In this talk, we present an application of the method of brackets for definite integrals involving Bessel functions. This is a procedure for the evaluation of definite integrals over the positive real line based on the expansion of the integrand as a power series. The primary object of this method is the so-called bracket series, which is produced and evaluated according to a small number of rules.

Bessel functions are a family of functions with important applications in the study of wave propagation. These functions possess several properties and a power series representation which is suitable for the application of the method of brackets.

We will illustrate the successful application of the method through several examples, based on problems taken from the Table of Integrals, Series, and Products.

Chromatic polynomials are nice polynomials counting the number of ways one can color a graph with n colors. We will see some nice properties of chromatic polynomials and that they are special cases of a bigger family of polynomials.

Compressed sensing is a technique from signal processing which allows signals to be recovered with fewer samples than required by the Nyquist-Shanon theorem by solving an undetermined system of linear equations. Clearly such an undetermined system of linear equations cannot be arbitrary. In this talk, we will learn about what conditions under which the recovery of a signal is possible and about applications of compressed sensing. 

Rotation synchronization is a problem that occurs in 3D reconstruction problems. Given a set of relative rotations between pairs of images, which are modeled in a graph with the nodes as images and edges as 3D rotation matrices that may be noisy and corrupted, the problem is to find the correct absolute rotations for each image upto a global rotation. Our method builds upon cycle-edge message passing (CEMP), which gives estimates of the corruption or error levels of each relative rotation and then uses a minimum spanning tree (MST) with the error levels as the edge weights to get the absolute rotations. In our methods, we remove some edges of the graph based on certain criteria involving the error level of an edge relative to that of its neighbors, and then use CEMP+MST. We use two separate criteria in our work. We look at the accuracy of both methods compared to that of CEMP+MST on the whole graph for real image data. While our methods do not seem to consistently produce more accurate solutions, some ideas are proposed to pick better subgraphs for more accurate solutions.

The Poisson and Dirichlet problems are cornerstones of partial differential equations and harmonic analysis and have interested mathematicians and physicians alike for the last two centuries. In this study, we aim to connect the two problems and analyze their solutions in the upper half-space. Specifically, the Poisson problem can be reformulated as a linear superposition of two explicitly solvable problems, a Dirichlet-type problem, and the free-space Poisson equation. Using these results and the Carleson duality inequality, we provide estimates for the Poisson solution in terms of the initial data. Some ideas are also provided on how to elaborate these estimates to more complex domains.

Lie Algebras are important to math, but can be extremely confusing. In this talk, we will explain Lie Algebras as well as the topics required to understand them, including Manifolds, Vector Fields, and Lie Groups.

In this talk I give a brief intro to category theory, before discussing the current scope of computer implementations of categories. Then I will discuss a sample computational problem and go over the algorithms that I've developed to solve it.

In the early 20th century, mathematicians developed simplicial cohomology to classify topological spaces. In 1945, Gerhard Hochschild developed Hochschild cohomology to classify the deformations of algebras. Although the two cohomologies are developed from (figuratively) distinct fields, they display cochain-isomorphism when we narrow our scope to posets.

Along the way reaching this conclusion, we will briefly cover simplicial cohomology, Hochschild cohomology, and some concepts that are only tangible but worth mentioning.

LP algebras are a generalization of cluster algebras, first defined by Lam and Pylyavskyy in 2015. In this talk we will see a particularly nice class of LP Algebras called Graph LP Algebras along with some interesting conjectures about them. 

In this talk we present an application of the method of brackets for definite integrals involving Bessel functions. This is a procedure for the evaluation of definite integrals over the half line [0, ∞) based on the expansion of the integrand as a power series. The primary object in this method is the so-called bracket series, which is produced and evaluated according to a small number of rules.

Bessel functions are a family of functions with important applications in the study of wave propagation. These functions possess a number of properties and a power series representation which is suitable for the application of the method of brackets.

We will illustrate the successful application of the method through several examples, based on problems taken from Table of Integrals, Series, and Products. Also, we will present an intriguing example where the method does not lead directly to the value of the integral, but we propose a solution for this complication.

In this talk, we will begin by discussing Kazhdan-Lusztig polynomials, which are associated with a certain algebra on symmetric groups (or Coxeter groups more generally), called a Hecke algebra. Although these polynomials can be computed recursively by computers, the formulas are not suitable for humans, and the coefficients have no intuitive interpretation in general. In the special case of 321 pattern-avoiding permutations, however, we can compute Kazhdan-Lusztig polynomials in a quotient space called a Temperley-Lieb algebra in a straightforward way. Refinements to this method from my research will be presented at the end.

The Bergman projection is one of the richest and most interesting objects in multivariable complex analysis and is well-understood in many simpler domains. In this presentation, using tools such as Schur’s Test and Bell’s transformation formula, we study the boundedness of Toeplitz operators, which are weighted Bergman projections. In particular, we shall analyze the Bergman kernel on the symmetrized bidisc and obtain estimates for Toeplitz operators for symbols representing the distance to the boundary.

An introduction to symmetric functions.

A computer scientist student infiltrated URMS! I will show examples of computations using attribute grammars, their formalism, and a few properties time permitting. I'll also show off an application in translators and (the very recently developed) code prober.