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Series: Analysis Seminar

t-Haar multipliers are examples of Haar multipliers were the symbol depends both on the frequency variable (dyadic intervals) and on the space variable, akin to pseudo differential operators. They were introduced more than 20 years ago, the corresponding multiplier when $t=1$ appeared first in connection to the resolvent of the dyadic paraproduct, the cases $t=\pm 1/2$ is intimately connected to direct and reverse inequalities for the dyadic square function in $L^2$, the case $t=1/p$ naturally appears in the study of weighted inequalities in $L^p$. Much has happened in the theory of weighted inequalities in the last two decades, highlights are the resolution of the $A_2$ conjecture (now theorem) by Hyt\"onen in 2012 and the resolution of the two weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte Tuero in 2014. Among the competing methods used to prove these results were Bellman functions, corona decompositions, and domination by sparse operators. The later method has gained a lot of traction and is being widely used in contexts beyond what it was originally conceived for in work of Lerner, several of these new applications have originated here at Gatech. In this talk I would like to tell you what I know about t-Haar multipliers (some work goes back to my PhD thesis and joint work with Nets Katz and with my former students Daewon Chung, Jean Moraes, and Oleksandra Beznosova), and what we ought to know in terms of sparse domination.

Series: Research Horizons Seminar

A motivating problem in number theory and algebraic geometry is to find
all integer-valued solutions of a polynomial equation. For example,
Fermat's Last Theorem asks for all integer solutions to x^n + y^n = z^n,
for n >= 3. This kind of problem is easy
to state, but notoriously difficult to solve. I'll explain a p-adic
method for attacking Diophantine equations, namely, p-adic integration
and the Chabauty--Coleman method. Then I'll talk about some recent
joint work on the topic.

Series: Algebra Seminar

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere,
restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If
we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this
way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a
p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to
non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over
a more general ground field.

Series: Geometry Topology Seminar

We discuss the growth of homonoly in finite coverings, and show that the growth of the torsion part of the first homology of finite coverings of 3-manifolds is bounded from above by the hyperbolic volume of the manifold. The proof is based on the theory of L^2 torsion.

Series: AMS Club Seminar

Join us for a discussion of making professional mathematics diagrams and
illustrations with free vector graphics editing software Inkscape.
We'll discuss and tinker with Bezier curves, TexTex, and vectorization
of scanned images.

Series: GT-MAP Seminars

There is an increasing trend for robots to serve as networked mobile sensing platforms that are able to collect data and interact with humans in
various types of environment in unprecedented ways. The need for
undisturbed operation posts higher goals for autonomy. This talk reviews
recent developments in autonomous collective foraging in a complex
environment that explicitly integrates insights from biology with models
and provable strategies from control theory and robotics. The methods
are rigorously developed and tightly integrated with experimental effort with promising results achieved.

Friday, November 10, 2017 - 14:00 ,
Location: Skiles 154 ,
Rafael de la Llave ,
GT Math ,
Organizer: Jiaqi Yang

We consider Hamiltonian systems with normally hyperbolic manifold with a homoclinic connection. The systems are of the form H_0(I, phi, x,y) = h(I) + P(x,y) ,where P is a one dimensional system with a homoclinic intersection. The above Hamiltonian is a standard normal form for near integrable Hamiltonians close to a resonance. We consider perturbations that are time dependent and may be not Hamiltonian. We derive explicit formulas for the first order effects on the stable/unstable manifolds. In particular, we give sufficient conditions for the existence of homoclinic intersections to the normally hyperbolic manifold. Previous treatments in the literature specify the types of the unperturbed orbits considered (periodic or quasiperiodic) and are restricted to periodic or quasi-periodic perturbations. We do not need to distinguish on the perturbed orbits and we allow rather general dependence on the time (periodic, quasiperiodic or random). The effects are expressed by very fast converging improper integrals. This is joint work with M. Gidea. https://arxiv.org/abs/1710.01849

Friday, November 10, 2017 - 13:55 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we continue discussing branched covers of 3-manifolds and prove universal links exist.

Series: Stochastics Seminar

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications.
Joint work with Pascal Maillard.

Series: Combinatorics Seminar