Seminars and Colloquia by Series

Normally Elliptic Singular Perturbations and persistence of homoclinic orbits

Series
CDSNS Colloquium
Time
Monday, November 22, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Nan Lu – Georgia Tech
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.

Localized planar patterns

Series
CDSNS Colloquium
Time
Monday, November 15, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Björn Sandstede – Brown University
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.

Exact Theory of Solitary Waves on Water with Surface Tension

Series
CDSNS Colloquium
Time
Monday, November 8, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Shu-Ming Sun – Virginia Tech
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.

Dynamic Transition Theory and its Application to Gas-Liquid Phase Transitions

Series
CDSNS Colloquium
Time
Monday, October 25, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Shouhong Wang – Indiana University
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.

Asymptotic entropy drops and escape rates for Gibbs measures

Series
CDSNS Colloquium
Time
Monday, April 26, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mark Pollicott – University of Warwick
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.

A weak convergence for Approximation of American Option Prices

Series
CDSNS Colloquium
Time
Thursday, April 22, 2010 - 16:00 for 1 hour (actually 50 minutes)
Location
Skile 255
Speaker
Prof. Weiping Li – Oklahoma State University
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.

Turing patterns and standing waves of FitzHugh-Nagumo type systems

Series
CDSNS Colloquium
Time
Monday, March 15, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Chao-Nien Chen – National Changhua University, Taiwan
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.

Multidimensional chaotic maps with hyperbolic attractors

Series
CDSNS Colloquium
Time
Monday, February 15, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Vladimir Belykh – Nizhny Novgorod University
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.

Modeling cancer stem cell differentiation

Series
CDSNS Colloquium
Time
Monday, February 1, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
269 Skiles
Speaker
Peter Kim – University of Utah –
We improved a computational model of leukemia development from stem cells to terminally differentiated cells by replacing the probabilistic, agent-based model of Roeder et al. (2006) with a system of deterministic, difference equations. The model is based on the relatively recent theory that cancer originates from cancer stem cells that reside in a microenvironment, called the stem cell niche. Depending on a stem cell’s location within the stem cell niche, the stem cell may remain quiescent or begin proliferating. This emerging theory states that leukemia (and potentially other cancers) is caused by the misregulation of the cycle ofproliferation and quiescence within the stem cell niche.Unlike the original agent-based model, which required seven hours per simulation, our model could be numerically evaluated in less than five minutes. The results of our numerical simulations showed that our model closely replicated the average behavior of the original agent-based model. We then extended our difference equation model to a system of age-structured partial differential equations (PDEs), which also reproduced the behavior of the Roeder model. Furthermore, the PDE model was amenable to mathematical stability analysis, which revealed three modes of behavior: stability at 0 (cancer dies out), stability at a nonzero equilibrium (a scenario akin to chronic myelogenous leukemia), and periodic oscillations (a scenario akin to accelerated myelogenous leukemia).The PDE formulation not only makes the model suitable for analysis, but also provides an effective mathematical framework for extending the model to include other aspects, such as the spatial distribution of stem cells within the niche.

The emergence of travelling waves for reaction-diffusion equations under a co-moving change of coordinates

Series
CDSNS Colloquium
Time
Monday, November 30, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Maria Lopez – Consejo Superior de Investigaciones Cientificas Madrid, Spain
We introduce a change of coordinates allowing to capture in a fixed reference frame the profile of travelling wave solutions for nonlinear parabolic equations. For nonlinearities of bistable type the asymptotic travelling wave profile becomes an equilibrium state for the augmented reaction-diffusion equation. In the new equation, the profile of the asymptotic travelling front and its propagation speed emerge simultaneously as time evolves. Several numerical experiments illustrate the effciency of the method.

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