In calculus courses you spend a lot of timing learning how to use and compute Riemann integrals, but it turns out the Riemann integral has some *bad* properties. Over 100 years ago it was replaced in much of mathematics with something else: Lebesgue integrals, which have *very good* properties.

In this talk, we'll see what is bad about Riemann integrals, what Lebesgue integrals are, and why they are considered superior.

To understand this talk, you should know (besides calculus) the difference between countable and uncountable sets.


Note: Free refreshments. The talk starts at 5:40.

Many differential equations can be nearly impossible to solve directly. This naturally leads to the question: is there a way to approximate the solution to a differential equation numerically? The Finite Element Method (FEM) is one such method that is used very often in the real world.

In this talk, we will introduce a boundary value problem (an important type of differential equation) and discuss the existence and uniqueness of its finite element solution. Additionally, we will introduce the finite element space and its nodal basis functions. FEM Matlab code will be presented near the end. 

Some familiarity with linear algebra is recommended but not necessary.

Note: Free refreshments. The talk starts at 5:40.

For 2000 years, the only type of geometry that anyone could imagine was the one resembling our everyday experience, with its exposition by the ancient Greeks in Euclid's "Elements" becoming the model for rigorous reasoning.

In the 1800s, several mathematicians independently discovered new models for geometry where some familiar properties of Euclidean geometry are no longer true. These
non-Euclidean geometries helped mathematicians realize that the scope of mathematics does not have to be tied only to concepts resembling something in the physical world.


In this meeting we'll solve some problems together about hyperbolic geometry in 1 and 2 dimensions.



Note: Free refreshments.

For a positive integer \(d\), let \(p_d(n) := 1^d + 2^d + \cdots + n^d,\) i.e., \(p_d(n)\) is the sum of the first \(n\) \(d\)th powers for \(n \geq 1\). For example, \(p_1(n) = n(n+1)/2\) and \(p_2(n) = n(n+1)(2n+1)/6\).

For each \(d\), \(p_d(n)\) is a polynomial in \(n\) of degree \(d+1\). This can be proved by induction on \(d\), but the inductive step can be a challenge. In this talk, we will prove \(p_d(n)\) is a polynomial in \(n\) of degree \(d+1\) by a different method, using L’Hospital’s rule.

Note: Free refreshments. The talk starts at 5:40.

A sum is called telescoping when its successive terms cancel out except for the first and last term. This is met in calculus courses as a trick for evaluating a small number of series, but writing something as a telescoping sum can be used in a variety of interesting ways.

In this meeting we'll solve some problems together using telescoping sums.



Note: Free refreshments.

Many people think of cryptography in the context of spies, but today cryptography is used everyday by almost everyone to encrypt everything from bank transactions to text messages. The talk will be a historical and mathematical tour of cryptography, as well as be a promotional talk for a course (MATH 3094) on the mathematics of encryption that the speaker will teach next semester.


Note: Free refreshments. The talk starts at 5:40.

A graph is a set of nodes (points) connected by edges (paths). They can be used to describe many kinds of networks: travel routes between cities, people who follow each other on social media, and communication between different parts of the brain. Graph theory has many real-life applications, such as GPS travel directions, search engine optimization, flight scheduling by airlines, and genome structure.

In this meeting we'll work together on some problems in graph theory.


Note: Free refreshments.

The study of elliptic curves is a deep and rich field of research, whose most famous application is to the proof of Fermat’s Last Theorem. In this talk, we will discuss several problems that can be formulated in terms of elliptic curves, including Fermat’s Last Theorem. We will then discuss how the theory of elliptic curves helps us to solve these problems, and many others.

Note: Free refreshments. The talk starts at 5:40.

Fermat's Last Theorem says the equation \(x^n + y^n = z^n\) has no solution in positive integers when \(n \geq 3\).
This problem was posed by Fermat in the 1600s and a proof of it was announced by Andrew Wiles 30 years ago, on June 23, 1993. This news made the top of the front page of the New York Times, which is quite unusual for an article on mathematics.

In this meeting, we'll watch a documentary about this problem that features Wiles, his colleagues, and other mathematicians whose work played a role in the resolution of Fermat's Last Theorem.


Before the documentary there will be a brief presentation about the link between Fermat's Last Theorem and
elliptic curves, which was the topic of last week's math club meeting.


Note: Free refreshments.

TBA

Note: Free refreshments. The talk starts at 5:40.

If you are considering graduate school in mathematics or related areas after college, come to this panel discussion where you will hear from members of the UConn math department about their experiences planning for and applying to graduate school. The discussion will then be opened to answer your questions. A packet containing a suggested reading list and some general advice will be provided too.


Note: Free refreshments. The event starts at 5:40.