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Series: Geometry Topology Seminar

Series: IMPACT Distinguished Lecture

In
the latent voter model, which models the spread of a technology
through a social network, individuals who have just changed their
choice have a latent period, which is exponential with rate λ during
which they will not buy a new device. We study site and edge versions
of this model on random graphs generated by a configuration model in
which the degrees d(x) have 3 ≤ d(x) ≤ M. We show that if the
number of vertices n → ∞ and log n << λn
<< n then the latent voter model has a quasi-stationary state
in which each opinion has probability ≈ 1/2 and persists in this
state for a time that is ≥ nm
for any m <∞. Thus, even a very small latent period drastically
changes the behavior of the voter model.

Friday, March 17, 2017 - 14:00 ,
Location: Skiles 006 ,
John Etnyre ,
Georgia Tech ,
Organizer: John Etnyre

This will be a 1.5 hour (maybe slightly longer) seminar.

Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.

Series: ACO Student Seminar

Optimization problems arising in decentralized multi-agent systems have gained significant attention in the context of cyber-physical, communication, power, and robotic networks combined with privacy preservation, distributed data mining and processing issues. The distributed nature of the problems is inherent due to partial knowledge of the problem data (i.e., a portion of the cost function or a subset of the constraints is known to different entities in the system), which necessitates costly communications among neighboring agents. In this talk, we present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems which can significantly reduce the number of inter-node communications. Our major contribution is the development of decentralized communication sliding methods, which can skip inter-node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions.This is a joint work with Guanghui (George) Lan and Yi Zhou.

Series: Algebra Seminar

Error-correcting decoding is generalized to multivariate
sparse polynomial and rational function interpolation from
evaluations that can be numerically inaccurate and where
several evaluations can have severe errors (``outliers'').
Our multivariate polynomial and rational function
interpolation algorithm combines Zippel's symbolic sparse
polynomial interpolation technique [Ph.D. Thesis MIT 1979]
with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc.
SNC 2007], and removes outliers (``cleans up data'') by
techniques from the Welch/Berlekamp decoder for Reed-Solomon
codes.
Our algorithms can build a sparse function model from a
number of evaluations that is linear in the sparsity of the
model, assuming that there are a constant number of ouliers
and that the function probes can be randomly chosen.

Series: School of Mathematics Colloquium

We present algorithms for performing sparse univariate
polynomial interpolation with errors in the evaluations of
the polynomial. Our interpolation algorithms use as a
substep an algorithm that originally is by R. Prony from
the French Revolution (Year III, 1795) for interpolating
exponential sums and which is rediscovered to decode
digital error correcting BCH codes over finite fields (1960).
Since Prony's algorithm is quite simple, we will give
a complete description, as an alternative for Lagrange/Newton
interpolation for sparse polynomials. When very few errors
in the evaluations are permitted, multiple sparse interpolants
are possible over finite fields or the complex numbers,
but not over the real numbers. The problem is then a simple
example of list-decoding in the sense of Guruswami-Sudan.
Finally, we present a connection to the Erdoes-Turan Conjecture
(Szemeredi's Theorem).
This is joint work with Clement Pernet, Univ. Grenoble.

Series: Graph Theory Seminar

For a graph G, the Colin de Verdière graph parameter mu(G) is the maximum
corank of any matrix in a certain family of generalized adjacency matrices
of G. Given a non-negative integer t, the family of graphs with mu(G) <= t
is minor-closed and therefore has some nice properties. For example, a
graph G is planar if and only if mu(G) <= 3. Colin de Verdière conjectured
that the chromatic number chi(G) of a graph satisfies chi(G) <= mu(G)+1.
For graphs with mu(G) <= 3 this is the Four Color Theorem. We conjecture
that if G has at least t vertices and mu(G) <= t, then |E(G)| <= t|V(G)| -
(t+1 choose 2). For planar graphs this says |E(G)| <= 3|V(G)|-6. If this
conjecture is true, then chi(G) <= 2mu(G). We prove the conjectured edge
upper bound for certain classes of graphs: graphs with mu(G) small, graphs
with mu(G) close to |V(G)|, chordal graphs, and the complements of chordal
graphs.

Series: Stochastics Seminar

Estimation of the covariance matrix has attracted significant attention
of the statistical research community over the years, partially due to
important applications such as Principal Component Analysis. However,
frequently used empirical covariance estimator (and its modifications)
is very sensitive to outliers, or ``atypical’’ points in the sample.
As P. Huber wrote in 1964, “...This raises a question which could have
been asked already by Gauss, but which was, as far as I know, only
raised a few years ago (notably by Tukey): what happens if the true
distribution deviates slightly from the assumed normal one? As is now
well known, the sample mean then may have a catastrophically bad
performance…”
Motivated by Tukey's question, we develop a new estimator of the
(element-wise) mean of a random matrix, which includes covariance
estimation problem as a special case. Assuming that the entries of a
matrix possess only finite second moment, this new estimator admits
sub-Gaussian or sub-exponential concentration around the unknown mean in
the operator norm. We will present extensions of our approach to
matrix-valued U-statistics, as well as applications such as the matrix
completion problem.
Part of the talk will be based on a joint work with Xiaohan Wei.

Series: IMPACT Distinguished Lecture

The
use of evolutionary game theory biology dates to work of
Maynard-Smith who used it to explain why most fights between animals
were of the limited war type. Nowak and collaborators have shown that
a spatial distribution of players can explain the existence of
altruism, which would die out in a homogeneously mixing population.
For the last twenty years, evolutionary games have been used to model
cancer. In applications to ecology and cancer, the system is not
homogeneously mixing so it is important to understand how space
changes the outcome of these games. Over the last several years we
have developed a theory for understanding the behavior of
evolutionary games in the weak selection limit. We will illustrate
this theory by discussing a number of examples. The most recent work
was done in collaboration with a high school student so the talk
should be accessible to a broad audience.