## Academic Year 2017 - 2018

### Spring 2018

### Fall 2017

No Talk - First Week of Classes

**Cool Mathematics**

*Abstract: *I will in fact talk about some unsolved, or recently solved, problems in Mathematics.
The main purpose of the meeting, however, will be to organize the Undergraduate Colloquium
for those interested in taking it for credit (1 hour credit/no credit). For many participants
this will be the first class in which they have to write a technical report. This
is a complicated yet gratifying task. I will discuss some of the issues involved and
also give a first introduction to the use of LaTeX.

**Hotel Infinity**

*Abstract:* You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full.
A new guest arrives and insists you give her a room. How do you accommodate her? The
next day, a family with infinitely many members arrives, each of whom wants a private
room. The next day infinitely many families, each with infinitely many members, arrive.
Each family member insists on a private room. You can do it! Infinity is different.

**Pseudo-random numbers: {mostly} a line {of code} at a time**

*Abstract:* Random numbers have an amazing range of application in both theory and practice.
Approximately-random numbers generated on a computer are called pseudo-random. This
talk discusses how one generates and tests such numbers, and shows how this study
is related to important mathematics and statistics - the Central-Limit Theorem and
the Χ2 measure - that have broad applications in many fields. Come and find out what
the Birthday Paradox, Diehard batteries, gorillas, Euclid, French soldiers, a Persian
mathematician, Prussian cavalry, and Queen Mary have to do with random numbers.

**The Angel Problem and other Games of No Chance**

*Abstract:* I'll discuss John Conway's "Angel Problem," a simple sounding game for two players (the angel and the devil). The game was introduced in
1982, and the problem of finding a winning strategy remained unsolved until 2005.
We'll talk about several variants of the game, and how these can be solved. Finally,
we'll discuss one or two other games that fall under the umbrella of combinatorial
game theory.

Casey Johnson (Math Department Alumnus)

**Student Opportunities at Department of Defense (with a little bit of math)**

*Abstract*: We discuss at a high level a variety of security-related positions within the U.S.
Government that may

be of interest to mathematicians. This will include professional opportunities, as
well as internships

available to undergraduates. As time permits, we will turn our attention to several
related algebra

problems and show how they can be combined to construct a primitive computer.

**Complex and Tropical Nullstellensatze**

*Abstract*: If a system of linear equations has no common solution, then by eliminating variables,
one can obtain 1 = 0 as a linear combination of the given equations. Hilbert’s *Nullstellensatz *or “zero set theorem” says that the same thing happens with polynomial equations provided
that one allows for complex solutions (and polynomial combinations). In the world
of tropical numbers we consider polynomial *relations *and then there is a result of the same nature. We’ll talk about this and also about
why the Nullstellensatz is the foundation of Algebraic Geometry.

**Big Data Employer Panel**

## Location - LCB Loft (4th floor)

*Abstract: *Want to know what it's really like to work in a job relating to math, data, and statistics?
Do you have questions about what it takes to land a position in "the real world" or
what classes you should focus on now for your future career success? Hear from former
students and get your questions answered! Food and networking to follow the panel.

**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.

**Desargues's theorem**

*Abstract*: Desargues's theorem, published in 1648, is a result in geometry about triangles
in perspective. Desargues was one of the fathers of projective geometry, and his theorem
is best understood in that setting. We'll talk about Desargue's theorem, projective
space, and how to prove the theorem.

**Representation Theory and the Hydrogen Atom**

*Abstract*:** **This talk is an advertisement for a new course that will be running in Spring 2018
called Representation Theory Techniques in Quantum Physics. The course will tell the
story of representation theory's predictive role in quantum mechanics by closely examining
the most basic example: the hydrogen atom. This talk will start by examining the history
of representation theory and exploring how representation theory found its way to
the forefront of 20th century mathematics and physics. Then it will give a preview
of the material that we will cover in Math 5750. The talk will assume a background
in linear algebra, but no background in group theory or quantum mechanics is necessary.

No Talk - Happy Thanksgiving

There will be an optional meeting for any students enrolled in Math 3000 who want to talk about the required report due at the end of the semester.

**Waves in strings**

*Abstract*: In this talk, we will study the problem of a vibrating string. How does a wave propagate
in a homogeneous string when the string is plucked? This is a standard problem in
mechanics and it has a well-known solution. So we will change some of the basic assumptions
to turn it into a very challenging problem. First, we will see how the wave propagates
when the string is made of two different materials. The key point is to understand
what happens at the space-interface between the two materials. Then we will try to
solve the following problem: what if the properties of the string suddenly change
in time? In other words, what happens if a string made of a certain material, suddenly
turns into a string made of another material? What happens at the time-interface?
How is the propagation of the wave affected by this time-inhomogeneity?

**Why can you tune a guitar, but not a piano?**

*Abstract*: Musicians rely on being able to tune their instruments to create music that sounds
pleasing and harmonious. In this talk, we'll discuss the mathematics behind the music
created by vibrating strings and how guitarists use these frequencies, called "harmonics,"
to get in tune. Exploring this topic further leads to a surprising and unsettling
result: it is impossible to tune a piano! Once we prove this, I'll explain what strategy
piano tuners use to get around it. Partial differential equations and boundary conditions
will make an appearance, but no knowledge of differential equations or music is required.

**Past Colloquia**

(links to Fall 1999 - Spring 2017)