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Chassidy Bozeman, Iowa State University
Zero forcing on a simple graph is an iterative coloring procedure that starts by initially coloring vertices white and blue and then repeatedly applies the following color change rule: if any vertex colored blue has exactly one white neighbor, then that neighbor is changed from white to blue. Any initial set of blue vertices that can color the entire graph blue is called a zero forcing set. The zero forcing number is the cardinality of a minimum zero forcing set. A well known result is that the zero forcing number of a simple graph is an upper bound for the maximum nullity of the graph (the largest possible nullity over all symmetric real matrices whose (ij)-th entry (for distinct i and j) is nonzero whenever {i,j} is an edge in G and is zero otherwise). A variant of zero forcing, known as power domination (motivated by the monitoring of the electric power grid system), uses the power color change rule that starts by initially coloring vertices white and blue and then applies the following rules: 1) In step 1, for any white vertex w that has a blue neighbor, change the color of w from white to blue. 2) For the remaining steps, apply the color change rule. Any initial set of blue vertices that can color the entire graph blue using the power color change rule is called a power dominating set. We present results on the power domination problem of a graph by considering the power dominating sets of minimum cardinality and the amount of steps necessary to color the entire graph blue.
Guy David, Courant Institute, New York University
Since the work of Cheeger, many non-smooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner.
Julia Bergner, University of Virginia
The notion of a 2-Segal space was recently defined by Dyckerhoff and Kapranov, and independently by Galvez-Carrillo, Kock, and Tonks under the name of decomposition space. Unlike Segal spaces, which encode the structure of a category up to homotopy, 2-Segal spaces encode a more general structure in which composition need not exist or be unique, but is still associative. Both sets of authors above proved that the output of the Waldhausen $S_\bullet$-construction is a 2-Segal space. In joint work with Osorno, Ozornova, Rovelli, and Scheimbauer, we look at a discrete version of this construction whose output is a 2-Segal set. We show that, via this construction, the category of 2-Segal sets is equivalent to the category of augmented stable double categories. In this talk, I'll introduce 2-Segal sets and spaces, discuss this result and a conjectured homotopical generalization, and, time permitting, look at some other interesting features of 2-Segal spaces.
Shirshendu Ganguly, Berkeley
The upper tail problem in the Erdös-Rényi random graph $G \sim G(n,p)$, where every edge is included independently with probability $p$, is to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1 + d$. The arithmetic analog considers the count of arithmetic progressions in a random subset of $Z/nZ$, where every element is included independently with probability $p$. In this talk, I will describe some recent results regarding the solution of the upper tail problem in the sparse setting i.e. where $p$ decays to zero, as $n$ grows to infinity. The solution relies on non-linear large deviation principles developed by Chatterjee and Dembo and more recently by Eldan and solutions to various extremal problems in additive combinatorics.
Davi Maximo, University of Pennsylvania
Abstract: In this talk we will give a precise picture on how a sequence of minimal surfaces with bounded index might degenerate on a given closed three-manifold. As an application, we prove several compactness results.
Hermann Mena, Yachay Tech, Ecuador
Leslie McClure, Professor and Chair of the Department of Epidemiology and Biostatistics, Drexel University.
There are no conferences this week.