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Series: Algebra Seminar

TBA

Monday, April 1, 2019 - 12:45 ,
Location: Skiles 006 ,
Ahmad Issa ,
University of Texas, Austin ,
Organizer: Jennifer Hom

Series: CDSNS Colloquium

<p>One of the characteristics observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. We introduce a model of network growth based on this notion of specialization and show that as a network is specialized its topology becomes increasingly modular, hierarchical, and sparser, each of which are properties observed in real networks. This model is also highly flexible in that a network can be specialized over any subset of its components. By selecting these components in various ways we find that a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions and clustering coefficients. This growth model also maintains the basic spectral properties of a network, i.e. the eigenvalues and eigenvectors associated with the network's adjacency network. This allows us in turn to show that a network maintains certain dynamic properties as the network's topology becomes increasingly complex due to specialization.</p>

Series: CDSNS Colloquium

Series: Algebra Seminar

This is a two day conference (March 30-31) to be held at Georgia Tech on algebraic geometry and related areas. We will have talks by Sam Payne, Eric Larson, Angelica Cueto, Rohini Ramadas, and Jennifer Balakrishnan. See https://sites.google.com/view/gattaca/home for more information.

Series: Combinatorics Seminar

Series: Stochastics Seminar

Series: Graph Theory Seminar

Reed
and Wood and independently Norine, Seymour, Thomas, and Wollan showed
that for each $t$ there is $c(t)$ such that every graph on $n$ vertices
with no $K_t$ minor has
at most $c(t)n$ cliques. Wood asked in 2007 if
$c(t)<c^t$ for some absolute constant $c$. This problem was recently
solved by Lee and Oum. In this paper, we determine the exponential
constant. We prove that every graph on $n$ vertices
with no $K_t$ minor has at most $3^{2t/3+o(t)}n$ cliques. This bound is tight for $n \geq 4t/3$.
We use the similiar idea to give an upper bound on the number of cliques in
an $n$-vertex graph with no $K_t$-subdivsion. Easy computation will
give an upper
bound of $2^{3t+o(t)}n$; a more careful examination gives an upper bound
of $2^{1.48t+o(t)}n$. We conjecture that the optimal exponential
constant is $3^{2/3}$ as in the case of minors.
This is a joint work with Jacob Fox.

Series: School of Mathematics Colloquium

The Remez inequality for polynomials quantifies the way the maximum of a polynomial over an interval is controlled by its maximum over a subset of positive measure. The coefficient in the inequality depends on the degree of the polynomial; the result also holds in higher dimensions. We give a version of the Remez inequality for solutions of second order linear elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of a solution. We discuss other results on quantitative unique continuation for solutions of elliptic PDEs and their gradients and give some applications for the estimates of eigenfunctions for the Laplace-Beltrami operator. The talk is based on a joint work with A. Logunov.

Series: High Dimensional Seminar

TBA