Title: A result by Davenport and Lewis on additive equations.
Abstract: We will present a result by Davenport and Lewis which states that an additive form with coefficients in $\mathbb{Q}_p$ of degree $d$ in $s>d^2$ variables has a non- trivial $p$-adic solution. No knowledge of the $p$-adics is necessary.
Pizza at 4:00 p.m., Talk at 4:15 p.m.
Title: Efficient Solutions of Large Saddle-Point Systems
Abstract: Summer projects are great; here is one such project. Linear systems of saddle-point type arise in a range of applications including optimization, mixed finite-element methods for mechanics and fluid dynamics, economics, and finance (basically everywhere). Due to their indefiniteness and generally unfavorable spectral properties, such systems are difficult to solve, particularly when their dimension is very large. In some applications - for example, when simulating fluid flow over large periods of time - such systems have to be solved many times over the course of a single run, and the linear solver rapidly becomes a major bottleneck. For this reason, finding an efficient and scalable solver is of the utmost importance. In this project, we examined various solution strategies for saddle-point systems.
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: Terraces, Latin squares, and the Oberwolfach problem
Abstract: A terrace is an arrangement of the elements of a finite group in which differences between adjacent elements adhere to certain restrictions. We introduce terraces and a number of related objects, including R-terraces and directed terraces, and discuss conjectures concerning the groups for which we can construct terraces. We also consider applications of terraces to problems in the areas of combinatorial design and graph theory - namely, the construction of row-complete Latin squares and solutions to some particular cases of the Oberwolfach problem.
Title: On the flag enumeration of the subspace lattice
Abstract: We consider the q-analogue of the Boolean algebra: the lattice of subspaces of an n-dimensional vector space over the finite field of q elements. Using the quasi-symmetric function of this lattice, we can evaluate a q-analogue of the classical descent set statistic in two cases. In one case, we express the values in terms of the classical descent set statistic and find the maximal value, extending De Bruijn and Niven's results in permutation enumeration. In the other, we compute the values for certain descent sets and conjecture when the maximum is obtained. Finally, when evaluating the quasi-symmetric function using a root of unity, we obtain a version of the cyclic sieving phenomenon on the Boolean algebra, due to Reiner, Stanton and White. This talk, which is based on joint work with Richard Ehrenborg, will include abundant examples and be accessible to anyone who has a minimal knowledge of combinatorics.
Title: Galois theory over the p-adic numbers
Abstract: The inverse Galois problem asks whether every finite group appears as the Galois group of some finite Galois extension of the rational numbers. One can ask the same question for other fields. In this talk, we will discuss Galois theory over the p-adic numbers. We will see that ramification groups provide a useful tool for analyzing the structure of these Galois groups. Using ramification groups, we will find limitations on the finite groups which can occur as Galois groups over the p-adic numbers. A review of the necessary facts about the p-adic numbers will be included. This talk is meant to be accessible to anyone with a basic understanding of Galois theory over the rationals.
Title: Fixed Points in Partially Ordered Sets
Abstract: In this talk we will look at what conditions we need to impose on posets to ensure automorphisms of the poset have fixed points. We will prove, and use, a discrete version of the famous Hopf-Lefshetz fixed point theorem from topology. Examples will be emphasized.
Title: Applying Bernstein's Theorem to Matrices
Abstract: Bernstein's Theorem is a theorem that bounds the best approximation error of a smooth function in terms of an ellipse where the function is analytic in its interior. In this talk, we will review this theorem and apply it to symmetric banded matrices to show that the entries of the resulting matrix (after a smooth function is applied) are bounded in an exponentially decaying manner away from the main diagonal.
Title: Strings of composite numbers
Abstract: Given d and e in {1,...,9} we consider the set of positive integers k that have the following property: k with d appended on the right n times is composite for all positive integers n. We also consider the set of positive integers k that have the following property: k with d and e appended alternately on the left and right (i.e. k,kd,ekd,ekdd...) is always composite regardless of the number of appended digits. We will discuss methods for finding elements in these sets and also consider finding their minimum elements.
Title: Another Friendly Talk About Cobordism
Abstract: In this talk I will present some basic ideas about cobordism. In particular, we will discuss manifolds, an equivalence relation, and some rings.
Haven't had Topology I yet, you say? This talk is still for you. You'll be fine.
