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Series: CDSNS Colloquium

In this talk, we generalize the classical Kupka-Smale theorem for ordinary differential
equations on R^n to the case of scalar parabolic equations. More precisely, we show
that, generically with respect to the non-linearity, the
semi-flow of a reaction-diffusion equation defined on a bounded domain
in R^n or on the torus T^n has the "Kupka-Smale" property, that is, all the
critical elements (i.e. the equilibrium points and periodic orbits) are hyperbolic and
the stable and unstable manifolds of
the critical elements intersect transversally. In the particular case of T1, the
semi-flow is generically Morse-Smale,
that is, it has the Kupka-Smale property and, moreover, the
non-wandering set is finite and is only composed of critical
elements. This is an important property, since Morse-Smale semi-flows are structurally
stable. (Joint work with P. Brunovsky and R. Joly).

Series: CDSNS Colloquium

They may flow like fluids but under constraints of mechanical energies from their crystal aspects. As a result, they exhibit very rich phenomena that grant them tremendous applications in modern technology. Based on works of Oseen, Z\"ocher, Frank and others, a continuum theory (not most general but satisfactory to a great extent) for liquid-crystals was formulated by Ericksen and Leslie in 1960s. We will first give a brief introduction to this classical theory and then focus on various important special settings in both static and dynamic cases. These special flows are rather simple for classical fluids but are quite nonlinear for liquid-crystals. We are able to apply abstract theory of nonlinear dynamical systems upon revealing specific structures of the problems at hands.

Series: CDSNS Colloquium

In this talk, I will explain the correspondence between the Lorenz periodic solution and the topological knot in 3-space.The effect of small random perturbation on the Lorenz flow will lead to a certain nature order developed previously by Chow-Li-Liu-Zhou. This work provides an answer to an puzzle why the Lorenz periodics are only geometrically simple knots.

Series: CDSNS Colloquium

An optimal transport path may be viewed as a geodesic in the
space of probability measures under a suitable family of metrics. This
geodesic may exhibit a tree-shaped branching structure in many
applications such as trees, blood vessels, draining and irrigation
systems. Here, we extend the study of ramified optimal transportation
between probability measures from Euclidean spaces to a geodesic metric
space. We investigate the existence as well as the behavior of optimal
transport paths under various properties of the metric such as
completeness, doubling, or curvature upper boundedness. We also introduce
the transport dimension of a probability measure on a complete geodesic
metric space, and show that the transport dimension of a probability
measure is bounded above by the Minkowski dimension and below by the
Hausdorff dimension of the measure. Moreover, we introduce a metric,
called "the dimensional distance", on the space of probability measures.
This metric gives a geometric meaning to the transport dimension: with
respect to this metric, the transport dimension of a probability measure
equals to the distance from it to any finite atomic probability measure.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.