Seminars and Colloquia by Series

Monday, December 6, 2010 - 11:00 , Location: Skiles 169 , John Mallet-Paret , Brown University , Organizer: Chongchun Zeng
We examine a variety of problems in delay-differential equations. Among the new results we discuss are existence and asymptotics for multiple-delay problems, global bifurcation of periodic solutions, and analyticity (or lack thereof) in variable-delay problems. We also plan to discuss some interesting open questions in the field.
Monday, November 29, 2010 - 11:00 , Location: Skiles 169 , Federico Bonetto , Georgia Tech , Organizer: Chongchun Zeng
Modern Economic Theory is largely based on the concept of Nash Equilibrium. In its simplest form this is an essentially statics notion. I'll introduce a simple model for the use of money (Kiotaki and Wright, JPE 1989) and use it to introduce a more general (dynamic) concept of Nash Equilibrium and my understanding of its relation to Dynamical Systems Theory and Statistical Mechanics.
Monday, November 22, 2010 - 11:00 , Location: Skiles 169 , Nan Lu , Georgia Tech , Organizer: Chongchun Zeng
We consider a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. We first derive and justify the limit system of the slow variables. Assuming a steady state persists, we construct the stable, unstable, center-stable, center-unstable, and center manifolds of the steady state of a size of order $O(1)$ and give their leading order approximations. Finally, using these tools, we study the persistence of homoclinic solutions in this type of normally elliptic singular perturbation problems.
Monday, November 15, 2010 - 11:00 , Location: Skiles 169 , Björn Sandstede , Brown University , Organizer: Chongchun Zeng
In this talk, I will discuss localized stationary 1D and 2D structures such as hexagon patches, localized radial target patterns, and localized 1D rolls in the Swift-Hohenberg equation and other models. Some of these solutions exhibit snaking: in parameter space, the localized states lie on a vertical sine-shaped bifurcation curve so that the width of the underlying periodic pattern, such as hexagons or rolls, increases as we move up along the bifurcation curve. In particular, snaking implies the coexistence of infinitely many different localized structures. I will give an overview of recent analytical and numerical work in which localized structures and their snaking or non-snaking behavior is investigated.
Monday, November 8, 2010 - 11:00 , Location: Skiles 169 , Shu-Ming Sun , Virginia Tech , Organizer: Chongchun Zeng
The talk concerns the mathematical aspects of solitary waves (i.e. single hump waves) moving with a constant speed on water of finite depth with surface tension using fully nonlinear Euler equations governing the motion of the fluid flow. The talk will first give a quick formal derivation of the solitary-wave solutions from the Euler equations and then focus on the mathematical theory of existence and stability of two-dimensional solitary waves. The recent development on the existence and stability of various three-dimensional waves will also be discussed.
Monday, October 25, 2010 - 11:00 , Location: Skiles 114 , Shouhong Wang , Indiana University , Organizer: Chongchun Zeng
Gas-liquid transition is one of the most basic problem to study in equilibrium phase transitions. In the pressure-temperature phase diagram, the gas-liquid coexistence curve terminates at a critical point C, also called the Andrews critical point. It is, however, still an open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. To answer this basic question, using the Landau's mean field theory and the Le Chatelier principle,  a dynamic model for the gas-liquid phase transitions is established. With this dynamic model, we are able to derive a theory on the Andrews critical point C: 1) the critical point is a switching point where the phase transition changes from the first order with latent heat to the third order, and 2) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the liquid-gas phase transition going beyond the Andrews point. In addition, the study suggest an asymmetry principle of fluctuations, which appears also in phase transitions in ferromagnetic systems. The analysis is based on the dynamic transition theory we have developed recently with the philosophy to search the complete set of transition states. The theory has been applied to a wide range of nonlinear problems. A brief introduction for this theory will be presented as well. This is joint with Tian Ma.
Monday, April 26, 2010 - 11:00 , Location: Skiles 269 , Mark Pollicott , University of Warwick , Organizer: Yingfei Yi
We consider a shift transformation and a Gibbs measure and estimate the drop in entropy caused by deleting an arbitrarily small (cylinder) set. This extends a result of Lind. We also estimate the speed at which the Gibbs measure escapes into the set, which relates to recent work of Bunimovich-Yurchenko and Keller-Liverani. This is joint with Andrew Ferguson.
Thursday, April 22, 2010 - 16:00 , Location: Skile 255 , Prof. Weiping Li , Oklahoma State University , Organizer: Haomin Zhou
Based on a sequence of discretized American option price processes under the multinomial model proposed by Maller, Solomon and Szimayer (2006), the sequence converges to the counterpart under the original L\'{e}vy  process in distribution for almost all time. We prove a weak convergence in this case for American put options for all time. By adapting Skorokhod representation theorem, a new sequence of approximating processes with the same laws with the multinomial tree model defined by Maller, Solomon and Szimayer (2006) is obtained. The new sequence of approximating processes satisfies Aldous' criterion for tightness. And, the sequence of filtrations generated by the new approximation converges to the filtration generated by the representative of L\'{e}vy process weakly. By using results of Coquet and Toldo (2007), we give a complete proof of the weak convergence for the approximation of American put option prices for all time.
Monday, March 15, 2010 - 11:00 , Location: Skiles 269 , Chao-Nien Chen , National Changhua University, Taiwan , Organizer: Yingfei Yi
There are many interesting patterns observed in activator-inhibitor systems. A well-known model is the FitzHugh-Nagumo system. In conjunction with calculus of variations, there is a close relation between the stability of a steady state and its relative Morse index. We give a sufficient condition in diffusivity for the existence of standing wavefronts joining with Turing patterns.
Monday, February 15, 2010 - 16:30 , Location: Skiles 255 , Vladimir Belykh , Nizhny Novgorod University , Organizer: Yingfei Yi
In this lecture, I will discuss a class of multidimensional maps with one nonlinearity, often called discrete-time Lurie systems. In the 2-D case, this class includes Lozi map and Belykh map. I will derive rigorous conditions for the multidimensional maps to have a generalized hyperbolic attractor in the sense of Bunimovich-Pesin. Then, I will show how these chaotic maps can be embedded into the flow, and I will give specific examples of three-dimensional piece-wise linear ODEs, generating this class of hyperbolic attractors.