Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

Date:  Monday, February 16, 2015
Location:  1866 East Hall (3:00 PM to 4:00 PM)

Title:  Direct Scattering for the Benjamin-Ono Equation with Rational Initial Data

Abstract:   The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable equation, which means that it can be solved using the inverse scattering transform (IST). In practice, this requires knowledge of a Lax pair, which is studied in lieu of the original PDE, to recover the so-called “scattering data”

In this talk, I will briefly introduce the general IST theory for the BO equation. I will then show a construction procedure for the scattering data of the BO equation with a rational initial condition under mild restrictions. The construction procedure uses the analyticity properties of the Lax pair solutions to recover a reflection coefficient, eigenvalues, and phase constants. Lastly, time permitting, I will show that this procedure validates well-known formal asymptotic results in the zero-dispersion limit.
Speaker:  Alfredo Wetzel


Activity of Inhibitory Neural Networks Generated by the Small World Paradigm

Date:  Monday, February 9, 2015
Location:  1866 East Hall (3:10 PM to 4:00 PM)

Title:  Activity of Inhibitory Neural Networks Generated by the Small World Paradigm

Abstract:   While the behaviors of excitatory neural networks have been throughly studied, similar networks with inhibitory signaling have not received the same meticulous treatment in the literature. Motivated by future research into hippocampus-like networks, in this talk we will discuss the activity of inhibitory neural networks by analyzing how changing the cell type (Type I vs. Type II) causes nearly antithetical behavior in nearest neighbor networks. Additionally, by utilizing the Small World Network Paradigm famously articulated by Watts and Strogatz, we will see how the behavior changes as we systematically alter the network topology from nearest neighbor to sparse random connectivity. Finally, we will examine the somewhat counterintuitive result that, unlike all-to-all networks, inhibitory networks with sparse random connectivity tend not to synchronize better for lower connection strength and duration of synaptic signaling.
Speaker:  Scott Rich

The Barycentric Formula for Interpolation and Integration

Date:  Monday, January 26, 2015
Location:  1866 East Hall (3:10 PM to 4:00 PM)

Title:  The Barycentric Formula for Interpolation and Integration

Abstract:   We will introduce the barycentric formula for polynomial interpolation, which leads to an excellent interpolation algorithm that is fast and stable. Similar formula can be derived for the Cauchy integral formula to provide an algorithm that solves the boundary value problem of Laplace equation. Such algorithm proves to be spectrally accurate everywhere in the domain including points that are almost touching the boundary.
Speaker:  Bobbie Wu

Fast and slow coupling in the mammalian circadian clock

Date:  Monday, December 08, 2014
Location:  3096 East Hall (3:00 PM to 4:00 PM)

Title:  Fast and slow coupling in the mammalian circadian clock

Abstract:   The mammalian circadian (daily) clock is controlled by the ~20,000 neurons of the suprachiasmatic nuclei (SCN). While the molecular mechanisms for generating rhythms in individual SCN neurons are well characterized, single-cell rhythms are weak and noisy, and it is still unknown how they are integrated to generate robust rhythms at the tissue level. To investigate this, we develop a highly detailed, multicellular, and multi-scale model of the SCN. The model is used to investigate, and make predictions about, the differential roles of fast GABAergic synaptic coupling, and slower paracrine signaling through VIP, which couple neurons in the SCN.
Speaker:  Dan DeWoskin

Fast, high-order algorithms for simulating vesicle flows through constrained geometries

Date:  Monday, November 24, 2014
Location:  3096 East Hall (2:30 PM to 3:30 PM)

Title:  Fast, high-order algorithms for simulating vesicle flows through constrained geometries

Abstract:   Locally inextensible vesicles and membranes suspended in a viscous fluid appear often in biological systems. Red blood cells, for example, share the same biconcave equilibrium conformation as vesicles and exhibit similar tank-treading and tumbling dynamics. Artificial vesicles are used in drug-delivery systems, and periodic cylindrical lipid bilayers are observed in cellular organelles such as mitochondria, chloroplasts, and endoplamic reticula. A boundary integral method based approach is provided for modeling these inextensible membranes. The numerical scheme is developed for the case of a 2D periodic lipid bilayer suspended in a viscous flow. In addition, a new algorithm is provided for modeling vesicle flows through arbitrary, periodic geometries.
Speaker:  Gary Marple

Introduction to the Mathematics of Infectious Diseases

Date:  Monday, November 10, 2014
Location:  3096 East Hall (2:30 PM to 3:30 PM)

Title:  Introduction to the Mathematics of Infectious Diseases

Abstract:   The earliest mathematical model of infectious disease was developed by Bernoulli in the 1760s to support smallpox inoculation. Developed by Kermack and McKendrick in the 1920-30s and popularized and extended by Anderson and May in the late 1970-80s, the basis for much of mathematical epidemiology is the SIR (Susceptible-Infected-Recovered) model. This talk will serve as an introduction to the mathematics of infectious diseases, covering topics including the next generation matrix and basic reproductive number, growth rate analysis, and identifiability, using the SIR model as an example. I will also briefly discuss some results from my research on the dynamics of HPV infection.
Speaker:  Andrew Brouwer

Introduction to Optimal Control

Date:  Monday, November 03, 2014
Location:  3096 East Hall (2:30 PM to 3:30 PM)

Title:  Introduction to Optimal Control

Abstract:   Optimal control theory, only became a separate branch in applied mathematics in the mid-20th century, has a long history that can date back to the famous Brachystochrone problems in the late 17th. Since it was formalized, optimal control problems popped up in almost all areas of engineering, economy and social science.

In this talk, a historical review of optimal control is first given. Then the formal optimal control problem is introduced, and some theories for its solution are highlighted, including Calculus of Variations, Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman Equation. Numerical techniques, such as dynamic programming, direct collocation method are shown with examples from engineering.

Speaker:  Chaozhe He
Institution:  UM Engineering