other libraries. And if we have time: 4. How to make a poster and cheaply print it.

Series: Algebra Seminar

The structure of non-archimedean curves X and their tame covers f:Y-->X is well understoodand can be adequately described in terms of a (simultaneous) semistable model. In particular, asindicated by the lifting theorem of Amini-Baker-Brugalle-Rabinoff, it encodes all combinatorialand residual algebra-geometric information about f. My talk will be mainly concerned with the morecomplicated case of wild covers, where new discrete invariants appear, with the different function being the most basic one. I will recall its basic properties following my joint work with Cohen and Trushin,and will then pass to the latest results proved jointly with U. Brezner: the different functioncan be refined to an invariant of a residual type, which is a (sort of) meromorphic differential form on the reduction, so that a lifting theorem in the style of ABBR holds for simplest wild covers.