Math Club

Calculus courses present the derivatives of all the basic trigonometric functions, but they present the power series of sin \(x\) and cos \(x\) while ignoring the power series of the other trigonometric functions. Why is that?

In this talk we will discuss the power series of tan \(x\), whose coefficients turn out to have connections to the binomial theorem and combinatorics.

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

The two main techniques of integration taught in calculus courses are integration by substitution and integration by parts. This talk will describe and illustrate a third technique of integration, almost never taught in math courses, called differentiation under the integral sign. It can handle integrals that appear inaccessible to simpler methods. The physicist Richard Feynman had great affection for differentiation under the integral sign, writing once “I caught on how to use that method, and I used that one damn tool again and again.”

Some familiarity with calculating partial derivatives from multivariable calculus will be assumed.

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

A random infinite graph is formed by starting with a countably infinite set of vertices and then flipping a coin for each pair of vertices to determine whether or not to place an edge between those vertices. This talk is about the kinds of graphs you get by this process and it will illustrate a nifty connection between probability, graph theory, and logic. 

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

The factorials \(n! = 1 \cdot 2 \cdot 3 \cdots n\), which count how many ways \(n\) objects can be arranged, show up anywhere that rearrangements have to be counted, such as combinatorics, probability, thermodynamics, statistical mechanics, and quantum mechanics. The numbers \(n!\) grow very rapidly, e.g., 100! has 158 digits. For applications in chemistry, \(n!\) may occur for \(n\) on the order of Avogadro’s number (about \(6.02 \times 10^\)), for which an exact factorial calculation is out of the question. When exact values are computationally inaccessible, it’s natural to seek approximations to the values.

Stirling’s formula is the standard way to estimate \(n!\) when \(n\) is large. In this talk we will see what Stirling’s formula is and how to derive it using tools from calculus.

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

A new mathematical description of a physical theory should be consistent with an earlier description of the theory in settings where the earlier one has been well-tested by experiments. This often means that the earlier mathematical description is a limiting case of the new one at certain parameter values or that the new description is a “deformation” of the old one.

As a prototypical example, the ancient idea of a flat Earth is a limiting case of the  more accurate sphere model when we look at a sphere up close. We also can interpret quantum mechanics as a deformation of classical mechanics via a mathematical technique called “deformation quantization.” Here, quantum mechanics involves Planck’s constant \(\hbar\) and its limiting behavior as \(\hbar \to 0\) leads us back to classical mechanics. This talk will give an introduction to what deformation quantization is all about.

The talk will assume the audience has seen partial derivatives and eigenvalues.

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

Composition of functions usually depends on the order in which it is done: $f(g(x))$ usually is not \(g(f(x))\). Sometimes, however,
\(f(g(x)) = g(f(x))\) for all \(x\). We then say \(f(x)\) and \(g(x)\) commute.

In this talk, we will see how polynomials can satisfy \(f(g(x)) = g(f(x))\). It turns out that there are only two interesting ways this can happen, and one way has an unexpected relationship to trigonometry.

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

The math club is hosting a panel discussion on undergraduate mathematics research opportunities. The panelists will discuss how to become involved in mathematics research as an undergraduate and what the research process is like.

A famous summation identity is \(1 + 2 + \cdots + n = n(n+1)/2\) when \(n\) is a positive integer. In this talk we will see many ways to explain this, some of which can be extended to work out a formula for \(1^2 + 2^2 + \cdots + n^2\) and higher power sums.

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

The calculus of finite differences is an analogue of ordinary calculus for sequences instead of functions. It is used in both pure and applied math, providing theorems and methods for evaluating infinite sums based on discrete analogues of derivatives, integrals, and the Fundamental Theorem of Calculus.

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

At this meeting, Prof. Conrad will answer any question you have about math (except homework questions). This is a chance to find out more about any problems, concepts, examples, historical events, etc. in math that you’ve heard or read about but don’t understand as well as you’d like.

Have you ever seen a mosquito and wanted to shoo it away? Certainly, if you used a nuclear bomb, this would solve the issue, despite there being many simpler ways to approach the problem. We will discuss mathematical equivalents of this: ways to prove elementary facts using overly complicated methods.

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