The Abstracts

Bounding the Largest Inhomogeneous Approximation Constant by Bishnu Paudel

Bounding the Largest Inhomogeneous Approximation Constant For an irrational real α and γ ̸∈ Z + Zα it is well known that M(α, γ) := lim inf |n|→∞ |n|||nα − γ|| ≤ 1/4 . In this talk, I will discuss upper and lower bounds depending on whether R := lim infi→∞ a_i is odd or even, where a_i are the partial quotients in the negative (i.e., the ‘round-up’) continued fraction expansion of α. I will give an upper bound of the form M(α, γ) ≤ C_0(R) with C_0(R) optimal for R ≥ 3. Then lower bound that there is always a γ such that M(α, γ) ≥ C_1(R) with C_1(R) optimal when R is even and asymptotically optimal when R is odd. In particular, for any R ≥ 3 there is a γ /∈ Z + Zα with M(α, γ) ≥ 1/( 6 √ 3 + 8) = 1 18.3923... .

Orbital Stability of Smooth Periodic Traveling Wave Solutions of the b-family of Camassa-Holm Equations by Brett Ehrman

We derive the conditions for orbital stability of smooth (spatially) periodic traveling wave solutions of the b-family of Camassa-Holm equations, with b > 1. We find that there exists a three-parameter family of such solutions of the form φ(x; a, E, c). We rewrite the PDE as an infinite-dimensional Hamiltonian system using a change of variables: m = u−uxx. Then we use spectral analysis to show orbital stability of µ := φ−φxx. This allows us to obtain orbital stability of φ with respect to perturbations in H^3_{per}[0, T], where φ is T-periodic. For general b > 1, we must use a Hamiltonian structure that is commonly disregarded in the integrable cases where b ∈ {2, 3}, due to the fact that our Hamiltonian structure is relatively messy. It remains to be verified whether or not the stability criteria are satisfied. Proving or disproving the criteria is my next objective

Isometry Killing Hyperbolization of 3-manifolds

by Malcolm Gabbard

An area of interest in low-dimensional topology and geometry is the study of ”equivalent” 3-manifolds. In this talk I will begin by defining one notion of equivalence, that of homology cobordism. One interesting result is that every 3-manifold is equivalent (homology cobordant) to a hyperbolic 3-manifold. More interesting is that if you have a group G acting on your initial manifold, you can find an equivalent hyperbolic manifold such that G acts on this new manifold by isometries. My work begins here and works to find equivalent hyperbolic manifolds who’s isometries are entirely described by subgroups of G.

Shear Flows of Liquid Crystal Models by Katheryn A. Beck

While in school we are taught that there are three states for matter to take: solid, liquid, and gas. However, this is an oversimplified view of the world. Liquid crystal material, for instance, is not one of the three states, which exhibits an intermediate phase between the solid crystal state and the isotropic liquid state. Thus, liquid crystals not only possess many of the mechanical properties of a liquid, but also are similar to crystals. Therefore, liquid crystal, as a kind of typical soft material, has been widely and successfully applied in life and industry. To understand the models associated with this liquid crystal material we first investigate various characteristic (including stability, steady-states, and hysteresis) related to a simplified version of the Ericksen-Leslie model. We then discuss some preliminary characteristics of the full Ericksen-Leslie model and the corresponding motivation from the simplified model.

How will quantum computers affect global internet security? by Humberto Bautista Serrano

In this talk we will discuss how the security of the current cryptographic protocols works and how it depends on the speed limitations of the classical computers. Although no prior knowledge of quantum mechanics or computation is assumed, we will give a mathematical introduction to the main cryptographic protocols and explain how these are being modified or replaced to face the challenges that quantum computers impose in order to reach secure transmission of information.

On the Chromatic Symmetric Function of Trees

by Enrique Salcido

A symmetric function F is a power series in countably many commuting indeterminates, such that permuting the indeterminates leaves F unchanged. The chromatic symmetric function (CSF) of a graph is a symmetric function that generalizes its chromatic polynomial. In this talk we will present the information that can be obtained from the CSF of a tree, such as the degree sequence, path sequence, and matching polynomial.

An Approximation of the Modulus of the Family of Edge Covers by Adriana Ortiz-Aquino

The discrete p-modulus is a very flexible and general tool for measuring the richness of families of objects defined on a graph. Modulus has been studied for specific families of graph objects and it has been shown to generalize well-known graph theoretic quantities such as shortest path, max flow/min cut, and effective resistance. Our focus is on the p-modulus of the family of edge covers and on the more general family of fractional edge covers on an unweighted, undirected graph. Through the theory of Fulkerson blocking duality, every family of objects has a corresponding dual family whose modulus is closely related to the modulus of the original family. Our results show that the dual family of fractional edge covers is the family of stars, which greatly reduces the number of constraints for the p-modulus problem. With this, we give an approximation for the modulus of edge covers using the modulus of fractional edge covers.

AdrianaKMGSC_Talk.pdf

A Property Shared by the Class of Schr¨odinger Operators

by Ryan Frier

In this talk we will discuss the class of Schrodinger operators iu_t − |∇|^αu = 0 with initial conditions u(x, 0) = f(x) ∈ L^2 . Here we are only considering α to be a positive, and at some point we will have to restrict α even more. A brief discussion on the regularity and sharpness of the regularity will be provided, and then we will discuss rapid L^2 decay of the bilinear estimate, which in turn implies a bootstrapping bound, that leads us to the fact that extremizers must be analytic.

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