# Research

## Complex Systems: Mathematics, Computation & Science

### Tuesday, November 26, 2002

Title: Crossover From Poisson to Wigner-Dyson Level Statistics in One-Dimensional Systems With Integrability Breaking
Speaker: David Rabson, Physics Department
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

Arrange a chain of atoms in a ring, and allow the free electrons on the lattice to hop from site to neighboring site. If only nearest-neighbor electrons interact, the system is known to be integrable: there are as many integrals of the motion as there are solutions. If we then apply an irrelevant perturbation (such as a magnetic flux through the center of the ring), energy levels will wander independently as a function of perturbation. Each such level can carry a current; the transport is termed ballistic. According to the standard view of random-matrix theory, even an infinitesimal second-neighbor interaction will destroy the integrability; energy levels will repel as a function of the perturbation, and states will carry no ballistic current. The transport is called diffusive. The probability distribution of energy spacings should follow the Wigner-Dyson form.

I will present some numerical evidence that this latter picture is wrong when the chain is finite and speculate on the possible implications for quantum computing.

### Tuesday, November 19, 2002

Title: Ozone Production in the Troposphere
Speaker: Noreen Poor, College of Public Health
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

Formation and destruction of ozone in the lower atmosphere is a cyclical process driven by sunlight and the availability of nitrogen oxides and volatile organic compounds. The ambient temperature, degree of atmospheric mixing, and build-up of pollutants in the atmosphere are factors that contribute to elevated concentrations of ozone close to the Earth's surface. The complexity of the physical and chemical processes associated with ozone production make challenging regional control of ozone to levels below the National Ambient Air Quality Standard.

### Tuesday, November 5, 2002

Topic: Planning Session
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

The organizers of this seminar invite all interested faculty to join us in a planning session for the next semester. There are several proposals for a new format next semester, and we are interested in hearing your ideas.

### Tuesday, October 29, 2002

Title: Mathematics for a Biological Mystery
Speaker: W. Richard Stark
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

How can a totally disorganized system exhibit organized behavior at the global level? Many mathematicians, the author included, have used irregular networks of automata to model systems (idealized tissues, insect colonies) which most clearly present this mystery. A version of calculus (on a domain not related to the real numbers) has recently yielded some success.

### Tuesday, October 15, 2002

Title: The Prostate, Mathematics, and Computers
Speaker: Edgar Sanford
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

Problems in Clinical Medicine will be presented which result from minimal utilization of fundamental mathematics as well as computer science. The present status of prostate disease will be used as an example of clinical issues that could be more rapidly addressed by using mathematical and computer applications. Frankly, this will be a plea for help from our academic colleagues in disciplines not obviously related to health care.

### Tuesday, October 8, 2002

Title: Formal Language Theory Models for DNA Recombinant Processes
Speaker: Nataša Jonoska
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

We will describe the theoretical model of splicing systems which was originally developed in the late 80's by T. Head. This model uses the sequence of DNA nucleotides to treat molecules as realizations of abstract strings or words. The detailed definition of the splicing concept comes from considerations of the cut and paste activity of a ligase and restriction enzymes on double stranded DNA molecules.

### Tuesday, October 1, 2002

Title: Birth, death, and the risk of population extinction in heterogeneous populations
Speaker: Gordon Fox, Department of Biology
Time: 1:00pm–2:00pm
Place: PHY 109

### Tuesday, September 24, 2002

Title: Community Networks
Speaker: Gary Huxel, Department of Biology
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

My talk will focus on factors that regulate stability in large networked systems. Specifically I will examine dynamics and stability of food webs. The approach can be broadly applied to biological, economical, and social systems. Furthermore, these highly complex systems represent significant challenges to computational systems.

### Tuesday, September 17, 2002

Title: Knots and DNA
Speaker: Masahiko Saito
Time: 1:00pm–2:00pm
Place: PHY 109

### Tuesday, September 10, 2002

Title: The Mathematics of Threshold Phenomena
Speaker: Greg McColm
Time: 1:00pm–2:00pm
Place: PHY 109

#### Abstract

Many experiments consist of sitting and watching some value $$N$$ slowly increase, and waiting for a system to somehow react.

For example, slowly increase the temperature, and see at what temperature a snowflake melts. Or give a rat increasing amounts of a toxin, and see when it gets sick.

If $$T$$ is the time when the system reacts, then $$T$$ has a very small variance when the system is highly predictable (snowflake), but lower variance if the system is less predictable (rat).

There is a well-developed theory of “stopping time random variables” $$T$$ for very nice systems. But many systems are quite complicated, and in this talk we will take a look at how to understand the predictability of such complicated (and often poorly understood) systems.