Department of Mathematics
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Tuesdays, 4:30 - 5:30pm in JWB 335
Math 6960-3 (1 - 3 credit hours)
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.
Talks will be held on Tuesdays at 4:30pm in JWB 335, unless otherwise
Tentative speakers: Fred Alder, Nick Cogan, Ken Golden, James Keener, Andy Oster, Greg Piepmeyer
Aug 27 (39)
Speaker: Paul Bressloff
Title: The Hallucinating Brain
Geometric visual hallucinations are seen by many observers after
taking hallucinogens such as LSD or cannabis, on viewing bright flickering
lights, on waking up or falling asleep, in "near death" experiences, and
in many other syndromes. The resulting images were organized by Klüver, a
Chicago neurologist, into four groups called "form constants": (1)
tunnels and funnels, (2) spirals, (3) lattices, including honeycombs and
triangles, and (4) cobwebs. In this talk we present a theory of the
origin of these hallucinations in the early visual processing centers of
the cerebral cortex of the brain. The basic idea is that the ingestion of
a drug leads to the release of chemicals that destabilize the visual part
of the brain inducing spontaneous patterns of cortical activity. We show
that the mapping between visual space and cortical space together with the
intrinsic architecture of cortex (how the brain is wired up) determine the
geometry of the visual images.Visual hallucinations thus provide a window
into the internal structure of the brain that can help us to understand
how the brain processes images under normal conditions.
Sep 3 (29)
Speaker: Alastair Craw
Title: Calculating G-Hilb via "It's a knock-out!"
A well known result from algebraic geometry "resolves a
singularity" via a cute procedure involving continued fractions. I'll
describe this result before introducing a recent generalisation involving
several continued fractions competing against each other (hence the silly
title) that resolves a much more complicated singularity. The method is
easy to describe and involves lots of great pictures drawn on fabulous
graphics packages, so TeXies will leave the talk exhausted but satisfied.
Sep 10 (29)
Speaker: Joshua Thompson
Title: Normal Curves and Surfaces in Ideal Triangulations
A curve is "Normal" if it fits "nicely" within a given triangulation
of a surface. Splitting a surface along these Normal Curves leaves us with the
same surface, but now in simpler pieces. This is very useful in classifying
all 2-manifolds. Similarly, a 3-manifold can be split (along Normal Surfaces)
into simpler pieces. We examine the existence of these Normal Curves and
Surfaces within a special "Ideal" triangulation.
Sep 17 (28)
Speaker: Brad Peercy
Title: Dropping Acid: Quantification of Hydrogen Movement in Cardiac Cells
Hydrogen, H+, is an important biological ion. Many proteins in a cell
behave differently in the presence of a high rather than a low
concentration of hydrogen, [H+]. Along with normal fluctuations of [H+],
the acidity or [H+] can change dramatically under abnormal conditions.
During a heart attack [H+] can increase by an order of magnitude inside of
Until recently, little has been done to quantify even normal H+ movement
in cardiac cells. In this talk I will discuss experiments which have been
performed to quantify H+ movement in rabbit cardiac cells. I will also
discuss the role mathematical modeling had in aiding the experimentalists
and derive the diffusion based model.
Sep 24 (22)
Speaker: Andrejs Treibergs
Title: Geometry Affects the Fundamental Frequency of a Manifold
Oct 1 (Cancelled)
Oct 8 (25)
Speaker: Brynja Kohler
Title: Muscle Physiology and Dynamics in the Stretch Reflex
In this talk I will describe the physiology of skeletal muscle, and
present A.F. Huxley's (1957) model of muscle contraction. This
mathematical model is based on the microscopic structure of protein
filaments within muscle cells, and yet it can very accurately model
macroscopic dynamic properties of muscle. I will explain the
force-velocity relationship as it is derived from the Huxley model,
which is useful in formulating models of phenomena from systems
physiology such as reflex pathways. I will describe the one of the
simplest and most important muscle reflexes -- the stretch reflex -- and
show some results of different behaviors which can emerge under different
Oct 15 (25)
Title: Complete Conics
Q: What's a conic?
A: The set of solutions (in the xy-plane) of a quadratic equation:
ax^2 + bxy + cy^2 + dx + ey + f = 0.
Q: How many conics are there?
A: Looks like a 6 dimensional vector space, corresponding
to the choices of a,b,c,d,e,f. But we don't consider the solutions to
0 = 0 to be a conic, and if we multiply each a,b,c,d,e,f by a fixed
constant k, then we get the same conic! So there
is a 5 dimensional PROJECTIVE space of conics.
Q: Are you pulling my leg?
A: Well, yeah. If we work with real numbers, then we'd
have to consider the empty set (x^2 + y^2 + 1 = 0) to be
a conic, which is pretty dumb. So we'll work with complex conics.
This makes things a lot better, but even there we'd have to consider
lines (like x^2 = 0) to be conics, which is still not good.
Q: How do we fix this?
A: Complete conics. Come to the talk and watch me blow up projective space.
Oct 22 (25)
Speaker: John Zobitz
Title: Pascal Matrices and Differential Equations
As any graduate student knows, solving differential equations
can be a difficult task. Even the "simpler" ones with constant
coefficients become challenging when nonhomogeneous equations arise.
Unfortunately, methods to solve these equations (variation of parameters,
annihilator method) are not very "user-friendly".
This talk develops a novel method to solve nonhomogeneous differential
equations with constant coefficients using matrices. I will show how to
reduce any differential equation of this type into a simple non-singular
matrix equation. What is interesting about the solution is the mixed bag
of tricks one needs to arrive there: fundamental calculus
ideas, linear algebra, and a touch of combinatorics. Along the way we
will encounter "Pascal Matrices"--lower triangular matrices with entries
that correspond to Pascal's triangle--and prove a nice result about such
Oct 29 (28)
Title: A Special Hour with Relativity
I will explain the difficulties physicists were encountering with Newtonian
mechanics at the turn of the last century and what prompted them to explore
new theories of mechanics. As you know, it was Einstein's Theory of Special
Relativity that successfully resolves those difficulties.
I will explain what assumptions Einstein made on spacetime, what led him
to make the two bold postulates of special relativity and how the entire
theory follows along with its many astonishing predictions, all of which
have now been routinely verified experimentally. The legendary formula
E = m c ^2 of course will be derived and the famous phenomena of length
contraction and time dilation will be explained. I will close the talk, time
permitting, with the Twin Paradox: One twin takes a space trip travelling
nearly at light speed and returns to Earth to find the other twin much much
older than herself!
Nov 5 (25)
Speaker: Matthew Clay
Title: Coxeter groups
Groups generated by elements of order two with relations only
for pairs of generators are called Coxeter groups. If you relax your
definition of reflection, you can view these as a group of reflections on
a linear space. Most of the finite Coxeter groups can be viewed as the
symmetry group of some shape, and doing so will give us a triangulation of the
sphere. These triangulated spheres are the building blocks (called
apartments) of spherical buildings. Even though buildings seem to have a
very rigid structure, they turn up in a lot of places.
Nov 12 (Cancelled)
Nov 19 (31)
Title: Hebbian Learning: How V1 Got Its Stripes
The architecture of the primary visual cortex (V1) is rich in patterns.
Neurons in V1 are found to have the property of ocular dominance, i.e. a
neuron only receives input from one eye. It is of interest to speculate
how this comes about. We will present a phenomological model to explain
this property. Also, it is found that neurons with like ocular dominance
properties tend to group together. Many different patterns arise in the
distribution of these ocular dominance regions. Specifically we will
examine ocular dominance stripes, which occur in macaque monkey.
Nov 26 (22)
Speaker: Nancy Sundell
Title: Testing for Genetic Differences Between Populations
A question of interest to a wide variety of biologists is the extent
to which natural populations that are geographically separated are
genetically different. I'll present a (relatively) new family of
statistics that can be used to test for genetic differentiation, along
with some basic population genetics models that can be used to examine
the power of these statistics. Some applications to real data sets
will also be presented.
Dec 3 (20)
Past Graduate Colloquia
Speaker: Stefan Folias
Title: Making Waves
Waves are an interesting type of pattern formation which many equations
are capable of generating. From vibrating strings and water pulses in
channels to the electrical activity of light, the heart, and the brain,
these physical contexts provide motivation for the study of traveling
waves in all sorts of linear and nonlinear differential and
integro-differential equations. This talk will illustrate some of the
challenges involved in realizing and analyzing traveling waves.
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Department of Mathematics
University of Utah
155 South 1400 East, JWB 233
Salt Lake City, Utah 84112-0090
Tel: 801 581 6851, Fax: 801 581 4148