Chair's Message
Dear Alumni and Friends of the Department of Mathematics: Ivan Yotov
In this issue of By the Numb3rs you can read about recent developments in the Department as well as some accomplishments by our students and faculty. We are pleased to inform you that Dr. Michael Neilan has won the Leslie Fox Prize in Numerical Analysis. Dr. Jonathan Rubin recently spent his sabbatical working on a research project in New Zealand at the University of Auckland. Undergraduate students are working with Dr. Bard Ermentrout on several interesting projects and Jeff Connors, Ph.D., class of 2010, has accepted a tenure-track Assistant Professor position at the University of Connecticut.
We are pleased to share with you that our faculty received three new research awards from the National Science Foundation. We have highlighted some of the excellent research being done by Dr. Neilan in computational and applied mathematics with an emphasis on numerical methods for partial differential equations, in particular, finite elements methods. With the continued support of the Math Research Center, with generous funding from the Dietrich School of Arts and Sciences, we have organized several workshops and conference and several more are scheduled for the Spring 2014 term.
We have changes and additions to our department. Dr. Paul Gartside has stepped down as Graduate Director after 4 years of dedicated service. We'd like to extend our thanks for his hard work and dedication as Graduate Director. Dr. David Swigon has been named as the new Graduate Director. Dr. Hao Xu joins us as a new Assistant Professor and Ms. Pat Markham joins us as the Graduate Student Administrator.
As always, please keep in touch. We would love to hear from you. Visit our web page www.mathematics.pitt.edu for information on how to contact us and for the latest news about the Department.
Sincerely yours,
Ivan Yotov, Chair
Department of Mathematics
Assistant Professor Michael Neilan wins the Leslie Fox Prize
Michael Neilan
On June 24, 2013 Michael Neilan won the Leslie Fox Prize in numerical analysis with the talk, "Conforming and divergence-free Stokes elements on general triangular meshes." This talk is based on the paper joint with Johnny Guzman (Brown University) with a similar title appearing in the journal Mathematics of Computations.
Computed solution of an academic test problem with an incompatible finite element pair
The overall theme of this work was to design numerical methods that closely mimic the physical and mathematical structures of the underlying system of equations describing the motion of fluids, in particular, the Stokes/Navier-Stokes equations. These partial differential equations are ubiquitous in mod A compatible finite element pair satisfying incompressibility weakly
Except for simple and unrealistic settings the Stokes problem cannot be solved in closed form and computational methods are the only viable way to compute (approximate) solutions. The finite element method is an established and ubiquitous tool to solve partial differential equations, and these methods have been developed for A compatible finite element pair satisfying incompressibility exactly
In the Leslie Fox Talk, Dr. Neilan discussed the first class of finite element spaces for the Stokes problem that are both inf-sup stable and satisfy the incompressibility condition exactly on general simplicial partitions. The key idea in the construction is to use cohomological techniques with a discrete smooth de Rham complex as their main tool. Starting with a class of H2 conforming finite element spaces developed 45 years ago, Neilan and Guzman used the complex as a guide to construct finite element pairs with the desired properties.
Prof. Rubin Travels to New Zealand for Sabatical
New Zeala Downtown Auckland from Mt. Victoria, across the Waitemata Harbor
The Department of Mathematics at the University of Auckland provided an excellent setting for sabbatical research combining a mathematical focus together with relevance to biological systems. The department boasts a first‐class, highly active applied dynamical systems group. Both discrete and continuous dynamics are represented, and the group even has a healthy interaction with the department’s topologists on the study of various complex mappings and manifold structures. Most of its members have some interest in biological dynamics, engaging in interdisciplinary collaborations while maintaining an emphasis on mathematical issues, which was exactly what I was seeking for my sabbatical.
Cathedral Cove near Hahei, North Island, NZ
The starting point for the multiple time scale project is the observation that interacting biological processes often After the Haka, we were part of the tribe (Waitangi Treaty Grounds, North Island, NZ)
In the bursting project, we are considering systems with more traditional two time scale separation. The systems we study generate outputs known as bursts, featuring phases of rapid oscillations alternating with quiet phases of little activity. Bursts in some neurons help to generate basic activity such as breathing, and bursts also can show up in certain neural diseases. Typically, one part of fast-‐slow analysis is to treat a slow variable as a parameter and to consider how the behaviors of fast variables change as this parameter is varied. Aided by computer simulations, we carried out a version of this approach aimed at fitting the output of a model system with two slow variables to experimentally observed bursting patterns. The idea is that we can first find a curve in two‐slow-variable space such that forcing the slow variables to travel along this curve causes the fast variables to generate the patterns that we want, and then we can find parameters that cause the slow variables to follow close to this curve on their own. As a result, we can use the fast‐slow time scale separation to gain efficiency in fitting model behaviors to data.
Another major challenge of the visit had more to do with time zones than time scales: moving a family of four, Watch out for the orcs on the Pelennor Fields (on the South Island of New Zealand)
After six months in New Zealand, we exchanged islands and accents, moving on for a month in Australia. We spent most of that time in Sydney, where I worked with Martin Wechselberger at the University of Sydney on a ma Jumping croc on the Adelaide River, Australia
Featured Alumni
Jeffrey Connors, PhD 2010
Jeffrey Connors finished a position as a postdoctoral research associate at Lawrence Livermore National Laboratory (LLNL) over the summer and started a tenure-track professorship at the University of Connecticut - Avery Point campus in the department of Mathematics. At LLNL he has been focusing on methods to calculate estimates of numerical errors for multi-physics computer simulations to help understand uncertainties in simulation outputs. Now that he has started as a professor at University of Connecticut, he will continue some of this work but will focus more on the problem of numerical modeling of atmosphere-ocean interaction.
Undergrate Research
Cutaneous Rabbit
Dr. Bard Ermentrount is working with Evan Cresswell, a senior Mathematics Undergraduate Major on the Cutaneous Rabbit. The illusion called the cutaneous rabbit, works as follows. A subject is tapped two times on the wrist and a third time at the crook of the elbow. His eyes are covered. The illusion is that the second tap is felt somewhere between the wrist and the upper forearm. There have been a few very abstract models of this using priors and Bayesian inference.
Ermentrout and Cresswell created a mechanistic model for this illusion. The model is based on a neural network representing the spatial location on the arm. This is coupled with a second process that recognized what the stimulus was but this has a delay associated with it. The delayed recognition of the "what" coupled with the rapid recognition of "where" gives the illusion of the location of the second tap to be somewhere in between depending on the timing. The model is able to fit psychophysical data. They are now trying to see how the illusion depends on different parameters in the model. Cresswell will present this work in the form of a poster at a math meeting in January 2014.
First Experiences in Research
Sophomore neuroscience major, Sarah Miller is continuing to work on a project started in the Spring term as part of the First Experiences in Research (she liked it enough that she wanted to continue). This project is a bit more abstract than the others. Here She and Dr. Ermentrout are looking at the dynamics of locally connected oscillators in square and cubic lattices. Such networks will frequently synchronize, but can also produce more exotic patterns such as rotating waves and various other complicated patterns. Last semester they characterized all the nonsynchronous patterns in 6x6 and 8x8 arrays of oscillators. They are now trying to understand the patterns that occur in NxNxN arrays for small N. The motive for this is that such nonsynchronous dynamics is seen in two-dimensions in various experimentally induced seizures on slabs of cortical tissue and three-dimensional waves are seen during arrythmias in the heart.