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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

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.

Series: Job Candidate Talk

The mean field variational inference is widely used in statistics and
machine learning to approximate posterior distributions. Despite its
popularity, there exist remarkably little fundamental theoretical
justifications. The success of variational inference
mainly lies in its iterative algorithm, which, to the best of our
knowledge, has never been investigated for any high-dimensional or
complex model. In this talk, we establish computational and statistical
guarantees of mean field variational inference. Using
community detection problem as a test case, we show that its iterative
algorithm has a linear convergence to the optimal statistical accuracy
within log n iterations. We are optimistic to go beyond community
detection and to understand mean field under a general
class of latent variable models. In addition, the technique we develop
can be extended to analyzing Expectation-maximization and Gibbs sampler.

Wednesday, January 31, 2018 - 13:55 ,
Location: Skiles 006 ,
Sudipta Kolay ,
GaTech ,
Organizer: Anubhav Mukherjee

The Jordan curve theorem states that any simple closed curve decomposes the 2-sphere into two connected components and is their common boundary. Schönflies strengthened this result by showing that the closure of either connected component in the 2-sphere is a 2-cell. While the first statement is true in higher dimensions, the latter is not. However under the additional hypothesis of locally flatness, the closure of either connected component is an n-cell. This result is called the Generalized Schönflies theorem, and was proved independently by Morton Brown and Barry Mazur. In this talk, I will describe the proof of due to Morton Brown.

Series: Research Horizons Seminar

In recent years the problem
of low-rank matrix completion received a tremendous amount of
attention. I will consider the problem of exact low-rank matrix
completion for generic data. Concretely, we start with a
partially-filled matrix M, with real
or complex entries, with the goal of finding the unspecified entries
(completing M) in such a way that the completed matrix has the lowest
possible rank, called the completion rank of M. We will be interested in
how this minimal completion rank depends on the
known entries, while keeping the locations of specified and unspecified
entries fixed. Generic data means that we only consider partial fillings
of M where a small perturbation of the entries does not change the
completion rank of M.

Wednesday, January 31, 2018 - 11:00 ,
Location: Skiles 006 ,
Prof. Mansoor Haider ,
North Carolina State University, Department of Mathematics & Biomathematics ,
Organizer: Sung Ha Kang

Many biological soft tissues exhibit complex interactions between passive biophysical or biomechanical mechanisms, and active physiological responses. These interactions affect the ability of the tissue to remodel in order to maintain homeostasis, or govern alterations in tissue properties with aging or disease. In tissue engineering applications, such interactions also influence the relationship between design parameters and functional outcomes. In this talk, I will discuss two mathematical modeling problems in this general area. The first problem addresses biosynthesis and linking of articular cartilage extracellular matrix in cell-seeded scaffolds. A mixture approach is employed to, inherently, capture effects of evolving porosity in the tissue-engineered construct. We develop a hybrid model in which cells are represented, individually, as inclusions within a continuum reaction-diffusion model formulated on a representative domain. The second problem addresses structural remodeling of cardiovascular vessel walls in the presence of pulmonary hypertension (PH). As PH advances, the relative composition of collagen, elastin and smooth muscle cells in the cardiovascular network becomes altered. The ensuing wall stiffening increases blood pressure which, in turn, can induce further vessel wall remodeling. Yet, the manner in which these alterations occur is not well understood. I will discuss structural continuum mechanics models that incorporate PH-induced remodeling of the vessel wall into 1D fluid-structure models of pulmonary cardiovascular networks. A Holzapfel-Gasser-Ogden (HGO)-type hyperelastic constitutive law for combined bending, inflation, extension and torsion of a nonlinear elastic tube is employed. Specifically, we are interested in formulating new, nonlinear relations between blood pressure and vessel wall cross-sectional area that reflect structural alterations with advancing PH.