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.
Can You Hear the Shape of a Manifold?
Closed, curved surfaces in three space and rectangles with periodic boundary conditions are examples of manifolds. If a manifold vibrates according to the wave equation, what frequencies occur as overtones, and how are the frequencies influenced by the geometry of the manifold? Knowing that some geometric quantities, such as the volume, can be determined from the frequencies, Mark Kac asked in 1966 whether the manifold itself (or a drum) can be completely determined knowing its frequencies. By the work of Milnor and Gordon-Webb-Wolpert the question has been resolved in the negative.
By separating variables, the frequencies turn out to be given by the eigenvalues of the Laplacian (the spectrum) of the manifold. I'll describe some results that relate the geometry to the spectrum including the variational formulation for eigenvalues and Weyl's asymptotic formula.
Random Search Strategies for Delivering Cellular Resources to Active
Synapses in Neurons
A synapse is a connection between two neurons that chemically propagates a
nerve impulse from one neuron to another. It is thought that changing the
strength of the synaptic connections between certain neurons is the
mechanism underlying learning and memory. New studies have also
implicated this mechanism in drug addiction. I will present recent work
I've done with Paul Bressloff using random search theory to help
understand part of this story. We model the transport of building
materials necessary to strengthen a synaptic connection using a jump
Markov process with both discrete and continuum states.
A Number by Another Name
In this talk I'll explore some ideas relating to how we describe
numbers. I'll first cover the notion of the set of describable
numbers, and then touch upon some of the stranger, more subtle
aspects of the real numbers, including the fact that "most" of the
real numbers cannot be described individually. I'll then discuss the
concept of Kolmogorov complexity, and ideas about how we measure how
hard it is to describe a number.
Path-Finding Slime Mold: A-MAZE-ing!
In this talk, I will discuss the paper by Atsushi Tero, et al.
called "A mathematical model for adaptive transport network in path
finding by true slime mold". When the true slime mold Physarum
polycephalum is spread over a maze, with food placed at the maze entrance
and exit, the slime mold rearranges itself until it connects the two food
sources by the shortest path. This process of path-finding is attributed
to an underlying physiological mechanism that we will explore in the talk.
The main focus of the talk will be the mathematical model that the
authors create to help explain how this amoeboid organism "solves" a maze.
A Topic in Invariant Theory
Symmetry on Acid
Symmetric causes in nature have long been known to produce
symmetric effects (Curie, 1894). Group theory can be used to quantify the
symmetry of equations governing a physical system and therefore predict the
symmetry of its solutions. We review an example of such techniques in a
model of neural activity in primary visual cortex (V1). Drug-induced
geometric visual hallucinations are thought to arise spontaneously in V1 due
to its functional architecture (Bressloff et al, 2001; Bressloff and
Kilpatrick, 2008). Using symmetric bifurcation theory, perturbation methods,
and Fourier analysis we can characterize steady-states of the model that
match quite well with experimentally observed contoured hallucinations.
A partition of a positive integer n is a non-increasing sequence
a_1, ..., a_k of positive integers such that n = a_1 +...+ a_k, and the
number of possible partitions of n is denoted p(n). For example, the
partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, so p(4)=5. To study the
partition function p(n), many tools and techniques have been developed using
combinatorics, generating functions, and modular forms. Using combinatorics
and generating functions, we will discuss various properties of p(n) like
Euler's Theorem and the Ramanujan congruences, and we'll talk about the
exact formula for p(n) due to Hardy and Ramanujan.
Foundations of Statistical Mechanics
In statistical physics one is faced with the problem of assigning
probabilities to events based on a few significant bits of information. In
practice this information is far from sufficient to obtain objective nor
unique probabilities. It is common to use the concept of entropy in order
to develop a theoretical description of the macroscopic properties of a
system, based on its underlying microscopic properties, which are often not
I will discuss how the entropy maximization approach to statistical
mechanics allows one to derive the system probability distribution and the
first law of thermodynamics without any assumptions about the nature of the
system or its evolutions, hence may be applied to many Hamiltonian systems.
What is a braid in math? Braid groups were introduced explicitly by
Emil Artin in 1925 although the idea was implicit in Adolf Hurwitz's
work (1891). Artin found a beautiful presentation of the braid groups
in terms of generators and relations. Moreover, he proposed to use
braids to study knots and links. There are now applications involving
braid groups in several mathematical fields such as topology (knots
and links), geometry (mapping class group) and dynamical systems.
In this talk I will define braid groups and I will present Artin's
theorem and few applications of braid groups.
Dimension Estimation and Manifold Learning
In many modern applications, researchers are faced with very
high-dimensional data which is not compatable with tradition methods of analysis. In many of
these applications, however, the data can be thought of as living on a much lower dimensional manifold embedded in a high-dimensional Euclidean space. In this talk, I will discuss what one means by local Hausdorff dimension and
how local dimension can be used to constuct an estimator for the intrinsic
dimension of such a manifold without prior knowledge of the sampling method used to
collect the available data.
Birational Algebraic Geometry
In this talk, I will illustrate several ideas from birational algebraic geometry. I will then show how some of the geometric techniques can be used to study interesting problems in algebra.
Dangers in Key Reuse: WEP is old and broken
WEP is a common protocol used to encrypt communications on wireless
networks. Due to a collection of small breaks, modern methods have
completely eliminated the security once promised by WEP. In this talk we will explore
the nature of these breaks and show how they can be used together to perform a
key recovery attack with high probability in short time using common equipment.
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