Math Department Events Listing

Come schmooze with your fellow mathematicians-at-arms. Enjoy fine cookies, teas, and conversation, or try your hand at a variety of games.

One of the commonly used equations describing nematic liquid crystals is the so-called Doi-Smoluchowski equation. In essence, it is a kinetic equation for evolution of the orientation probability density of the system. I will present an analogue of this equation for spatially inhomogeneous systems and will discuss the associated problems of moment closure and reduction to Ginzburg-Landau type dynamics.

I will first talk about the growth rate of the class number in a geometric $\Z_p$-extension of a function field. Then the $p$-part of the class group and the Iwasawa theory for geometric $\Z_p$-extensions will be considered. Lastly I will present some results about the class number of a geometric $(\Z_p)^d$-extension of a function field.

Results of NSF Audit and Effort Reporting will be discussed.

In the 1970's the physicists discovered that quantum field theories have hidden symmetries which are very different from the symmetries hitherto known. The usual symmetries are governed by the actions of Lie groups on geometric objects but the new symmetries, called super symmetries by Salam and others, are governed by a new type of geometric structure. The theory of super manifolds an super Lie groups emerged out of these attempts and has been the object of intense scrutiny by the physicists in the past 30 years. The new collider being readied at CERN may decide if Nature is (or was at the early stages of the universe) super symmetric. In this lecture I shall give a brief introduction to super manifolds and super Lie groups and try to describe recent results and open problems. Much of the work was done (and is being done) with the collaboration of Gianni cassinelli, Alessandro Toigo, Claudio Carmeli, and Luigi Balduzzi, of the Genoa theoretical physics group. Familiarity with differentiable manifolds and Lie groups is welcome but I shall try to be as elementary as possible.

In 1973 Karl von Frisch was awarded the Nobel Prize in Physiology for his work decoding the “language” of bees. The language of bees, also known as the “waggle dance,” is used by foraging honeybees to indicate the distance and direction of a food source to other bees in the hive. Since then we have learned from numerous behavioral studies that a honeybee’s judgment of distance is based on a visual estimate of speed. This estimate is relatively independent of spatial frequency, contrast, and direction of motion and can even be made with only monocular input.

Despite our thorough understanding of the many behaviors that rely on visual speed estimation, we still lack a solid understanding of the underlying neural processes responsible for the estimate. My research has focused on evaluating various models of visual speed estimation based on their effectiveness for replicating, in simulation, the behaviors observed in honeybees. I then use the results from the simulations to generate hypotheses and experiments to further refine the models. In addition, I have recently begun my own behavioral work to gather more detailed information on the properties of the honeybee speedometer.

First, I will give a relatively gentle introduction to cluster expansions, which refers to a technique for controlling measures on spaces of large dimension that are defined by a density with respect to a reference measure. After that, I will present a connected graph identity that may lead to an alternative proof of the Fernandez-Procacci result.

In the late 1980's Igor Volovich introduced a bold idea that the geometry of space-time could be non-Archimedean at ultra-small distances and times. The reasoning behind this hypothesis is based on the idea that at distances of the order of the Planck length no measurements are possible. In this lecture I shall describe some recent work of mine with my student Jukka Virtanen in which we discuss projective unitary representations of the Poincare group over local fields and their relationship to the unitary representations of the conformal group. Some familiarity with p-adic fields would be welcome but no technical knowledge is demanded. I shall try to explain the basic ideas as simply as possible.

In this talk, I will explain the ins and outs of videos and what considerations should be taken into account when making them. More specifically, I will focus on constructing videos from a series of still images in order to visualize a simulation. And if you want to find out how to seamlessly present your videos in LaTeX presentations, I'll tell you how to do that too.

Matrix maps describing the discrete time dynamics of (demographically) structured populations have a long tradition in population dynamics. I will summarize a general theory of nonlinear matrix population models that takes a bifurcation theoretic point-of-view and utilizes the inherent net reproductive number R0 as a key parameter. I will then present new results concerning an important special class of matrix models that fails to fall under the purview of the general theory. This special class is biologically important because it describes the dynamics of semelparous populations. Semelparous populations consist of individuals that reproduce only once before they die; examples include annual plants and many species of insects (e.g., cicadas). The mathematically degenerate bifurcation at R0 = 1 that occurs for these models can lead to unusual types of oscillations and to attractors with temporally separated generations (i.e., cohort waves). Mathematical techniques used in the analysis, besides bifurcation theory methods, include monotone flows and average Lyapunov functions.

Come schmooze with your fellow mathematicians-at-arms. Enjoy fine cookies, teas, and conversation, or try your hand at a variety of games.

Red blood cell movement, deformation, and partitioning in small diverging microvessel bifurcations are simulated using a two-dimensional, flexible-particle model. For isolate red blood cell movement, while simulated red blood cell trajectories tend to follow background fluid streamlines, significant deviations from these streamlines can occur because of red blood cell migration towards vessel centerlines and red blood cell obstruction of downstream vessels. The net effect of these behaviors is explored in symmetric and asymmetric vessels to produce results comparable with experiment. In addition, preliminary results and insights are presented for multiple red blood cell motion in straight vessels and in bifurcations.

In 2003, Zerjal et al suggested that ~8% of Y chromosomes in Asia trace their origin to Genghis Khan and his male relatives. Similarly, over 100 published anthropological studies have linked male status and dominance to increased reproductive success. If dominant men frequently reproduce more than others and pass this advantage to their sons, this should leave a detectable signal in the genome. In this talk, we will look at genetic data from 41 Indonesian communities and discuss the probabilistic models and statistical methods used to analyze this data. We will apply Prof. Watkins' results from last week to help determine just how skewed the history of male reproduction really is.

Euler was the first mathematician who realized that it would be very important to have a theory of divergent series in which such series have sums associated to them. He discovered several of the modern methods of summation. He applied his methods to derive the functional equation for the Riemann zeta function one hundred years before Riemann. His work on the summation of the factorial series was the forerunner to what eventually emerged as the Borel summability in the twentieth century. The factorial series has zero radius of convergence and hence cannot be summed by most methods and so Euler called it the divergent series par excellence. His work has led to problems that are still open and relate such divergent series to differential equations and continued fractions. This lecture is an attempt to describe these aspects of Euler's work. No special background is needed.

In developing software for scientific computation, one has typically been forced to choose between the fast development but slow program execution associated with high-level programming languages and the fast execution but slow development associated with low-level languages like FORTRAN. Another alternative has been the use of commercial software such as MATLAB® or Mathematica®, but these applications are expensive and closed-source, thereby inhibiting sharing and collaboration with those portions of the scientific community who can’t afford these applications. Such commercial codes are also highly specialized, making it difficult to extend them to wider problem domains.

The recent arrival of the SciPy package of high-level scientific computation modules for the Python programming language allows for the development of scientific computations suffering from none of these drawbacks. Python is a very high-level language, allowing for extremely rapid development of robust and sophisticated software. The speed-critical components of SciPy (linear algebra, FFTs, numerical quadrature, etc.) are called from within Python but are implemented in FORTRAN or C, enabling rapid execution speed. Furthermore, all of Python and SciPy are completely free and open-source, enabling scientific codes developed in Python to be distributed freely. And by using Python to develop scientific computations, one has access to a huge library of other general functionality, enabling for example easy development of software that has sophisticated graphical user interfaces (GUIs) and interfaces with databases, or that uses the internet to send e-mail or retrieve data.

The talk will provide a brief high-level overview of the Python programming language, followed by several examples of the ease with which powerful scientific computations can be performed using SciPy. Examples will include image processing, signal processing, solution of PDEs using sparse matrix solvers, advanced scientific visualization, and interactive distributed and parallel supercomputing.

We study the spectrum of a linear advection-diffusion equation in a periodic domain, where the diffusion coefficient changes its sign. We prove that the spectrum of an associated linear operator consists of an infinite set of simple eigenvalues on the imaginary axis and the set of corresponding eigenfunctions is complete. However, we also show, assisted with numerical approximations, that the complete set of linearly independent eigenfunctions does not form a basis in a space of square integrable functions and that the Cauchy problem for the advection-diffusion equation is ill-posed.