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Graduate Colloquia
Fall 2002
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 noted.

Tentative speakers:   Fred Alder, Nick Cogan, Ken Golden, James Keener, Andy Oster, Greg Piepmeyer

Aug 27 (39)

Speaker: Paul Bressloff
Title: The Hallucinating Brain
Abstract:  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!"
Abstract:  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
Abstract:  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
Abstract:  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 cardiac cells.
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)
Speaker:  N/A
Title:  N/A
Abstract:  N/A

Oct 8 (25)
Speaker:  Brynja Kohler
Title:  Muscle Physiology and Dynamics in the Stretch Reflex
Abstract:  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 experimental conditions.

Oct 15 (25)
Speaker:  Aaron Bertram
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
Abstract:  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 matrices.


Oct 29 (28)
Speaker:  Kenneth Chu
Title:  A Special Hour with Relativity
Abstract:  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
Abstract:  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)
Speaker:  N/A
Title:  N/A
Abstract:  N/A

Nov 19 (31)
Speaker:  Andrew Oster
Title:  Hebbian Learning: How V1 Got Its Stripes
Abstract:  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
Abstract:  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)
Speaker:  Stefan Folias
Title:  Making Waves
Abstract:  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.

Past Graduate Colloquia

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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