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Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

We
will discussing the wobbling of some pedestrian bridges induced by
walkers when crowded and show how this discussion leads to several
problems that can be studied with the help of mathematical modeling,
analysis
and simulations.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

In this talk, we start with the mathematical modeling of air-water interaction in the framework of the interface problem between two incompressible inviscid fluids under the influence of gravity/surface tension. This is a nonlinear PDE system involving free boundary. It is generally accepted that wind generates surface waves due to the instability of shear flows in this context. Based on the linearized equations about shear flow solutions, we will discuss the
classical Kelvin--Helmholtz instability etc. before we illustrate Miles' critical layer theory.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We analyze this stochastic model by applying truncation approximations to the exact master equations describing the evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis.

Series: Research Horizons Seminar

Food and Drinks will be provided after the seminar.

In this seminar, Prof. John Etnyre
will begin this talk by discussing a classical question concerning
periodic motions of particles in classical physics. In trying to better
understand this question we will develop the notion of a symplectic
structure. This is a fundamental geometric
object that provides the "right way" to think about classical mechanics,
and many many other things too. We will then indicate how modern ideas
can be used to give, at least partial, answers to our initial naive
questions about periodic motions.

Series: Research Horizons Seminar

Many physical models without dissipation can be written in a Hamiltonian
form. For example, nonlinear Schrodinger equation for superfluids and
Bose-Einstein condensate, water waves and their model equations (KDV,
BBM, KP, Boussinesq systems...), Euler equations for inviscid fluids,
ideal MHD for plasmas in fusion devices, Vlasov models for collisionless
plasmas and galaxies, Yang-Mills equation in gauge field theory etc.
There exist coherent structures (solitons, steady states, traveling
waves, standing waves etc) which play an important role on the long time
dynamics of these models. First, I will describe a general framework to
study linear stability (instability) when the energy functional is
bounded from below. For the models with indefinite energy functional
(such as full water waves), approaches to find instability criteria will
be mentioned. The implication of linear instability (stability) for
nonlinear dynamics will be also briefly discussed.

Series: Research Horizons Seminar

Please note the delayed start for this week only.

The main focus of this talk is a class of asymptotic methods called
averaging. These methods approximate complicated differential equations
that contain multiple scales by much simpler equations. Such
approximations oftentimes facilitate both analysis and computation. The
discussion will be motivated by simple examples such as bridge and
swing, and it will remain intuitive rather than fully rigorous. If time
permits, I will also mention some related projects of mine, possibly
including circuits, molecules, and planets.

Series: Research Horizons Seminar

In this talk, we will discuss what entails being a front-office
quant at JP Morgan in the Emerging Markets group. We discuss
why Emerging Markets is viewed as its own asset class and what there is to
model. We also give practical examples of things we look at on a daily
basis. This talk aims to be informal and to appeal to a wide audience.

Series: Research Horizons Seminar

We will discuss methods for solving polynomial equations with integer solutions using the loops on the space of all complex
solutions to the same equations. We will then state generalizations of this method due to A. Grothendieck.

Series: Research Horizons Seminar

There is a beautiful idea that one can study spaces by
studying associated geometric objects. More specifically one can
associate to a manifold (that is some space) a symplectic or contact
manifold (that is the geometric object). The question is how useful is
this idea. We will discuss this idea and related questions for subspaces
(that is immersions and embeddings) with a focus on curves in the plane
and knots in three space. If time permits we will discuss powerful new
tools from contact geometry that allow one use this idea to construct
invariants of knots and more generally embeddings and immersions in any
space.

Series: Research Horizons Seminar

This talk is intended to be a cocktail of many things. I will start
with standard random matrices (called GUE in the slang) and formal
computations which leads one to the main problem of counting planar
diagrams. This was done by physicists, though the main computation of
generating functions for such planar diagrams go through an analytic
tools. Here I will change the topic to analysis, and get through with
the help of Chebyshev polynomials and how these can be used to solve a
minimization problem and then from there to compute several generating
functions of planar diagrams. Then I will talk about
tridiagonalization which is a main tool in matrix analysis and point out
an interesting potential view on this subject.