The goal of this Colloquium is to encourage interaction among graduate
students, specifically between graduate students who are actively researching
a problem and those who have not yet started their research. Speakers will
discuss their research or a related introductory topic on a level which
should be accessible to nonspecialists. The discussions will be geared
toward graduate students in the beginning of their program, but all are
invited to attend. This invitation explicitly includes undergraduate students.
No Talk - Introductory Meeting
Theorem of the Highest Weight
The "Theorem of the Highest Weight" gives a remarkably explicit
parameterization of irreducible, finite-dimensional representations of a
complex semisimple Lie algebra (up to some natural notion of equivalence).
In addition to explaining what some of these words mean, the goal of this
talk is to explain this important theorem primarily by considering the
special case of sl(2,C). The only prerequisites should be a good background
in linear algebra.
An Introduction to Brownian Motion
This talk will mainly focus on the history of Brownian Motion. I
will briefly discuss it's origin as well as the contributions of Einstein,
Bachelier, Wiener and Donsker towards a mathematical model and understanding of
Brownian Motion. I will discuss some of the interesting properties of Brownian
Motion and if there is time, I will motivate some applications using Brownian
Motion. I will not assume prior knowledge of measure theory and any concepts
needed from probability will be provided during the talk.
A Brief Intro to Direct Methods in the Calculus of Variations
Most natural systems seek to minimize their free energy, and it quite
often happens that that energy can be expressed in terms of an integral
functional. This suggests that functions which describe physical systems should
be minimizers of the energy functional over an admissible class of functions.
The direct method in the Calculus of Variations gives us tools to determine when
such a minimizer exists. We will discuss the general framework of the Calculus
of variations and discuss how the minimization process relates to PDE.
The geometry of commutative algebra
Commutative algebra is closely linked with algebraic geometry. One area
studies rings and the other studies spaces, but we will discuss how they are
related by studying something as elementary as polynomials. We will see that
certain properties of rings correspond to certain geometric properties of
spaces. During the talk we will also introduce a conjectures which, though
easy to understand, has been open since 1939.
A Model of Respiratory Inflammation
Chronic inflammation occurs when the immune system is unable to return to a
neutral state. This inflammation can be devastating to the health of the
individual. Mathematical models can provide insight to controlling unwanted
inflammation and even lend some ideas about the underlying causes. I will
introduce a model of respiratory inflammation and give a quick overview of
the immune system. The analysis of this model identifies some conditions
commonly seen in diseases which are associated with chronic inflammation.
Fractal Dust and Schottky Dancing
The Book Indra's Pearls is a visually stunning story of a computer aided
exploration of unusually symmetrical shapes which arise when two special
spiral motions interact. In other words, the book has lots of pictures
of 'fractals' in it - and it describes these images in an incredibly
accessible way. I will shamelessly use slides created by the book's
author (with permission) to convey some of the mathematics behind the
striking images. Beautiful pictures are guaranteed, a good description
of them is likely. This talk should be accessible to mathematical people of all varieties.
Biophysical model of AMPA Receptor Trafficking and its Regulation
AMPA receptors mediate the majority of fast excitatory synaptic
transmission in the central nervous system, and evidence suggests that AMPA
receptor trafficking regulates synaptic strength, a phenomenon implicated in
learning and memory. There are two major mechanisms of AMPA receptor
trafficking: exo/endocytic exchange of surface receptors with intracellular
receptor pools, and the lateral diffusion or hopping of surface receptors
between the postsynaptic density and the surrounding extrasynaptic membrane.
I will present a biophysical model of these trafficking mechanisms under
basal conditions and during the expression of long term potentiation (LTP)
and depression (LTD). I will show that the model reproduces a wide range of
physiological data, and how the model reveals features that a synapse must
have to be "physiological."
Linear and Arboreal Galois Representations
This talk will introduce the idea of a Galois representation and
explain why these objects are of central importance in algebraic number
theory. The connections between Galois representations and other objects,
such as elliptic curves and modular forms, will then be discussed as further
motivation for studying these objects. The last part of the talk will focus
on recently developed arboreal Galois representations that can be understood
by constructing trees with a natural Galois action.
If you take the unit square and glue parallel sides, you get a surface known
as the torus. What if you start with a parallelogram that is not a square? You get a different kind of torus. In fact, there is an entire parameter
space of these tori. In this talk, we will start to get a basic understanding of what this space of tori looks like. For example, we will
determine its dimension and decide if it is a connected space.
How to Stream and Collide Gracefully
The Lattice Boltzmann Method (LBM) is a relatively new computational
approach to solving fluid flow problems. Instead of assuming that the fluid
is a continuum, as in the Navier Stokes equations, the Lattice Boltzmann
Method uses a mesoscopic particle based technique. One major advantage of
this is that it allows for more complex boundary conditions.
In this talk I briefly discuss the derivation of the LBM from kinetic theory
and describe its numerical ``stream and collide'' algorithm. I then show how
to apply this method to biofluid flow problems, such as blood moving through
an artery. Since blood is comprised of macroscopic particles such as red
blood cells and platelets moving within plasma, it is a non-Newtonian fluid.
This property makes the computational flow problem difficult to solve
The Witt Ring
With a little bit of linear algebra we shall capture an
overview of E. Witt's written paper of 1936, discussing the construction
of the Witt Ring via the Grothendiek Ring. In a nutshell of
unintelligible words: the Witt ring is a ring of classes of
non-degenerate quadratic forms on finite-dimensional vector spaces over
a field, modulo hyperbolic planes. From here, I would like to consider
two cases: the Witt Ring over the real numbers and over finite fields.
Talk cancelled this week
The inspiration for the construction
of the classical resultant is the following simple observation: A polynomial
is completely determined by its coefficients, and the coefficients of a
polynomial have a very special relationship with the roots of that polynomial.
Consequently, if two polynomials have a common root, it is not unreasonable to
expect that we should be able to detect this by some relationship among the
coefficients of those polynomials. The resultant precisely describes this relationship.
In this talk, we will define the classical resultant, explain the generalization to n+1
polynomials in n variables, as well as a formalization of the construction, and state a
geometric application of the formalism.
An Examination of the Limitations of Classical Probability Theory
Classical probability theory was developed largely during the latter
part of the 19th century. This theoretical framework provides many tools that
allow us to analyze both discrete and absolutely continuous random variables.
These tools, however, are insufficient for even the most rudimentary analysis of
continuous random variables that are not absolutely continuous. In this talk
I'll briefly review the "tools" under consideration, explore the nature of these
limitations by way of a standard example, and introduce the alternative
framework on which we build "Modern" probability theory. Some familiarity with
undergraduate probability and/or basic measure theory will be helpful but not
required as the talk is (should be!) self contained.
155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148