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Wednesday, February 7, 2018 - 13:55 ,
Location: Skiles 006 ,
Hyun Ki Min ,
GaTech ,
Organizer: Anubhav Mukherjee

The figure 8 knot is the simplest hyperbolic knot. In the late 1970s, Thurston studied how to construct hyperbolic manifolds from ideal tetrahedra. In this talk, I present the Thurston’s theory and apply this to the figure 8 knot. It turns out that every Dehn surgery on the figure 8 knot results in a hyperbolic manifold except for 10 exceptional surgery coefficients. If time permits, I will also introduce the classification of tight contact structures on these manifolds. This is a joint work with James Conway.

Series: Analysis Seminar

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? I will briefly describe how to establish the existence of such functions via a precompactness argument for maximizing sequences. Then for exponents $q\in(3,\infty)$ sufficiently close to even integers, we identify the maximizers and prove a quantitative stability theorem.

Series: PDE Seminar

This is a joint work with Piermarco Cannarsa and Wei Cheng. We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian. The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions. We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

Monday, February 5, 2018 - 13:55 ,
Location: Skiles 005 ,
Mark A. Davenport ,
Georgia Institute of Technology ,
Organizer: Wenjing Liao

The discrete prolate spheroidal sequences (DPSS's) provide an efficient
representation for discrete signals that are perfectly timelimited and
nearly bandlimited. Due to the high computational complexity of
projecting onto the DPSS basis - also known as the Slepian basis - this
representation is often overlooked in favor of the fast Fourier
transform (FFT). In this talk I will describe novel fast algorithms for
computing approximate projections onto the leading Slepian basis
elements with a complexity comparable to the FFT. I will also highlight
applications of this Fast Slepian Transform in the context of
compressive sensing and processing of sampled multiband signals.

Series: CDSNS Colloquium

We present a discrete setting for the viscous Hamilton Jacobi equation, and prove convergence to the continuous case.

Series: CDSNS Colloquium

In this talk I will present some results concerning the existence and the stability of quasi-periodic solutions for quasi-linear and fully nonlinear PDEs. In particular, I will focus on the Water waves equation. The proof is based on a Nash-moser iterative scheme and on the reduction to constant coefficients of the linearized PDE at any approximate solution. Due to the non-local nature of the water waves equation, such a reduction procedure is achieved by using techniques from Harmonic Analysis and microlocal analysis, like Fourier integral operators and Pseudo differential operators.

Series: Research Horizons Seminar

Some basic problems, notions and results of the Ergodic theory will be
introduced. Several examples will be discussed. It is also a
preparatory talk for the next day colloquium where
finite time properties of dynamical and stochastic systems will be
discussed rather than traditional questions all dealing with asymptotic
in time properties.

Friday, February 2, 2018 - 15:00 ,
Location: Skiles 271 ,
Gladston Duarte ,
University of Barcelona & GT ,
gladston@maia.ub.es ,
Organizer: Jiaqi Yang

In a given system of
coordinates, the Restricted Three-Body Problem has some interesting
dynamical objects, for instance, equilibrium points, periodic orbits,
etc.
In this work, some connections between the stable and unstable manifolds
of periodic orbits of this system are studied. Such connections let one
explain the movement of Quasi-Hilda comets, which describe an orbit
that sometimes can be approximated by an ellipse of semi-major axis
greater than Jupiter's one, sometimes smaller.
Using a computer algebra system, one can compute an approximation to
those orbits and its manifolds and investigate the above mentioned
connections.
In addition, the Planar Circular model is used as a base for the fitting
of the orbit of comet 39P/Oterma, whose data were collected from the
JPL Horizons system. The possibility of using other models is also
discussed.

Series: ACO Student Seminar

Physical sensors (thermal, light, motion, etc.) are becoming ubiquitous and offer important
benefits to society. However, allowing
sensors into our private spaces has resulted in considerable privacy
concerns. Differential privacy has been developed to help alleviate
these privacy
concerns. In this
talk, we’ll develop and define a framework for releasing physical data
that preserves both utility and provides privacy. Our notion of
closeness of physical data will
be defined via the Earth Mover Distance and we’ll discuss the
implications of this choice. Physical data, such as temperature distributions, are often only accessible to us via a linear
transformation of the data.
We’ll analyse the implications of our privacy definition for linear inverse problems, focusing on those
that are traditionally considered to be "ill-conditioned”. We’ll
then instantiate our framework with the heat kernel on graphs and
discuss how the privacy parameter relates to the connectivity
of the graph. Our work indicates that it is possible to produce locally
private sensor measurements that both keep the exact locations of the
heat sources private and permit recovery of the ``general geographic
vicinity'' of the sources. Joint
work with Anna C. Gilbert.

Friday, February 2, 2018 - 10:10 ,
Location: Skiles 254 ,
Marc Härkönen ,
Georgia Tech ,
harkonen@gatech.edu ,
Organizer: Kisun Lee

Differential operator rings can be described as polynomial rings over differential operators. We will study two of them: first the relatively simple ring of differential operators R with rational function coefficients, and then the more complicated ring D with polynomial coefficients, or the Weyl algebra. It turns out that these rings are non-commutative because of the way differential operators act on smooth functions. Despite this, with a bit of work we can show properties similar to the regular polynomial rings, such as division, the existence of Gröbner bases, and Macaulay's theorem. As an example application, we will describe the holonomic gradient descent algorithm, and show how it can be used to efficiently solve computationally heavy problems in statistics.