The eighth meeting of the Carolina Mathematics Seminar took place at University of South Carolina Lancaster, Lancaster, SC. This is a list of our speakers, titles and abstracts.
Julia Smith (Student at USC Sumter)
Title: Inverse functions and ciphers.
Abstract: A cipher is a method for creating secret messages. The purpose of using a cipher is to exchange information securely. Throughout history, many different methods have been created. I will discuss a cipher that depends on a linear function and the use of its inverse to decode a secret message. Basic modular arithmetic will be used throughout the talk.
Linyuan Lincoln Lu (USC Columbia)
Title: A Fractional Analogue of Brooks' Theorem.
Abstract: Let Δ be the maximum degree of a connected graph G. Brooks' theorem states that the only connected graphs with chromatic number Δ+1 are complete graphs and odd cycles. Here we proved a fractional version of Brooks' theorem: we classified all connected graphs G with the fractional chromatic number χ f(G)≥ Δ. (Joint work with Xing Peng).
Naima Naheed (Benedict College)
Title: Convex Minorant of the Nonconvex Thomas-Fermi Energy Functional.
Abstract: Mathematically rigorous versions of Thomas-Fermi theory and its generalizations were developed in the 1970s and 1980s by Lieb, Simon, Benilan, Brezis, Gisele and Jerome Goldstein and others. Later Phillippe Benilan, Gisele and Jerome Goldstein (BGG) incorporated Fermi-Amaldi correction into the Thomas-Fermi energy functional. As a result convexity is lost.
The theory which will be presented here includes a convex minorant of the nonconvex Thomas-Fermi energy functional. The corresponding Euler-Lagrange equation will become a nonlinear elliptic system involving measures, which will be solved using the methods of BGG. Then the existence of a ground state Thomas-Femi density will be obtained using, among other things, topological degree theory.
Xing Peng (USC Columbia)
Title: The minimum number of monochromatic short progressions in Zn
Abstract: For any n≥k≥3, let Mk(n) be the minimum number of monochromatic k-term progressions in any 2-coloring of Zn. We studied asymptotic bounds of Mk(n) for k =3,4 and large n. For k=3, we show random colorings achieve the minimum number of monochromatic 3-term progressions. For k =4, we construct a 2-coloring of Zn. with few 4-term progressions than a random coloring has. Our upper bound and lower bound for M4(n) improve the previous result given by Wolf on M4(p) for prime p. (Joint work with Linyuan Lu).
Julian Buck (Francis Marion University)
Title: The relationship between Topological Dynamics and C*-Algebras.
Abstract: The classification program for C*-algebras is one of the main current branches of research in abstract functional analysis. C*-algebras that arise by looking at dynamical systems on topological spaces have provided especially good examples for the purpose of classification theory, where one can start with commutative C*-algebras (essentially, function spaces) and construct new noncommutative examples through a universal construction (the crossed product). This produces a sort of 3-tiered system: the base dynamical system, its associated function algebra, and the new crossed product C*-algebra. In this talk I will describe some of the interplay between the dynamical system and its crossed product, and how this is exploited to show the crossed product has a suitably nice structure for the classification program.
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