Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Given some matrices, one aspect of them we might want to understand is where they move the points of the vector space. That is, what does the collection of orbits look like? A classical approach to this problem is to consider the functions that stay the same after applying these matrices. Sometimes finding all such functions is difficult, so one could restrain their ambition and instead hope to find enough such functions to distinguish the orbits. In this talk, we will discuss some classical results regarding the collection of invariants, and, if time permits, analogues pertaining to collections of invariants that separate orbits, all the while focusing on examples.

Over the integers/rationals/reals/complex numbers, there is a long history of applying analytic techniques (e.g., differentiation and integration, and if you are really fancy, resolution of singularities) to answer questions regarding the complexity (read: "badness") of a given polynomial. Though less well-known, there are ways to do the same for polynomials over finite fields; in this setting, we aren't able to use analytic techniques, and instead must rely on the most important function in characteristic $p>0$, the Frobenius function (or p^th power map). In this talk, I will outline some of these approaches, and discuss a beautiful connection between them. *Note:* I will do my best to keep things as concrete and simple as possible; in particular, I won't assume any prior knowledge of characteristic p methods. As such, I hope that this talk will be accessible to students in both pure and applied areas.

Abstract: When a group acts on a geometric object, we can learn about the algebraic structure of the group. Buildings are a tool used in many different areas of mathematics to study a variety of groups. This talk will give a brief introduction to buildings and some of their nice properties through specific examples.

Abstract: The Navier-Stokes equations are widely used in engineering and science to model the flow of fluids. Mathematically, they are a system of nonlinear partial differential equations for the unknown fluid velocity and fluid pressure. They are also the subject of one of the Millennium Prize Problems. A correct proof of (or counter-example to) the problem statement set out by the Clay Mathematics Institute would yield a $1,000,000 prize. This talk will give a brief introduction to the Navier-Stokes equations, the Millennium Prize Problem statement, and some recent theoretical results.

Speaker: TBA

Abstract

Cellular Stress: The Development of an ODE model describing PKR Induced Translation Attenuation in response to Pox Virus Infection

Cells regularly face a variety of stressors from external and internal sources. Examples of stressors include the accumulation of misfolded proteins, viral infection, nutrient starvation and heme deficiency. Interestingly cells respond to each of these four very different stressors with the shared strategy of temporarily shutting down the translation of proteins. In this talk we will discuss the biology background necessary to understand the mechanism of translation attenuation and place it in the greater context of the cellular response to pox virus infection. We will then discuss the development of an ODE model describing the system and describe preliminary results in the context of Pox virus infection.

Counting Lizard spoons.

Burnside's orbit counting lemma gives us a nice way to count the number of configurations up to symmetry. I will give some examples from real world situations such as tic-tac-toe, putting lizard spoons in bowls, and painting coins. If time allows, I will also discuss generating functions and their usefulness in organizing combinatorial information. We will see an elementary example in detail and then glimpse a deeper example from Gromov-Witten theory.

Continued Fractions

A continued fraction is a natural way of representing real numbers that is in many aspects more useful than our standard decimal notation. I will describe the construction of continued fractions and their building blocks (called convergents), and discuss some of their many applications in fields such as number theory, chaotic dynamical systems, and piano tuning

Mapping Class Groups

What sorts of things can we do to a surface that will give us the same surface back? Can we spin them, twist them, cut them apart andglue them back together? We'll look at homeomorphisms of surfaces and use them to form groups. These groups, called mapping class groups, have applications to knot theory, braid groups, and Teichmuller space.

Abstract:Tropical geometry is the geometry of polynomials over the tropical semi-ring. Tropical curves can be thought of as metric graphs satisfying certain conditions and can be classified by their combinatorial type. Much of the study of tropical curves can be reduced to the study of these combinatorial types. We will discuss a way to parametrize all subdivisions of a given polygon in Z^2 and see how this is the same thing as parametrizing all types of tropical curves belonging to a certain family.

Abstract: The phenomenon of synchrony is a pervasive force in the natural world, and it sometimes pops up in places we don't expect. What are the minimum requirements for synchrony? Can inanimate objects synchronize spontaneously? By analyzing basic oscillator models, we can answer these questions and gain a deeper insight into this emergence of order.

Speaker: TBA

Abstract