Department of Mathematics
https://as.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityWed, 18 Oct 2017 15:07:20 +0000en-UShourly1https://wordpress.org/?v=4.7.2Department Tea
https://as.vanderbilt.edu/math/2017/10/department-tea-14/
Tue, 17 Oct 2017 21:33:50 +0000https://as.vanderbilt.edu/math/?p=5660Hamiltonicity of 3-Connected, Planar, K_{1,1,5}-Minor-Free Graphs
https://as.vanderbilt.edu/math/2017/10/hamiltonicity-of-3-connected-planar-k_115-minor-free-graphs/
Mon, 16 Oct 2017 21:10:47 +0000https://as.vanderbilt.edu/math/?p=5655In 2014, Ellingham, Marshall, Ozeki, and Tsuchiya proved that all 3-connected, planar, K_{2,5}-minor-free graphs are Hamiltonian. They also constructed an infinite family of 3-connected, planar, K_{2,6}-minor-free graphs which are not Hamiltonian. In this talk we examine the class of 3-connected, planar, K_{1,1,5}-minor-free graphs, which is a superset of 3-connected K_{2,5}-minor-free graphs and a subset of 3-connected K_{2,6}-minor-free graphs. We prove the existence of induced fans and show that all graphs in this class are Hamiltonian, with exactly one exception.

]]>Coefficients of Certain Unitary Representations of Thompson Groups
https://as.vanderbilt.edu/math/2017/10/talk-title-tba-192/
Wed, 11 Oct 2017 22:10:46 +0000https://as.vanderbilt.edu/math/?p=5267There is a general construction, essentially due to Ore, of a group of quotients of certain rather special categories. If the category is planar rooted binary forests (under stacking), the group of quotients is Thompsons group F. Actions of these groups of fractions can be constructed whenever there is a functor from the underlying category to another category that admits direct limits. I will briefly review these constructions and show how to construct many actions of F, including unitary representations. Irreducibility is then a major question. As a first attempt to tackle this problem I will focus on the coefficients of these unitary representations. Calculation of coefficients led to a construction of knots and links but we will see that it also leads to the study of iteration of dynamical systems which in the simplest cases are rational functions on ℂ P^1. We will show some pictures of Julia and Fatou sets of these maps.
]]>Department Tea
https://as.vanderbilt.edu/math/2017/10/department-tea-13/
Wed, 11 Oct 2017 21:33:44 +0000https://as.vanderbilt.edu/math/?p=5600Teaching Computers to See: Categorizing Images with Matrices and Derivatives
https://as.vanderbilt.edu/math/2017/10/teaching-computers-to-see-categorizing-images-with-matrices-and-derivatives/
Wed, 11 Oct 2017 00:00:13 +0000https://as.vanderbilt.edu/math/?p=5589Many technologies we dream of require the solution to the problem of computer vision. That is, we need to be able to teach a computer to recognize what it sees. We can use this to help medical professionals make diagnoses, to allow self-driving cars to distinguish between a plastic bag and a person, or to eliminate the task of hand-sorting objects on a conveyor belt in a factory. In this talk we explore how exactly computers work with images while also investigating and developing one of the leading algorithms used in classify images. We will discover that an image is nothing more than a matrix and that the algorithm we desire is an application of a technique learned in a student’s first semester of calculus.
]]>Department Tea
https://as.vanderbilt.edu/math/2017/10/department-tea-12/
Tue, 10 Oct 2017 21:33:14 +0000https://as.vanderbilt.edu/math/?p=5597Circuit Graphs and Relative Connectivity
https://as.vanderbilt.edu/math/2017/10/circuit-graphs-and-relative-connectivity-2/
Mon, 09 Oct 2017 21:10:23 +0000https://as.vanderbilt.edu/math/?p=5595Induction arguments for k-connected graphs can be difficult because subgraphs of a k-connected graph are not necessarily k-connected. In 1966 David Barnette found a way to get around this for 3-connected planar graphs. He showed that every 3-connected planar graph has a spanning tree of maximum degree at most 3 by working with a more general class of graphs known as “circuit graphs”, which behave well in induction proofs. Similar ideas have been used in proofs of other results on hamiltonicity, spanning trees, and related concepts. We show how these ideas can be extended in a general way. In particular, if G is a graph and S is a subset of V(G), we can define G to be “k-connected relative to S”, or “(k,S)-connected”, if certain conditions hold. We establish some basic properties of this concept and illustrate how they can be used. This is joint work with Dan Biebighauser of Concordia College in Minnesota, and is an update on a similar talk Paul H. Edelman gave in spring 2015.
]]>Spatial Spread of Epidemic Diseases in Geographical Settings: Seasonal Influenza Epidemics in Puerto Rico
https://as.vanderbilt.edu/math/2017/10/spatial-spread-of-epidemic-diseases-in-geographical-settings-seasonal-influenza-epidemics-in-puerto-rico/
Fri, 06 Oct 2017 22:10:46 +0000https://as.vanderbilt.edu/math/?p=5334Deterministic models are developed for the spatial spread of epidemic diseases in geographical settings. The models are focused on outbreak that arise from a small number of infected hosts imported into sub-regions of the geographical settings. The goal is to understand how spatial heterogeneity influences the transmission dynamics of the susceptible and infected populations. The models consist of systems of partial differential equations with diffusion terms describing the spatial spread of the underlying microbial infectious agents. Applications are given to seasonal influenza epidemics in Puerto Rico.
]]>Planar Algebras for the Drinfeld Centres of the Even Parts of the ADE Subfactors
https://as.vanderbilt.edu/math/2017/10/talk-title-tba-197/
Fri, 06 Oct 2017 22:10:24 +0000https://as.vanderbilt.edu/math/?p=5301Consider a subfactor whose even and dual even parts are both equivalent to some unitary tensor category C. Such subfactors are classified by braided auto-equivalences of the Drinfeld centre of C. In this talk I’ll give a presentation of the (unshaded) planar algebras associated to the centres of the even parts of the ADE subfactors. I’ll then explain how we can use these planar algebras to compute the braided auto-equivalences of the corresponding centres.
]]>Genuine Equivariant Operads
https://as.vanderbilt.edu/math/2017/10/talk-title-tba-201/
Fri, 06 Oct 2017 21:10:33 +0000https://as.vanderbilt.edu/math/?p=5326A classic result by Elmendorf states that G-spaces and G-coefficient systems are Quillen equivalent. Moreover, this result remains true if one replaces spaces with other well-behaved categories (including simplicial sets and categories). However, when considering G-operads, this equivalence fails to capture certain desired subtleties. In this talk, I will introduce (G-)operads and review this failure. I will then define a new algebraic gadget called genuine G-operads (playing the role of coefficient systems), and state an Elmendorf-type result in this context. This is joint work with L. Pereira.
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