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Series: Combinatorics Seminar

Among the most studied tree growth processes there are recursive trees and
linear preferential attachment trees. The study of these two models is
motivated by the need of understanding the evolution of social networks. A
key feature of social networks is the presence of vertices that serve as
hubs, connecting large parts of the network. While such type of vertices
had been widely studied for linear preferential attachment trees, analogous
results for recursive trees were missing.
In this talk, we will present joint laws for both the number and depth of
vertices with near-maximal degrees and comment on the possibilities that
our methods open for future research.
This is joint work with Louigi Addario-Berry.

Series: Stochastics Seminar

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the
edges with common distribution F. For which F is there an infinite self-avoiding path with
finite total weight? This question arises in first-passage percolation, the study of the
random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2
(there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there
is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably
more difficult due to the presence of finite paths of zero-weight edges on all scales. I will
discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient
conditions on F for the existence of an infinite finite-weight path. The methods involve
comparing the model to another one, invasion percolation, and showing that geodesics
in first-passage percolation have the same first order travel time as optimal paths in an
embedded invasion cluster.

Series: Graph Theory Seminar

Let $G$ be a graph containing 5 different vertices $a_0, a_1, a_2, b_1$ and $b_2$. We say that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible if $G$ contains disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We give a characterization for $(G,a_0,a_1,a_2,b_1,b_2)$ to be feasible, answering a question of Robertson and Seymour. This is joint work with Changong Li, Robin Thomas, and Xingxing Yu.In this talk, we will discuss the operations we will use to reduce $(G,a_0,a_1,a_2,b_1,b_2)$ to $(G',a_0',a_1',a_2',b_1',b_2')$ with $|V(G)|+|E(G)|>|V(G')|+E(G')$, such that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible iff $(G',a_0',a_1',a_2'b_1',b_2')$ is feasible.

Series: School of Mathematics Colloquium

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random particle simulations, flocks and mills, and their qualitative behavior. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. Compactly supported global minimizers determine the flock patterns whose existence is related to the classical H-stability in statistical mechanics and the classical obstacle problem for differential operators.

Series: Research Horizons Seminar

SPORT
is a 12-week *PAID* summer internship offered by the National Security
Agency (NSA) that provides 8 U.S. Citizen graduate students the
opportunity to apply their technical skills to current, real-world
operations research problems at the NSA. SPORT
looks for strong students in operations research, applied math,
computer science, data science, industrial and systems engineering, and
other related fields.
Program Highlights:
-- Paid internship (12 weeks, late May to mid-August 2018)
-- Applications accepted September 1 - October 31, 2017
-- Opportunity to apply operations research, mathematics, statistics, computer science, and/or engineering skills
-- Real NSA mission problems
-- Paid annual and sick leave, housing available, most travel costs covered
-- Flexible work schedule
-- Opportunity to network with other Intelligence Agencies

Series: Analysis Seminar

The
classical Balian-Low theorem states that if both a function and it's
Fourier transform decay too fast then the Gabor system generated by this
function (i.e. the system obtained from this function by taking integer
translations and integer modulations) cannot be an orthonormal basis or a Riesz basis.Though it provides for an
excellent `thumbs--rule' in time-frequency analysis, the Balian--Low
theorem is not adaptable to many applications. This is due to the fact
that in realistic situations information about a signal is given by a
finite dimensional vector rather then by a function over the real line.
In this work we obtain an analog of the Balian--Low theorem in the
finite dimensional setting, as well as analogs to some of its
extensions. Moreover, we will note that the classical Balian--Low
theorem can be derived from these finite dimensional analogs.

Series: Geometry Topology Seminar

Series: Combinatorics Seminar

This talk will focus on tree automata, which are tools to analyze existential monadic second order properties of rooted trees. A tree automaton A consists of a finite set \Sigma of colours, and a map \Gamma: \mathbb{N}^\Sigma \rightarrow \Sigma. Given a rooted tree T and a colouring \omega: V(T) \rightarrow \Sigma, we call \omega compatible with automaton A if for every v \in V(T), we have \omega(v) = \Gamma(\vec{n}), where \vec{n} = (n_\sigma: \sigma \in \Sigma) and n_\sigma is the number of children of v with colour \sigma. Under the Galton-Watson branching process set-up, if p_\sigma denotes the probability that a node is coloured \sigma, then \vec{p} = (p_\sigma: \sigma \in \Sigma) is obtained as a fixed point of a system of equations. But this system need not have a unique fixed point. Our question attempts to answer whether a fixed point of such a system simply arises out of analytic reasons, or if it admits of a probabilistic interpretation. I shall formally defined interpretation, and provide a nearly complete description of necessary and sufficient conditions for a fixed point to not admit an interpretation, in which case it is called rogue.Joint work with Tobias Johnson and Fiona Skerman.

Series: Stochastics Seminar

We discuss two recent results concerning disease modeling on networks. The infection is assumed to spread via contagion (e.g., transmission over the edges of an underlying network). In the first scenario, we observe the infection status of individuals at a particular time instance and the goal is to identify a confidence set of nodes that contain the source of the infection with high probability. We show that when the underlying graph is a tree with certain regularity properties and the structure of the graph is known, confidence sets may be constructed with cardinality independent of the size of the infection set. In the scenario, the goal is to infer the network structure of the underlying graph based on knowledge of the infected individuals. We develop a hypothesis test based on permutation testing, and describe a sufficient condition for the validity of the hypothesis test based on automorphism groups of the graphs involved in the hypothesis test. This is joint work with Justin Khim (UPenn).