Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

P-adics: A Real Alternative

Most of us have begun a class by showing our students how we build sets of numbers– the natural numbers, integers, rationals, and reals. In this talk, we will look at the p-adic numbers, a different way to complete the rationals. Though they are less familiar, the p-adic numbers often have neater properties than the reals. I will define the p-adic numbers and give examples of their uses in analysis, algebra, and geometry.

What is curl of the curl? Discussion on Maxwell's equations.

A struggle to understand all the terms and material properties appearing in Maxwell's equations. Comparison with heat, wave an Navier Stokes equations. What is curl of the curl? What is sigma, mu and epsilon? Are there magnetic monopoles? Is there magnetic field at all?

A Glimpse of Tropical Geometry

We will talk about what happens when multiplication is replaced by addition and addition is replaced by taking the minimum. We will answer questions like: What are some natural situations where you might use this "semi-ring"? Can you do geometry with these funny operations? and What is so tropical about it? The answer to the second question is yes, and we will give some cool examples to compare and contrast this geometry with classical geometry. I will review the necessary basic classical geometry, so the only prerequisites for this talk are that you can multiply, add, and take the minimum of two numbers.

Vesicles and the Calculus of Variations

Inside each cell, there are many small sacks called vesicles. These vesicles are mainly made up of lipids and are often used to transport cellular machinery. Experiments have shown that vesicles can exhibit a variety of different shapes depending on what they are made of and what their environment is like. In this talk, we’ll look at a vesicle made up of two different types of lipids and we’ll try to understand the free energy of the vesicle. Then we’ll talk about how we can use the calculus of variations to minimize the free energy and find the corresponding shape of the vesicle.

A new Spin on 2-adic Galois extensions

I'll consider an inverse Galois problem over the 2-adic numbers. We'll reduce this problem to a question about quadratic forms which, in turn, will boil down to a Gauss sum computation. Clifford algebras, Weil indices, and elliptic curves will appear along the way.

Stochastic Processes and Stochastic Calculus

One of the gifts of 20th century Mathematics was the invention of stochastic calculus and a particular application. With this talk I plan on introducing one-dimensional Brownian motion as an example of a continuous everywhere but nowhere differentiable function. I'll discuss stochastic processes and stochastic calculus and integration, providing definitions and a useful, but maybe familiar, technique. If time permits, I'll discuss the Black-Scholes-Merton model used quite a bit in financial mathematics.

An Introduction to Tropical Mathematics

In Tropical Mathematics we replace the addition of real numbers by "maximum" and the multiplication by "addition" and then we do math on this semifield. I start with three examples and compare the "classical mathematics" with "tropical mathematics": Algebra (Fundamental Theorem of Algebra), Geometry (graph of a line), and Analysis (integral and Fourier Transform). Then I try to explain that we can think of the tropical mathematics as Moslov dequantization of the classical one. If there was enough time, as an application, I show how we can use the tropical mathematics to solve problems like the Hilbert's sixteen problem. we do not prove any something, so no prerequisites!

When Can You Fold a Map?

There is a lot that you can do with a pile of scratch paper sitting on your desk (besides filling it up with Math of course)! I will address some special cases of the following open problem : Given a map (a rectangle) partitioned into an m в n regular grid of squares, with each non boundary grid edge assigned to be either a mountain or valley crease, can the map be folded flat into one square, respecting the creases? I will also show some amazing things that can be done with a piece of paper.

Why our immune system doesn't kill us...most of the time.

Our immune system is incredibly powerful and, if misdirected, can wreak havoc on our bodies. How is that, in most people most of the time, our immune system can do such a good job of killing the appropriate cells? In this talk I will describe some of the proposed mechanisms and then put them to the test with a mathematical model. The results will give us some key insights into how the immune system works.

The Mathematics of Democracy

You may have heard the phrase "It's not the people who vote that count. It's the people who count the votes." This may be true, but a mathematician could argue that what matters is not only who counts the votes, but how they count the votes. In this talk we'll examine a number of different voting procedures, each one plausibly fair, and see how the outcome of an election can be highly dependent upon the voting procedure. Then, we'll discuss and prove Arrow's theorem, a theorem that can be interpreted as saying some rather unsettling things about democracy.

Interesting Properties of the Superlens

The best resolution a conventional optical lens can achieve is about equal to the wavelength of the light used to form the image. Such lenses always have a positive index of refraction (a number that characterizes how the speed of light changes as it moves through the lens). In 1968, the Russian physicist Victor Veselago studied the properties of a material with a negative index of refraction. Although no such materials existed in 1968, Dr. Veselago pointed out that such materials would have surprising and desirable properties. Recently, scientists have created materials with a negative index of refraction (known as metamaterials). In 2000, Sir John Pendry suggested that the negative index materials studied by Veselago would be able to beat the diffraction limit and create a perfect image!! In this talk, I will explore some of the interesting properties of the proposed superlens as discussed in the papers by Veselago and Pendry. I will also discuss some of the results of Graeme Milton and others—these results are perhaps the most surprising properties of the superlens.

The simple loop conjecture for PSL_2(R)

In this talk, we will briefly discuss the original Simple Loop conjecture, a statement about the kernel of non-injective homomorphisms from between surfaces groups. Then, we will give a counter-example to the generalization of this conjecture to PSL_2(R). This talk follows a recent short paper by Katie Mann.

Modelling Motors with Stochastics

ABSTRACT

Blowing Up Plane Curves

Plane curves with sharp corners or self-intersections tend to misbehave. Algebraic geometers are harsh masters: curves that refuse to be nice are repeatedly "blown up" until they capitulate. Starting from basic definitions, we'll try to understand how blowing up can be used to reform a curve. We'll see that blowing up requires an infinite bottle of glue, that nodal and cuspidal curves can learn to be as charming as a parabola, and, time permitting, that every curve is good at heart.