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New Perspectives in Geometric Analysis

Program


May 9-11, 2006
Math Department office: University Hall 2040
Lecture Room: Stranahan 0131
Schedule
Tuesday, May 9
Time
 Title
 Speaker
 Location
8:30a-9:20a Registration.
*Please register at Math department Office UH 2040 after 9:20 a.m.
   Stranahan 0131
9:20a-9:30a Opening Address
(Welcome to Toledo!)		 Dean Sue Ott Rowlands
Stranahan 0131
9:30a-10:30a Geometric Methods in Passive-blind Image Forensics
Presentation slides
 Tian-Tsong  Ng (Columbia) Stranahan 0131
10:30a-11:00a Break, refreshments served.   UH2040
11:00a-12:00p Geometric and Topological Issues for Conformally Mapping the Human Brain Monica Hurdal (Florida State) Stranahan 0131
12:00p-2:00p Lunch break  
2:00p-3:00p Adiabatic Elimination:
Stochastic Differential Equations in Physics and Economics 		Reza Kamaly (N & C LLC) Stranahan 0131
3:00p-3:15p Break, refreshments served.
UH2040
3:15p-4:15p Rigidity of CR morphisms between strongly pseudoconvex CR manifolds  Stephen Yau (UIC, Illinois) Stranahan 0131
4:15p-4:30p Break.
   UH2040
4:30p-5:30p Discussion   UH2210
6:30p-8.30p
 Reception
Libbey Hall
Wednesday, May 10
9:00a-9:30a Refreshments served   UH2040
9:30a-10:30a Geometric Analysis Methods Applied in Computer Science  David Xianfeng Gu (Stony Brook) Stranahan 0131
10:30a-11:00a Break, refreshments served.   UH2040
11:00a-12:00p Source Separation based on Morphological Diversity Jean-Luc Starck (Service d'Astrophysique, France) Stranahan 0131
12:00p-2:00p Lunch break   UH2040
2:00p-3:00p Geometry, Harmonic Analysis and Statistical Inference Mikhail Belkin  (OSU) Stranahan 0131
3:00p-3:15p Break, refreshments served.
UH2040
3:15p-4:15p Natural Images: From Multiscale Manifold Models to Compressive Imaging Richard Baraniuk (Rice)  Stranahan 0131
4:15p-4:30p Break   UH2040
4:30p-5:30p Discussion   UH2210
7.00-9.00
 Banquet

Thursday, May 11
9:00a-9:30a Refreshments served   UH2040
9:30a-10:30a Some aspects of Information Geometry Maung Min-Oo (Mcmaster) Stranahan 0131
10:30a-11:00a Break, refreshments served.   UH2040
11:00a-12:00p On quasi-local mass and its positivity
 		Mu-Tao Wang (Columbia) Stranahan 0131
12:00p-12:30p Conclusion.   Stranahan 0131




David Xianfeng Gu (Stony Brook)
Title:Geometric Analysis Methods Applied in Computer Science
Abstrac:
This talk introduces new advances for geometric analysis methods applied in computer graphics, computer vision, geometric modeling, medical imaging fields.

With the development of computer technologies, profound mathematics theories plays more and more vital roles. Many new algorithms, systems based on Geometric analysis have been invented and implemented, which has reshaped the fields of graphics, vision, and will reshape the animation, computer generated movie industry in near future.

This talk will summarize several novel algorithms developed in the last couple of years. The following topics will be discussed: first, heat flow method applied for conformal mapping; second, Hodge theory applied for computing conformal structures. third, Ricci flow applied for uniformization; fourth,  isothermal coordinates applied for fluid dynamics
on surfaces; fifth,  geometric structures for geometric modeling; and finally, Teichmuller space applied for shape analysis;

These powerful theoretic tools are implemented and applied for surface parameterization, surface matching, shape retriveal, animation, facial recognition, colonoscopy, brain mapping.

The fundamental problems in computer graphics will be briefly introduced.
Future directions will be discussed.
Monica Hurdal (Florida State)
Title:
Geometric and Topologicaly Issues for Conformally Mapping the Human Brain
Abstract:
The human brain is a highly convoluted organ with much of the processing occuring on the surface layer called the grey matter. The folding patterns of every indivudal are unique, making it difficulty to compare control groups to target populations. Due to the complex folding patterns of the brain, neuroscientists are interested in utilizing mathematical methods to quantify and characterize brain function and anatomy. I will present some
of the methods that I am using from topology, geometry and complex analysis to create conformal maps of the human brain from magnetic resonance imaging (MRI) data. I will also discuss some of the issues that arise and future directons that are needed to compare different population groups, including MRI data acquired from populations with schizophrenia, Alzheimer's and major depressive disorder.
Reza Kamaly (N & C LLC)
Title:
Adiabatic Elimination:
Stochastic Differential Equations in
Physics and Economics
Abstrac:
This paper contrasts two well–known stochastic differential equations —Langavin’s dynamical theory of Brownian motion and the economic theory of Geometric Brownian Motion— using the adiabatic elimination procedure. The linear Langevin equation is successful as a Markov process, but our investigation highlights difficulties in applying
Langevin’s method blindly to nonlinear GBM phenomenon. In Langavin’s analysis, Markovian properties are only attained after information loss due to spatial and temporal coarse–graining; but in the economics literature, paradoxically, the Markovian property is considered a product of no information loss and a direct result of market efficiency. In describing the dynamical properties of GBM, we prove that GBM suffers a plethora of mathematical and conceptual problems: GBM’s relevance to asset pricing models must be reevaluated. We conclude that appeals to mathematical formalism and Itˆo calculus are, alone, no substitute for cogent analysis.
Stephen Yau (UIC, Illinois)
Title:
Rigidity of CR morphisms between strongly pseudoconvex CR manifolds
Abstract: TBA
Richard Baraniuk (Rice)
Title: Natural Images: From Multiscale Manifold Models to Compressive Imaging
Abstract: TBA
Jean-Luc Starck (Service d'Astrophysique, France)
Title: Source Separation based on Morphological Diversity
Abstract:
The Morphological Component Analysis (MCA) is a a new method which allows us to separate features contained in an image when these features present different morphological aspects. We show that MCA can be very useful for decomposing images into texture and piecewise smooth (cartoon) parts or for inpainting applications. We
extend MCA to a multichannel MCA (MMCA) which leads to a new approach for blind source separation, based on the morphological diversity concept instead of the statistical independence of the source.
Mikhail Belkin  (OSU)
Title:Geometry, Harmonic Analysis and Statistical Inference
Abstract:
In my talk I will discuss geometric aspects of natural data. In particular, I will talk about the role of the Laplace-Beltrami operator in certain statistical inference tasks, how to reconstruct it when the underlying manifold is not known, and algorithms arising from
that point of view.
Tian-Tsong  Ng (Columbia)
Title: Geometric Methods in Passive-blind Image Forensics
Abstract:
In this talk, we will review the emerging research area, passive-blind imagegeometric methods in imageimage forensics is to find out the condition of an image without any prior information such as the pre-embedded digital watermark. The two main functions of image forensics are image forgery detection ( e.g., is an image fake or authentic?) and image source identification (e.g., is an image generated by a camera or a computer). We applied geometricimage forensic problems, i.e., distinguishing photograph and computer graphics, and estimating the camera response function from a single image. We model the differences between photograph and computer graphics using quantities in differential geometry and develop a set of geometricin-depth description of the two geometric methods. 
 Maung Min-Oo (Mcmaster)
Title:Some aspects of Information Geometry
Abstract: TBA
Mu-Tao Wang (Columbia)
Title:On quasi-local mass and its positivity
Abstract:The quasi-local mass is a quantity associated with a compact space-like region in the space-time. It is expected that this information can be derived from the boundary, which is a two-dimensional space-like surface.
We shall discuss some recent developments in this direction which include Shi-Tam's proof of the positivity of the Brown-York mass, Liu-Yau's quasi-local mass and its positivity, and a generalization by Wang-Yau to a quasi-local energy-momentum space-like vector. The construction relies on the solutions of canonically defined elliptic and parabolic equations associated to the geometry of the surface, and the application of the positive mass theorem.