Joshua Cooper -- Graph Pressing Sequences and Binary Matrix Algebra

One can construct a useful metric on genome sequences by computing minimal-length sortings of (signed) permutations by reversals. Hannenhalli and Pevzner famously showed that such sorting sequences are essentially equivalent to a certain sequences of operations -- ``vertex pressing'' -- on bicolored (aka loopy) graphs. We examine the matrix algebra over GF(2) that arises from the theory of such sequences, providing a collection of equivalent conditions for their existence and showing how linear algebra, poset theory, and group theory can be used to study them. We discuss enumeration, characterization, and recognition of uniquely pressable graphs (those with exactly one pressing sequence); a relation on pressing sequences that has a surprisingly diverse set of characterizations; and some open problems.

Keller VandeBogert -- Some Fun with The Basel Problem

The Basel Problem is a classic problem in mathematics, which asks one to compute the infinite sum of the reciprocals of the squares. It is well known what the answer is, and the problem was solved almost 300 years ago by Euler. In this talk, I want to give 3 brief proofs of the Basel problem: the first is the original proof employed by Euler himself, the second using Fourier series, and the third is completely out of left field. All of the methods technically require nothing more than standard calculus, but are still exceedingly clever.

Aditya Kiran -- An Optimization-Based Discontinuous-Galerkin Approach for High-Order Accurate Shock Tracking

Shock waves are ubiquitous in many applications such as multiphase, transonic and supersonic flows. Here, we present a novel discontinuity-tracking framework for resolving discontinuous solutions of conservation laws with high-order discontinuous-Galerkin methods. The method aims to align the inter-element boundaries with discontinuities by deforming the computational mesh. A discontinuity-aligned mesh ensures the discontinuity is represented through inter-element jumps while smooth basis functions interior to elements are only used to approximate smooth regions of the solution, thereby avoiding Gibbs’ phenomena that create well-known stability issues. The method recasts the conservation law as a PDE-constrained optimization problem that simultaneously solves the conservation law and aligns the mesh with discontinuities. We demonstrate optimal $O(h^{p+1})$ convergence rates. This is the work from my summer at the Berkeley lab. (P.S: There will be a lot of pictures)

Tracy Huggins -- Locales and Non-Spatial Geometries

Certain results in Non-Commutative Geometry suggest a need for geometric spaces that don't have any points. I will give an introduction to the theory of locales and give an example of such a "non-spatial" geometry.

Xiangcheng Zheng -- A spectral method for the boundary value problem of fractional diffusion equation

In this presentation we consider the approximation of a variable coefficient two-sided fractional diffusion equation (FDE) with homogeneous boundary conditions, having unknown $u(x)$. By introducing an intermediate unknown, $q(x)$, the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of $q(x)$, $q_N(x)$. The approximate solution to $u(x)$, $u_N(x)$, is obtained by post processing $q_N(x)$. An error analysis is given for $q(x) - q_N(x)$ and $u(x) - u_N(x)$. Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates.

Jacob Juillerat-- Some Fun with Newton Polygons

Newton polygons are a tool used to derive information about the factors of a polynomial. I will give an introduction to Newton polygons and a nice example that correlates to an attempt to better grasp the vanishing properties of the Fourier coefficients of powers of the Dedekind eta function.

John Burkardt -- The Shattered Urn

Sudoku, Instant Insanity, Tangrams, the Soma Cube, Pentomino Tiling and even logic puzzles like "Who Owns the Zebra" can all be thought of as tasks in which a "shattered" or disassembled object needs to be reconstructed from a collection of pieces. Whether you are fixing a broken Greek urn, or solving a puzzle, a standard technique involves backtracking, that is, making a series of guesses until you hit a dead end, and then backing up to the last choice you made and trying the next one. This is a steady and sure procedure, but can be slow, and doesn't provide much insight into the problem. Many of these problems can instead, almost magically, be turned into the task of solving an underdetermined linear system, something we know a lot about. I will concentrate on the particular case of tiling a region with polyominoes.

Patrick McFaddin -- Toric varieties and their derived categories

Toric varieties (defined over the complex numbers) have proved to be extremely useful test objects for various algebro-geometric questions, as many computations of interest may be phrased entirely in terms of combinatorial data, e.g., fans, cones, polytopes. For arithmetic questions, e.g., existence of rational points, one appropriately considers arithmetic toric varieties. A question of Esnault asks if the derived category, a powerful cohomological invariant, can detect rational points, and we put forth a means of studying derived categories of arithmetic toric varieties via exceptional collections. Such collections provide an analogue to (semi-)orthonormal bases of an inner-product space. In this talk, we will present various examples of arithmetic toric varieties and discuss an effective Galois descent result for exceptional collections together with applications.

Alicia Lamarche -- Abstract Nonsense

We will discuss basic category theory, the Yoneda perspective, and why we care about these things.