- You are here:
- GT Home
- Home
- News & Events

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

In this talk, we will discuss: (1) How geometry plays a role in machine learning/data science? (2) What it's like being a mathematician at a software company.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

Weak Galerkin (WG) is a new finite element method for partial differential equations where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation. In this talk, the speaker will introduce a general framework for WG methods by using the second order elliptic problem as an example. Furthermore, the speaker will present WG finite element methods for several model PDEs, including the linear elasticity problem, a fourth order problem arising from fluorescence tomography, and the second order problem in nondivergence form. The talk should be accessible to graduate students with adequate training in computational mathematics.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar

In this seminar,we will explain why and how unpredictable (chaotic) dynamics arises in deterministic systems. Some open problems in dynamical systems, probability, statistical mechanics, optics, (differential) geometry and number theory will be formulated.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.

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.