# Undergraduate Colloquium

**Spring 2017**

**Wednesdays 12:55 - 1:45**

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- January 11 No Talk

- January 18 Andrejs Treibergs
**Fractals, Self-Similarity and Hausdorff Dimension**

*Abstract:*Fractals are sets with fractional dimension and self-similar sets are those which are similar to a proper subset. An example of a fractal is the von Koch curve in the picture which is similar to the cyan subset. We shall show how to construct some self-similar fractals using iterated function systems. We shall discuss Hausdorff dimension, and how it can be estimated for such fractals. The von Koch curve turns out to have Hausdorff dimension 1.262.- January 25 Aaron Bertram
**Tropical Mathematics**

*Abstract:*Tropical arithmetic is "max-plus" arithmetic, in which the maximum of two numbers plays the role of addition and the sum of two numbers plays the role of multiplication. This satisfies all the properties of ordinary arithmetic except for one: there is no tropical subtraction. Nevertheless, we can look at tropical polynomials in one and more variables, and a sort of miracle happens. Even though they have no roots, tropical polynomials nevertheless can always be factored (the tropical quadratic formula is a piece of cake) and the "tropical curves" in the plane associated to polynomials in two variables (lines, conics, etc.) always intersect in the right number of points. What happens in higher dimensions? Well, it's complicated....- February 1 Peter Alfeld
**What can you do with a slide rule?**

*Abstract:*Back in the days when people first went to the moon, electronic calculators did not exist. Instead we used slide rules. They were indispensable for professionals, and students were required to own one and know how to use it. High Schools and Universities offered courses on the proper use of a slide rule. Just like calculators today, slide rules were mostly every day and commonplace instruments, but some were fancy, expensive, and treasured by their owner.

I'll describe how slide rules work, why they work, and what you can do with them. A typical slide rule has anywhere from ten to thirty scales, rather than just two, and there are thousands of mathematical expressions that you can evaluate just as easily as you can multiply or divide two numbers. On the other hand, you can't use a slide rule to add or subtract two numbers, and you need to understand your problem well enough to be able to figure out on your own the location of the decimal point in your answer.

You'll be able to examine several slide rules, and I'll tell you what's involved in being a slide rule collector.- February 8 Julia Inozemtseva
**Mathematical Models of Interactions between Species: Peaceful Co-existence of Vampires and Humans**

*Abstract:*Using differential equations to model interactions between species, we will enjoy imagining how our society would live with vampires among us. Being a part (not the pleasant one) of the predator-prey relationship, we want to know what conditions the human population would need to survive or even fight back when vampires multiply and attack. Do we actually need a vampire-slayer to protect us? It appears that several popular culture sources outlining the models describe plausible and peaceful vampire and human co-existence.

Slides are available: Mathematical Models of Interactions between Species- February 15 Daniel Smolkin
**The History of the Normal Distribution**

*Abstract:*The Normal Distribution, sometimes called the Bell Curve or the Gaussian distribution, is one of the most recognized curves in math and science. Heights, IQ scores, errors in measurements, and stock prices have all been assumed to be Normally distributed, sometimes with disastrous consequences. This talk will discuss the first derivation of the Normal distribution, which was not by Gauss, itâ€™s rediscovery decades later, and its rise to ubiquity.- February 22 Eric Bloomquist
**Navigating the Job/Internship Search Process**

*Abstract:*Do you have plans for after the semester wraps up in May? If you are graduating and haven't started your full-time job search, now is the time! If you aren't graduating (or are planning on grad school) and want to spend the summer doing something productive for your career, time to start planning! During this session, Eric Bloomquist--a Career Coach at the U--will walk through the typical job search process from sourcing potential leads all the way through acing the interview. Come prepared with questions you may have about any part of the process, from resumes, cover letters, interviews, LinkedIn, networking, salary negotiation.- March 1 Rex Butler
**Levenshtein Distance and Auto-Correct**

*Abstract:*Suppose one is helping create a mathematics app that lets anyone vote for their favorite mathematician. The options given are Euclid, Newton, Euler, and Gauss. Could one write an auto-correct algorithm that could guess what a user meant if they mistyped their answer? For example, could one write an algorithm to associate "Nwton" with Newton, "Eulker" with "Euler", and "Gavss" with "Gauss" on a mathematical basis?

In this context, we would like to use something like a 'closest' match, where the notion of closeness or similarity is given by distance defined between finite sequences of symbols. One such distance is called the Levenshtein distance, the basics of which we will cover in this talk.- March 8 Matt Cecil
**Some Irrational Thoughts about $\pi$ and $e$.**

*Abstract:*$\pi$ and $e$ show up in just about every math course. $\pi$ even has its own day on the calendar! They are both irrational numbers and hence have a non-repeating decimal expansion. In this talk, I will discuss how you might find their digits by using approximations derived from power series. I will also prove that they are irrational. This talk should be accessible to anyone who has taken or is currently taking Calculus II.- March 15 No Talk - Spring Break

- March 22 Weicong Su
**Two Equivalent Definitions of the Exponential Function**

*Abstract:*Two common definitions of the exponential function exp(x) are known to many, namely, the limit definition and the power series definition. Ever wonder why these two seemingly different forms of expressions give the same exponential function? In this talk, the speaker will investigate the origin of the exponential function from a theoretical point of view. We shall see proofs that both definitions are well-defined and a proof of the equivalence of the two definitions based on some elementary algebraic and analytic techniques. If you love mathematical reasoning, especially the theory of analysis, then this is the talk for you.- March 29 Bhargav Karamched
**The Princess Problem**

*Abstract:*The princess problem (also known as the secretary problem) is a famous problem in optimal stopping theory. Suppose that a princess is visited by n suitors. What strategy should she use to pick the best suitor, if she cannot bring back suitors she has already rejected? In this talk, we will derive the optimal strategy for the princess problem, then discuss variants of the problem and extensions.- April 5 Arjun Krishnan
**Prime Numbers, Combinatorics, Physics, and Probability**

*Abstract:*Understanding why certain types of randomness ---probability distributions--- appear again and again in seemingly unrelated contexts is a central theme in probability theory. For example, understanding the universality of the bell-curve or normal distribution is a classical achievement of probability that goes back to DeMoivre and Laplace. Over the last thirty years, a rather mysterious new class of universal distributions that are in some sense diametrically opposite to the normal distribution have emerged. I will talk about their appearance in the zeros of the zeta function, combinatorics, neutron scattering, and crystalline growth.- April 12 Tom Alberts
**Branching Processes**

*Abstract:*A branching process models the changes in a population level in which each individual produces a random number of offspring. They are a very simple but important part of probability theory and can be used to model reproduction within a bacteria colony, the spread of surnames in genealogy, or the propagation of neutron collisions in an atomic bomb. This talk will go over the basic models of branching processes and some interesting variants and then describe the beautiful mathematics behind one of the most important questions in the subject: what is the probability that the population ultimately goes extinct?- April 13 Sir Randolph Bacon, cousin-in-law to Colin Adams
**Special Colloquium, 4:00 PM, JWB 335**

**Blown Away: What Knot to Do When Sailing**

*Abstract:*Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed!

Please join us as professor Colin Adams of Williams College assumes the role of Sir Randolph Bacon III to teach the mathematical theory of knots. This talk is open to a general audience.- April 19 Dylan Zwick
**Real Numbers - The Work of the Devil**

*Abstract:*"God created the integers, all else in the work of man" - Leopold Kronecker

In this talk we'll take a whirlwind tour of the work of man, beginning with the natural numbers, and building to the reals. Much is swept under the rug in the construction of the real numbers, including some very interesting, even troubling, questions that we'll examine. After this talk, you'll never think about the real numbers in quite the same way.- April 26 No Talk - Reading Day