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UConn Math Club

Title: Hamiltonian Cycles and Their Uses

Speaker: Voula Collins (University of Connecticut)

Time: Wednesday, February 22, 2017 at 5:45 pm

Place: MONT 321Abstract: There are many different ways to think about traveling through a graph: minimizing how long it takes to get from vertex A to vertex B, the number of ways to get from A to B, whether you travel along every edge just once, whether you travel through every vertex just once, and many more. The last two types of paths in a graph are called Eulerian and Hamiltonian paths, respectively. We will see what is known about whether such paths in a graph exist, how to find them, and why we care about them, including a discussion of some useful and not-quite-so useful applications.Comments: Free pizza and drinks!

Title: Bernoulli numbers

Speaker: Keith Conrad (University of Connecticut)

Time: Friday, February 24, 2017 at 12:20 pm

Place: MONT 111Abstract: The Bernoulli numbers are a sequence of rational numbers that start out as 1, -1/2, 1/6, 0, -1/30, ... and first arose in formulas for the sum of powers of the first $n$ integers. They later found applications in number theory, numerical analysis, differential topology, Lie theory, and physics. We will explain what the Bernoulli numbers are and discuss some of their applications.

Title: Coupled supersymmetric quantum mechanics: unifying the harmonic oscillator and supersymmetric quantum mechanics

Speaker: Cameron Williams (University of Houston)

Time: Friday, February 24, 2017 at 1:30 pm

Place: MONT 226Abstract: In this talk, we will briefly cover the standard quantum mechanical harmonic oscillator and the usual ladder operator technique for solving it. The ladder operator technique will be used to motivate traditional supersymmetric quantum mechanics. Supersymmetric quantum mechanics is useful for finding excited state energies of quantum systems, but has inherent flaws. Exact solutions via supersymmetric techniques are hard to come by and the (super)algebra of supersymmtric quantum mechanics is not very rich. Supersymmetric quantum mechanics is meant to mimic the quantum mechanical harmonic oscillator, however in general no true ladder structure exists. Returning to a treatment of the quantum mechanical harmonic oscillator under the guise of supersymmetry leads to a natural new structure which has flavors of both supersymmetry and the usual ladder structure from the harmonic oscillator. We call this new structure coupled supersymmetric quantum mechanics (coupled SUSY for short). Some results regarding the (Lie) algebraic structure underlying coupled SUSY, uncertainty principles, and coherent states for coupled SUSY will be discussed.

Title: Mathematical Relativity and the Geometry of Initial Data Sets for the Einstein Equations II

Speaker: Armando Cabrera (University of Connecticut)

Time: Friday, February 24, 2017 at 3:30 pm

Place: MONT 111Abstract: The Einstein Equations, which describe the interaction between the matter and the geometry of the universe, can be interpreted as a complicated system of hyperbolic differential equations. One way to study its solutions is by considering an initial value problem, which naturally leads to the understanding of the geometry of special Riemannian manifolds with non-negative scalar curvature, called initial data sets. In this series of lectures, we will briefly describe some important concepts in mathematical relativity, and then we will focus on the geometry of initial data sets by going over some classical results and moving towards recent developments.