All are welcome to come mingle over coffe, tea, and cookies.

Title:  Palindromes, Lychrel Numbers, and the 196-Conjecture

Abstract:  A palindrome is any number that is the same when written backwards such as 123321 or 595. In this talk we’ll examine the reversal-addition algorithm for producing palindromes and discuss whether any natural number will produce a palindrome under the reversal-addition algorithm. In particular we will talk about the 196-Conjecture which is that 196 will never produce a palindrome under the reversal-addition algorithm. Finally we will look at a couple of ways to modify the reversal-addition algorithm to (possibly) make it so that any natural number will produce a palindrome

Title:  r-Stable Hypersimplices

Abstract:  The n,k-hypersimplices are a well-studied collection of polytopes.  Inside each n,k-hypersimplex we can define a finite nesting of subpolytopes that we call the r-stable n,k-hypersimplices.  In this talk, we will define the r-stable hypersimplices and then see that they share a nice geometric relationship via a well-known regular unimodular triangulation of the n,k-hypersimplex in which they live.  Using this relationship, we will then identify some geometric and combinatorial properties of the r-stable hypersimplices.  In doing so, we will see that a number of the properties of the n,k-hypersimplex also hold for the r-stable hypersimplices within.

Title:  Chromatic Levels in the Homotopy Groups of Spheres

Abstract:  Understanding the homotopy groups of spheres $\pi_nS^k$ is one of the great challenges of algebraic topology. One of the fundamental theorems in this field is the Freudenthal suspension theorem. It states that $\pi_{n+k}S^k$ is isomorphic to $\pi_{n+k+1}S^{k+1} $ when $k$ is large. Homotopy theorists call this phenomena \emph{stabilization}. The stable homotopy groups of spheres are defined to be these families of isomorphic groups. They form a ring, commonly denoted by $\pi_*S$. Despite its simple definition, this ring is extremely complex; there is no hope of computing it completely. However, it carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the \emph{chromatic layers}. There are many structural conjectures about the chromatic filtration. In this talk, I will give an overview of chromatic theory and talk about one of the structural conjectures, the \emph{chromatic splitting conjecture}.

Title:  Chip-Firing and Tropical Independence

Abstract:  We will discuss the basic theory of divisors on graphs, with a primary focus on concrete examples.  If time permits, we will describe how these tools are used to provide new proofs of some well-known theorems in algebraic geometry.

The iridates have become fertile ground for studies of new physics driven by spin-orbit coupling (SOC) that is comparable to the on-site Coulomb and other relevant  interactions.  This unique circumstance creates a delicate balance between interactions that drives complex magnetic and dielectric behaviors and exotic states seldom or never seen in other materials. A profound manifestation of this competition is the novel Jeff = 1/2 Mott state that was observed in the layered iridates with tetravalent Ir4+(5d5) ions. On the other hand, very little attention has been drawn to iridates having pentavalent Ir5+(5d4) ions, primarily because the strong SOC limit is expected to impose a nonmagnetic singlet ground state (Jeff  = 0). In this talk, we review the underlying physical properties of the iridates including perovskites, honeycomb lattices and double perovskites with pentavalent Ir5+ ions, and report results of our recent studies that emphasize spin-orbit-tuned ground states stabilized by chemical doping, application of pressure and magnetic field. In addition, we address the urgent question that the Jeff states may not survive in the presence of strong non-cubic crystal fields and/or exchange interactions.

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Cumulative Exam Review

Course Instructor: Dr. Anne-Frances Miller

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Topics will be announced at least one week in advance.  Check back soon for more details.

Course Instructor: Dr. Anne-Frances Miller