Graduate Colloquium Spring 2006

Spring 2006

Tuesdays, 4:35 - 5:35 PM, JWB 335

Math 6960-001, Class Number 4665

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 4:35 pm in JWB 335, unless otherwise
noted.

**January 24:**

**Speaker: **
Qiang Song

**Title:** What's a stack?

**Abstract:**

*
This talk is intended to be an introduction for everyone to the notion of
an algebraic stack, which was first developed by A.Grothendieck and M.Artin in
1960s in order to parameterize geometric objects varying in families. It's said
that now the stacks are outgrowing their basic function as a convenient framework
and language for studying moduli and are becoming an essential tool in everyday
research. However, it's not an easy concept to digest and it will take a lot of
time to set up those "spaces" in one's mind. As for me, I like the notion because
although I don't have much geometric intuition, learning stacks can help me study
the very interesting subject-- the moduli space of curves. (It gives me rigorous
proofs!)
If you are not interested in the moduli problems, you don't need worry since just
basic topology and algebra is assumed and you may get some new ideas about the
meaning of "space" after the talk.
*

Calcium plays an important role in muscle contraction. Tight regulation of calcium release is particularly necessary in cardiac muscle in order for blood to be pumped effectively. Recent evidence suggests that many disturbances in heart rhythm, cardiac arrhythmias, are due to abnormal calcium handling in cardiac cell. In particular, calcium waves in cardiac ventricle cell is found to be pathological. In this talk, we will explore the nature of excitation-contraction coupling in healthy cells through the calcium induced calcium release mechanism. Following that, we will study the nature of wave propagation in deterministic and stochastic models of cardiac cell strand with discrete release sites.

Liz:

Cardiac cells are excitable, which means that their membranes can undergo action potentials. As excitation spreads from cell to cell, over the myocardium, the muscle contracts and the heart pumps blood. In this talk, I will discuss the mechanisms by which action potentials are able to spread from one cell to the next. In particular, propagation is possible due to gap junction channels which connect the intracellular space of neighboring cells. These channels act as linear resistors to the spread of intracellular current, which is mathematically convenient. However, I will also discuss the possibility of a secondary mechanism for propagation. The "Electric Field Effect" was first proposed in the 50s by Nicholas Sperelakis, who suggested that propagation can occur via an electric field which forms in the narrow junctional cleft between neighboring cells. To better understand this mechanism, we can make a quasi-steady state approximation and study the dynamics of the system.

Rotating any face of the cube changes the position and orientation of the little pieces comprising it. These rotations generate a group we shall call "Rubik's Group", where two actions are equivalent if their effects on the solved cube are identical. Our goal is to analyse the structure of this group.

As a warmup, what is wrong with this link?

Good #1) AC is intuitively appealing (a lot of people will disagree).

Good #2) The use of AC is ubiquitous in mathematics.

Evil #1) AC has certain "absurd" consequences for which one might almost want to reject AC as an axiom.

You will see the proofs of a number of very elementary facts with the use of AC explicitly spelled out. You will see how many foundational results in areas ranging from abstract algebra to point-set topology to algebraic geometry to partial differential equations all rely crucially on AC. And yet, AC also has some very disturbing consequences, one of which being the famous Banach-Tarski Paradox, which Dr. Savin discussed in the Undergraduate Colloquium on April 4.

Come and hear about this fascinating axiom!

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148