In this talk, I'll introduce a family of related problems in metric geometry, the most famous of which is the Analyst's Traveling Salesman Problem. I'll highlight work on this topic by graduate students and postdocs at UConn. Most importantly, I'll spend time talking about open problems and possible directions for research.

In this expository talk, we will begin by defining elliptic curves, and stating the Mordell-Weil theorem for elliptic curves over number fields. We will then give a (limited) survey of the study of elliptic curves. Topics will include N-division fields, Galois representations, and the theory of complex multiplication. We will finish with a brief overview of ongoing research into Galois representations attached to elliptic curves with complex multiplication.

Statements such as "the essence of mathematics lies in proofs" (Ross, 1998) suggest the centrality of proof within mathematical practice. Due to its importance within the discipline, educators emphasize the importance of learning and instruction of proof. The National Council of Teachers of Mathematics lists "proof and reasoning" as one of the five process standards in K-12 mathematics instruction. Similarly, the Mathematical Association of America states that mathematics majors should be proficient at proof as they progress through the major. Despite this, there has been little discussion on the role and scope of proof in early undergraduate courses, most notably the calculus sequence. In this talk, I will propose a study that aims to understand mathematicians' perspectives towards proof in this setting.

Cryptography is the practice and study of techniques used for secure communication in the presence of adverse third parties. Although cryptography is of interest to computer scientists, there is a lot of mathematics being done behind the scenes. In this talk, we will explain the difference between private key cryptography and public key cryptography. We will also talk about some computationally hard math problems that are used in public key cryptography.

In this talk, we will give an introduction to the multigrid methods. The multigrid method is an optimal solver which was first proposed for solving elliptic boundary value problems. We will discuss the history of the multigrid methods, the advantages of using multigrid methods, the basic ideas behind multigrid methods and the future of multigrid methods.

The rational numbers have infinitely many completions to a locally compact field. The real numbers are the most familiar example. Each of the others is associated to a prime number \(p\) and is called the \(p\)-adic numbers. Inside the \(p\)-adic numbers, the integers are a bounded subset without isolated points. This talk, which is a preview of a course that will be taught in the fall, will introduce the \(p\)-adic numbers and use analysis (power series) on them to solve two problems whose statements don't mention \(p\)-adic numbers at all.

Trisecting angles, doubling cubes, and squaring circles using only a straight-edge and compass are three geometric constructions from ancient Greece that were first solved in the 1800s. This talk, in two parts, will explain the solutions by translating each question from geometry to algebra (the perspective of field extensions) and then we'll see a modern twist to these problems by including paper-folding in our toolbox.

The UDL (Universal Design for Learning) Framework is used to improve and optimize teaching and learning for all of the diverse learners that we encounter in our classes. In this talk, the fundamentals of this framework will be introduced along with an example of how it was implemented to reimagine MATH 3710(Introduction to Mathematical Modeling) to create a modular, inquiry-based course structure here at UConn. The basic process of research in math education will also be highlighted.

One of the major concerns of the study of reverse mathematics is determining the axiomatic strength of theorems. The focus of this talk is to discuss the axiomatic strength of a particular set of principles,
namely, what will be referred to as “maximal subfamily principles,” which assert the existence of a
maximal subfamily for which a particular \(\Sigma_0^1\) property holds for any pair of sets in the family. It has
already been shown that the addition of the 2 Intersection Principle (which states that every nontrivial family of sets has a maximal subset for which the intersection of each pair of sets is nonempty) to RCA\(_0\) (the subsystem of computable mathematics) is strictly stronger than RCA\(_0\) on its own. Since the 2 Intersection Principle (2IP) is itself a maximal subfamily principle defined by a \(\Sigma_0^1\) property, a natural question that arises is whether there is a principle of this type with a strength strictly above or between RCA\(_0\) and 2IP or if the strength of each maximal subfamily principle of this type is equal to either that
of RCA\(_0\) or 2IP.

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