Title:    Non-Vanishing Homology of the Matching Complex

Abstract:     A matching on a graph G is any subgraph where the maximum vertex degree is 1.  Since edge-deletion preserves the property of being a matching, the set of all matchings on G forms a simplicial complex M(G).  We will survey results on the lowest non-vanishing homology group for M(K_n) and, time permitting, discuss the extension of these results to more general graphs.  ​This is a practice talk for my qualifying exam, but no prior familiarity with simplicial complexes nor homology is assumed.

Pizza at 4:00 p.m., talk at 4:15 p.m.

Title:  Derangements, discrete Morse theory, and the homology of the boolean complex

Abstract:  The boolean complex is a construction associated to finite simple graphs. We summarize a matching which shows that this complex is homotopy equivalent to a wedge of spheres, and the number of these spheres is related to the boolean number, a graph invariant. To better understand this structure, we use a correspondence between derangements and basis elements and compute the homology of the boolean complex for several specific examples. A basic knowledge of discrete Morse theory may be helpful but is not necessary.

Pizza at 4:00 p.m., talk at 4:15 p.m.

Title:  An Introduction to Differential Forms

Abstract:  The theory of differential forms is a beautiful subject that has particular importance in geometry, topology and physics.  Our focus will be on how differential forms provide a better approach to multi-variable calculus.  Specifically, we will see that many ostensibly unrelated results are unified using differential forms.  Time permitted, we will introduce de Rham cohomology, a cohomology theory based on differential forms.

Title:  Gauss's Golden Theorem (Quadratic reciprocity)

Abstract:  For which odd primes is the congruence $x^2\equiv 0 \mod p$ soluble? The quadratic reciprocity is a marvelous theorem in number theory that deals with this question. We will present examples and a proof of the quadratic residue law (Time permitting).

Title:  Coding Theory and Subspace codes

Abstract:  In this talk I will give a basic introduction into the ideas of coding theory and subspace codes. Then I will give a few examples of constructions of subspace codes and show a theorem which can be used to link these codes. I will end on the idea of decoding particularly the linkage construction. This should be an introductory talk, no previous coding theory knowledge required.

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.