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

Tuesday, October 27, 2009 - 15:00 ,
Location: Skiles 269 ,
Piotr Kokoszka ,
Utah State University ,
Organizer: Liang Peng

The functional autoregressive process
has become a useful tool in the analysis of functional time series
data. In this model, the observations and the errors are curves,
and the role of the autoregressive coefficient is played by
an integral operator.
To ensure meaningful inference and prediction,
it is important to verify that this operator
does not change with time. We propose a method for testing
its constancy which uses the
functional principal component analysis. The test statistic is
constructed to have a Kiefer type asymptotic distribution. The
asymptotic justification of the procedure is very delicate and
touches upon central notions of functional data analysis.
The test is implemented using the
R package fda. Its finite sample performance is
illustrated by an application to credit card transaction data.

Series: Dissertation Defense

In this thesis, some eigenvalue inequalities for Klein-Gordon operators and restricted to a bounded domain in Rd are proved. Such operators become very popular recently as they arise in many problems ranges from mathematical finance to crystal dislocations, especially the relativistic quantum mechanics and \alpha-stable stochastic processes. Many of the results obtained here concern finding bounds for some spectral functions of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogues results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the operator Hm, are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved.

Series: Stelson Lecture Series

This lecture will be more for the mathematical audience

Whether the 3D incompressible Navier-Stokes equations can develop
a finite time singularity from smooth initial data is one of the
seven Millennium Problems posted by the Clay Mathematical Institute.
We review some recent theoretical and computational studies of the
3D Euler equations which show that there is a subtle dynamic depletion of
nonlinear vortex stretching due to local geometric regularity of
vortex filaments. The local geometric regularity of vortex filaments
can lead to tremendous cancellation of nonlinear vortex stretching.
This is also confirmed by our large scale computations for some of
the most well-known blow-up candidates. We also
investigate the stabilizing effect of convection in 3D incompressible
Euler and Navier-Stokes equations. The convection term is the main source
of nonlinearity for these equations. It is often considered destabilizing
although it conserves energy due to the incompressibility condition. Here
we reveal a surprising nonlinear stabilizing effect that the convection
term plays in regularizing the solution. Finally, we present a new class
of solutions for the 3D Euler and Navier-Stokes equations, which exhibit
very interesting dynamic growth property. By exploiting the special
structure of the solution and the cancellation between the convection
term and the vortex stretching term, we prove nonlinear stability and
the global regularity of this class of solutions.

Multiscale Modeling and Computation - The Interplay Between Mathematics and Engineering Applications

Series: Stelson Lecture Series

This lecture is more for the general audience. Reception following lecture. Organizers: Chongchun Zeng and Hao Min Zhou

Many problems of fundamental and practical importance contain multiple scale solutions. Composite and nano materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Direct numerical simulations using a fine grid are very expensive. Developing effective multiscale methods that can capture accurately the large scale solution on a coarse grid has become essential in many engineering applications. In this talk, I will use two examples to illustrate how multiscale mathematics analysis can impact engineering applications. The first example is to develop multiscale computational methods to upscale multi-phase flows in strongly heterogeneous porous media. Multi-phase flows arise in many applications, ranging from petroleum engineering, contaminant transport, and fluid dynamics applications. Multiscale computational methods guided by multiscale analysis have already been adopted by the industry in their flow simulators. In the second example, we will show how to develop a systematic multiscale analysis for incompressible flows in three space dimensions. Deriving a reliable turbulent model has a significant impact in many engineering applications, including the aircraft design. This is known to be an extremely challenging problem. So far, most of the existing turbulent models are based on heuristic closure assumption and involve unknown parameters which need to be fitted by experimental data. We will show that how multiscale analysis can be used to develop a systematic multiscale method that does not involve any closure assumption and there are no adjustable parameters.

Series: Geometry Topology Seminar

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

Monday, October 26, 2009 - 13:00 ,
Location: Skiles 255 ,
Chiu-Yen Kao ,
Ohio State University (Department of Mathematics) ,
kao@math.ohio-state.edu ,
Organizer: Sung Ha Kang

The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersivewave equation which was proposed to study the stability of one solitonsolution of the KdV equation under the influence of weak transversalperturbations. It is well know that some closed-form solutions can beobtained by function which have a Wronskian determinant form. It is ofinterest to study KP with an arbitrary initial condition and see whetherthe solution converges to any closed-form solution asymptotically. Toreveal the answer to this question both numerically and theoretically, weconsider different types of initial conditions, including one-linesoliton, V-shape wave and cross-shape wave, and investigate the behaviorof solutions asymptotically. We provides a detail description ofclassification on the results. The challenge of numerical approach comes from the unbounded domain andunvanished solutions in the infinity. In order to do numerical computationon the finite domain, boundary conditions need to be imposed carefully.Due to the non-periodic boundary conditions, the standard spectral methodwith Fourier methods involving trigonometric polynomials cannot be used.We proposed a new spectral method with a window technique which will makethe boundary condition periodic and allow the usage of the classicalapproach. We demonstrate the robustness and efficiency of our methodsthrough numerous simulations.

Monday, October 26, 2009 - 10:00 ,
Location: Skiles 255 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

We will focus on the "toy model" of bordered Floer homology. Loosely speaking, this is bordered Floer homology for grid diagrams of knots. While the toy model unfortunately does not provide us with any knot invariants, it highlights many of the key ideas needed to understand the more general theory.
Note the different time and place!
This is a 1.5 hour talk.

Series: ACO Distinguished Lecture

Preceded with a reception at 4:10pm.

To come to grips with consciousness, I postulate that living entities in
general, and human beings in particular, are mechanisms... marvelous
mechanisms to be sure but not magical ones... just mechanisms. On this
basis, I look to explain some of the paradoxes of consciousness such as
Samuel Johnson's "All theory is against the freedom of the will; all
experience is for it."
I will explain concepts of self-awareness and free will from a mechanistic
view. My explanations make use of computer science and suggest why these
phenomena would exist even in a completely deterministic world. This is
particularly striking for free will.
The impressions of our senses, like the sense of the color blue, the sound
of a tone, etc. are to be expected of a mechanism with enormously many
inputs categorized into similarity classes of sight, sound, etc. Other
phenomena such as the "bite" of pain are works in progress. I show the
direction that my thinking takes and say something about what I've found and
what I'm still looking for. Fortunately, the sciences are discovering a
great deal about the brain, and their discoveries help enormously in guiding
and verifying the results of this work.

Series: Other Talks

The spectral properties of a graph are intimately related to its structure. This can be applied in the study of discrete isoperimetric problems and in the investigation of extremal and algorithmic questions for graphs. I will discuss several recent examples illustrating this theme.

Series: Other Talks

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; price equilibria in markets; optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysis of the evolution of various types of dynamic stochastic models. It is not known whether these problems can be solved in polynomial time. Despite their broad diversity, there are certain common computational principles that underlie different types of equilibria and connect many of these problems to each other. In this talk we will discuss some of these common principles and the corresponding complexity classes that capture them; the effect on the complexity of the underlying computational framework; and the relationship with other open questions in computation.