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Colloquium: Spring 2012

(Tentative Schedule)

Time & location: All talks are on Thursday in Gibson 325 at 3:30pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Comments indicating vacations, special lectures, or change in location or time are made in green

Organizer: Mahir Can

January 11

Equivariant Lusternik-Schnirelmann Category of Compact Closed Orientable Surfaces and Related Topics

Waclaw MarzantowiczUAM (poland)

Abstract:

The problem of finding the equivariant Lusternik-Schnirelmann category of a compact manifold with an action of compact Lie group arises in a natural way, since similarly to the classical absolute case, it gives a lower estimate for the number of critical points of any equivariant $C^1$-function. In this elementary work we present a complete formula for the equivariant category of any orientation preserving action of a finite group $G$ on a closed surface. It turns out that, with some exceptions, it equals the number of singular orbits of the action. Some  direct applications are also given. Moreover, the gradient field of such a function defines a flow which generator is in the centralizer of any finite group of homeomorphisms. The cyclic group induced by this generator can be use to make shorter and more effective a  proof of theorem on the  homotopical theory of periodic points of homeomorphisms of finite order (or maps of homotopy finite order) of surfaces.

 

January 19

Partial Regularity of Weak Solutions of the Viscoelastic Navier-Stokes Equations

Ryan HyndNew York university

Abstract:

We prove an analog of the celebrated Caffarelli-Kohn-Nirenberg theorem for weak solutions of a system of PDE that model a viscoelastic fluid in the presence of an energy damping mechanism.  The system was recently introduced in a method of establishing the global in time existence of weak solutions of the well known Oldroyd model, which remains an open problem.

Location: Gibson 325

Time: 3:30 PM

January 24

Non Pattern Formation in Chemorepulsion

Kun ZhaoUniversity of iowa

Abstract:

In contrast to diffusion (random diffusion without orientation), chemotaxis is the biased movement of cells/particles toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. The former often refers to the attractive chemotaxis and latter to the repulsive chemotaxis. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures such as aggregates, fruiting bodies, clusters, spirals, spots, rings, labyrinthine patterns and stripes, which have been observed in experiments. In this talk, I will present some recent development on the rigorous analysis of a partial differential equation model arising from repulsive chemotaxis which is a system of conservation laws consisting of nonlinear and coupled parabolic and hyperbolic type PDEs. In particular, global wellposedness, large-time asymptotic behavior of classical solutions to such model are obtained which indicate that chemorepulsion problem of this type exhibits strong tendency against pattern formation. The results are consistent with general results for classical repulsive chemotaxis models.

January 26

The Role of Topology in the Theory of Integrable Systems

Leo ButlerCentral michigan university

Abstract:

At the end of the 19th century, H. Poincare began to build and use the tools of analysis situs in order to understand the qualitative properties of differential equations that defied exact solution. In this talk I will look at how this line of thought has developed, especially in connection to the theory of integrable, or exactly solvable, systems.

January 27

Quantitative Unique Continuation Results for  Anisotropic Operators

Tu NguyenUniversity of washington

Abstract:

We will describe the role of Carleman estimates in proving quantitative unique continuation results. In particular, recent results for anisotropic Maxwell systems and general parabolic operators will be discussed.

Location: Gibson 414

Time: 3:00 PM

February 1

Second and Fourth Order Nonlinear Eigenvalue Problems

Craig CowanStanford University

Abstract:

In this talk we will examine various second and fourth order nonlinear elliptic partial differential equations which contain a parameter which is varied.   Our interest is in how the value of the parameter effects the various properties of the solution(s) of the equation.  We will mostly concentrate on the question of the regularity of the so called extremal solution.

Location: Gibson 414

Time: 3:30 PM


February 2

From Swarming to Vortex Interaction:  Applications of Nonlocal PDEs

David UminskyUCLA

Abstract:

I will survey results in the analysis and simulations of a large class of nonlocal PDEs. The problems are of active scalar type and the applications are dictated by the nonlocal, interaction kernel. We will begin by considering nonlocal kernels with a gradient, attraction-repulsion structure which arise in minimal models of biological swarming and self-assembly. We use tools from dynamical systems and analysis to develop the mathematical theory for predicting which patterns will arise.  We also discuss solving the inverse problem of designing kernels for a given ground state structure. We conclude by turning to kernels which are incompressible in nature and discuss the numerical analysis of a new convergent, higher order deformable vortex method for simulating fluids.

Location: Stanley Thomas 101

Time: 2:00 PM


February 2

Structure-Preserving Integrators for Constrained Mechanical Systems

Dmitry ZenkovNC State University

Abstract:

Mechanical systems often have structures that are intrinsically preserved by systems' dynamics. Preserving these structures may be crucial for the quality of long-term numerical simulations of dynamics. This talk will introduce some of these structures, discuss their significance, and their discrete analogues for mechanical systems with velocity constraints.

February 7

The Geometry and Topology of Ricci Solitons

Ovidiu Muneanucolumbia university

Abstract:

"Ricci solitons are special Riemannian manifolds that arise in the study of Ricci flow, more prominently in the singularity analysis of the flow. The classification of Ricci solitons in three dimensions has been central for the Hamilton-Perelman proof of the Poincare conjecture. In this talk I will survey some recent development in the study of Ricci solitons in arbitrary dimension. I will begin with a brief introduction to Ricci flow and the role of solitons in this theory. Then the talk will focus on the structure of Ricci solitons and will include topics such as curvature and volume growth control and the topology at infinity. Understanding the solutions of certain partial differential equations is an important instrument for our study."

February 9

An Introduction to Cluster Algebras

Andrei ZelevinskyNortheastern University

Abstract:

Cluster algebras are commutative rings of a special kind discovered by Sergey Fomin and the speaker about a decade ago.  Since then they have made a surprising appearance in a variety of settings, including quiver representations, Poisson geometry, Teichmuller theory, non-commutative geometry, integrable systems, etc.  Their structure is governed by several discrete dynamical systems given by specific piecewise-polynomial and birational recurrences on a regular tree.  We will discuss foundations of the theory of cluster.  An exposition will be elementary and self-contained.

February 16

Optimal Stirring for Passive Scalar Mixing

Charlie Doeringuniversity of Michigan, ann arbor

Abstract:

We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a measure for mixing we adopt the H^{-1} norm of the scalar fluctuation field.  This 'mix-norm' is equivalent to (the square root of) the variance of a low-pass filtered image of the tracer concentration field, and is a useful gauge even in the absence of molecular diffusion.  This mix-norm's vanishing as time progresses is evidence of the stirring flow's mixing property in the sense of ergodic theory.  For the case of a periodic spatial domain with a prescribed instantaneous energy or power budget for the stirring, we determine the flow field that instantaneously maximizes the decay of the mix-norm, i.e., the instantaneous optimal stirring --- when such a flow exists.  When no such 'steepest descent' stirring exists, we determine the flow that maximizes that rate of increase of the rate of decrease of the norm.  This local-in-time stirring strategy is implemented computationally on a benchmark problem and compared to an optimal control approach utilizing a restricted set of flows.  This is joint work with Zhi Lin, Eveleyn Lunasin, and Jean-Luc Thiffeault.

Charles Doering is professor of mathematics at the University of Michigan, Ann Arbor. He is notable for his research that is generally focused on the analysis of stochastic dynamical systems arising in biology, chemistry and physics, to systems of nonlinear partial differential equations. Recently he has been focusing on fundamental questions in fluid dynamics as part of the $1M Clay Institute millennium challenge concerning the regularity of solutions to the equations of fluid dynamics. With J. D. Gibbon, he notably co-authored the book Applied Analysis of the Navier-Stokes Equations, published by Cambridge University Press.  

February 23

Deforming Associatives in G2 Manifolds

Selman Akbulutmichigan state university


Abstract:

A G2 structure on a 7-dimensional  manifold is a reduction of its tangent frame bundle to the Lie group G2. I will discuss G2 manifolds and their certain natural 3 and 4-dimensional submanifolds, which are called associative and coassociatives (they behave like holomorphic curves and Lagrangian submanifolds). I will describe various deformation spaces of associatives, discuss how they can be made smooth and relate this to their Seiberg-Witten invariants (joint work with S. Salur).
March 1

Studies on Curve Singularities

Claudia PoliniNotre Dame University

Abstract:

The goal of the talk is to relate the singularity types of a rational plane curve to the syzygies of the forms parametrizing it.  This is a report on joint work with Cox, Kustin, and Ulrich. More specifically, let C be a rational plane curve of degree d parametrized by three forms, which can be assumed to be of degree d as well. The syzygy matrix of this parametrization is a 2 by 3 matrix whose entries are forms of degrees d_1 and d_2, where d_1 + d_2=d. Among other things we consider curves of even degree d=2c; we show that if C has a singular point (including an infinitely near singular point) of multiplicity at least c, then the multiplicity of this singularity is exactly c and furthermore d_1 = d_2 =c. We establish, essentially, a correspondence between the constellation of multiplicity c singularitie on or infinitely near C on the one hand and the shapes of the syzygy matrices on the other hand. Using this, we give a stratification of the space of rational plane curves into irreducible locally closed sets, according to the constellation of singularities of maximal multiplicity c. 

March 8

Euclid and Dehn in the Operating Room

Margaret SymingtonMercer University

Abstract:

In this talk I will discuss an unlikely collaboration with a dermatologic surgeon -- the Euclidean geometry problem that emerged, and the little known kinship between dermatologists and geometric topologists it revealed. Along the way I will share some interesting topological results and pose a set of geometry questions that anyone can work on.

 

March 15

SPRING BREAK

March 22

Degenerate Diffusion Operator in Population Biology

Charles EpsteinUniversity of Pennsylvania

Abstract:

I describe recent joint work with Rafe Mazzeo proving existence, uniqueness and regularity results for degenerate diffusion operators that arise as infinite population limits of Markov Chain models in Population Genetics.  These problems are complicated because the configuration space is a manifold with corners and the principal symbol of the operator degenerates along the boundary. These existence results serve to prove the existence of a limiting Markov Process.

 

March 29

A Finiteness Theorem for W-Graphs

John StembridgeUniversity of michigan, ann arbor

Abstract:

A W-graph is an edge-weighted graph that encodes certain matrix representations of a Weyl group W or its associated Hecke algebra. In particular, the action of the Hecke algebra by left or right multiplication on its Kazhdan-Lusztig basis has this form.  If one is given the W-graph, it is easy to compute the Kazhdan-Lusztig polynomials, and these in turn carry useful information about the representation theory of the associated complex Lie groups.

In this talk we will point out a few basic features common to the W-graphs in Kazhdan-Lusztig theory (thereby isolating the class of "admissible" W-graphs), and explain how these features still manage to capture a large part of what is essential. For example, it turns out that there are only finitely many admissible W-graphs that are strongly connected, and the possibility of determining the full Kazhdan-Lusztig W-graph via combinatorics is tantalizingly close.

April 5

TBA

SpeakerUniversity

Abstract: TBA

April 12

New Techniques in Invariant Theory: Toric Deformation and Hibi Rings

Roger HoweYale university

Abstract:

In recent years, new techniques have been shedding light on long-standing issues and classical computationsin invariant theory and representation theory.  One new idea is ``toric deformation" - finding a simpler algebra that  retains the key properties of an algebra you are interested in.  This talk will review some key uses of toric deformation in invariant theory, highlighting the appearance of a particularly nice class of algebras known as Hibi rings.

April 19

Varieties in Flag Manifolds and Their Patch Ideals

Alexander YongUniversity of Illinois at urbana-champaign

Abstract:

This talk addresses the problem of how to analyze and discuss singularities of a variety X that ``naturally'' sits inside a flag manifold.

Our three main examples are Schubert varieties, Richardson varieties and Peterson varieties. The overarching theme is to use combinatorics and commutative algebra to study the *patch ideals*, which encode local coordinates and equations of X. Thereby, we obtain formulas and conjectures about X's invariants. We will report on projects with (subsets of) Erik Insko (U. Iowa), Allen Knutson (Cornell), Li Li (Oakland University) and Alexander Woo (U. Idaho).


April 26

Wavelet Leader Multifractal Analysis for Textures in Paintings:  Mandelbrot Meets Van Gogh

Patrice AbryENS Lyon, France

Abstract:

In image processing, texture characterization and classification is a classical theme that received continuous and significant attention. Amongst other properties, it is now well accepted that scale invariance (or simply scaling) provides practitioners with a fruitful paradigm to address texture characterization.  In this context, multifractal analysis consists of a mathematical theory aiming at characterizing precisely the fluctuations along space of the local regularity of a field.

Recently, a multifractal formalism (that is, a practical procedure to apply multifractal analysis to real world data) based on wavelet leaders has been proposed. It will be explained how and why it permits an accurate analysis of scale invariance in images and actually outperforms formalisms based on the sole wavelet coefficients.

Non parametric bootstrap procedures implemented in the wavelet domain further enable us to provide not only estimates for the multifractal attributes but also confidence intervals and to design hypothesis tests. These formalisms and procedures will be illustrated at work on the characterization of the textures of the paintings of different masters, a study conducted in the framework of the "Image Processing for Art Investigation" project. Notably, it will be illustrated how multifractal analysis enables to discriminate forgeries from original... or not!

(Joint work with S. Jaffard, Math Dept., Paris Est University and H. Wendt, ENSHEEIT, Toulouse, France)

May 3

Global Invariant Manifolds of Spike States

Peter Batesmichigan state university, east lansing

Abstract: 

Singularly perturbed elliptic equations often give rise to solutions that are almost constant except for one or more localized large amplitude excursions, so-called spike layers.  Here we consider the corresponding parabolic equations and show the existence of moving spike layer solutions – existing and retaining that shape for all negative as well as positive time.  In fact, the resulting dynamics is a rather nice purely geometric flow, which itself is of independent interest.

This PDE to ODE reduction result is made possible by first proving an abstract theorem on the existence of a normally hyperbolic invariant manifold for a semiflow in Banach space when one only has an approximation of such.  We believe that this abstract result is applicable in a wide variety of settings.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu