Colloquia 2012-2013: Difference between revisions
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problems. We will also discuss the analog of this problem in positive | problems. We will also discuss the analog of this problem in positive | ||
characteristic. | characteristic. | ||
===Fri, April 27: Gui-Qiang G. Chen (University of Oxford)=== | |||
''Nonlinear Partial Differential Equations of Mixed Type'' | |||
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include | |||
shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations), | |||
among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed. | |||
This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang. |
Revision as of 04:47, 20 April 2012
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Spring 2012
date | speaker | title | host(s) |
---|---|---|---|
Jan 23, 4pm | Saverio Spagnolie (Brown) | Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox | Jean-Luc |
Jan 27 | Ari Stern (UCSD) | Numerical analysis beyond Flatland: semilinear PDEs and problems on manifolds | Jean-Luc / Julie |
Feb 3 | Akos Magyar (UBC) | On prime solutions to linear and quadratic equations | Street |
Feb 8 | Lan-Hsuan Huang (Columbia U) | Positive mass theorems and scalar curvature problems | Sean |
Feb 10 | Melanie Wood (UW Madison) | Counting polynomials and motivic stabilization | local |
Feb 17 | Milena Hering (University of Connecticut) | The moduli space of points on the projective line and quadratic Groebner bases | Andrei |
Feb 24 | Malabika Pramanik (University of British Columbia) | Analysis on Sparse Sets | Benguria |
March 2 | cancelled | ||
March 9 | Eftychios Sifakis (UW-Madison, CS Dept.) | Numerical algorithms for physics-based modeling and interactive visual computing: Parallelism, scalability and their impact on theoretical research directions | Nigel |
Wednesday, March 14, 4PM | Sebastien Roch (UCLA) | Phase Transitions in Molecular Evolution: Relating Combinatorial and Variational Distances on Trees | |
March 16 | Burak Erdogan (UIUC) | Smoothing for the KdV equation and Zakharov system on the torus | Street |
March 19 | Colin Adams and Thomas Garrity (Williams College) | Which is better, the derivative or the integral? | Maxim |
March 23 | Martin Lorenz (Temple University) | Prime ideals and group actions in noncommutative algebra | Don Passman |
March 30 | Wilhelm Schlag (University of Chicago) | Invariant manifolds and dispersive Hamiltonian equations | Street |
April 6 | Spring recess | ||
April 13 | Ricardo Cortez (Tulane) | Introduction to the method of regularized Stokeslets for fluid flow and applications to microorganism swimming | Mitchell |
April 18 | Benedict H. Gross (Harvard) | The arithmetic of elliptic curves | distinguished lecturer |
April 19 | Benedict H. Gross (Harvard) | Arithmetic invariant theory | distinguished lecturer |
April 20 | Robert Guralnick (University of Southern California) | Maps from the Generic Riemann surface to the Riemann sphere | Shamgar |
April 27 | Gui-Qiang Chen (Oxford) | Nonlinear Partial Differential Equations of Mixed Type | Feldman |
Abstracts
Mon, Jan 23: Saverio Spagnolie (Brown)
"Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox"
The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability.
Fri, Feb 3: Akos Magyar (UBC)
On prime solutions to linear and quadratic equations
The classical results of Vinogradov and Hua establishes prime solutions of linear and diagonal quadratic equations in suﬃciently many variables. In the linear case there has been a remarkable progress over the past few years by introducing ideas from additive combinatorics. We will discuss some of the key ideas, as well as their use to obtain multidimensional extensions of the theorem of Green and Tao on arithmetic progressions in the primes. We will also discuss some new results on prime solutions to non-diagonal quadratic equations of suﬃciently large rank. Most of this is joint work with B. Cook.
Wed, Feb 8: Lan-Hsuan Huang (Columbia U)
Positive mass theorems and scalar curvature problems
More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.
Fri, Feb 10: Melanie Wood (local)
Counting polynomials and motivic stabilization
We will begin with the problem of counting polynomials modulo a prime p with a given pattern of root multiplicity. Here we will discover phenomena that point to vastly more general patterns in configuration spaces of points. To see these patterns, one has to work in the ring of motives--so we will describe this place where a space is equivalent to the sum of its pieces. We will then be able to describe how these patterns in the ring of motives are related to theorems in topology on the homological stability of configuration spaces. This talk is based on joint work with Ravi Vakil.
Fri, Feb. 17: Milena Hering (UConn)
The moduli space of points on the projective line and quadratic Groebner bases
The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood recently in work of Howard, Millson, Snowden and Vakil. They prove that the ideal of relations is generated by quadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Groebner basis.
Fri, Feb. 24: Malabika Pramanik (UBC)
Analysis on Sparse Sets
Fri, March 9: Eftychios Sifakis (UW-Madison, CS Dept)
Numerical algorithms for physics-based modeling and interactive visual computing: Parallelism, scalability and their impact on theoretical research directions
In recent years, computer graphics research and visual computing in general have become significantly more dependent on efficient and scalable numerical methods. Simulation of natural environments in the visual effects industry and interactive virtual environments for skill training are prime examples of numerically intensive visual tasks. The performance potential of modern hardware has also inspired current and emerging applications to expand their demands beyond photorealistic rendering and visual detail. A number of areas will now associate visual fidelity and appeal with the ability of computer-generated models to resolve the anatomical function of virtual human bodies, or the intricate dynamics of natural phenomena. In addition, the sensitivity of interactive applications to run-time performance warrants careful examination of the theoretical design choices for numerical techniques that maximize the benefit of modern parallel compute platforms.
This talk will highlight a number of numerical techniques where specific design choices have had significant performance and parallelism repercussions: discrete elliptic PDEs, high-order methods for interface problems on regular grids, multigrid methods for nonlinear problems, and preconditioning of Krylov methods. All these examples are drawn from computer graphics and physics-based modeling applications, and will be demonstrated in such context. I will particularly emphasize how various intricacies of computing platforms (such as bandwidth, vector width and synchronization considerations) often suggest nontrivial adjustments to theoretical approaches in order to maximize computational efficiency.
Wed, March 14: Sebastien Roch (UCLA)
Phase Transitions in Molecular Evolution: Relating Combinatorial and Variational Distances on Trees
I will describe recent results on a deep connection between a well-studied phase transition in Markov random fields on trees and two important problems in evolutionary biology: the inference of ancestral molecular sequences and the estimation of large phylogenies using maximum likelihood. No biology background will be assumed.
Fri, March 23: Martin Lorenz (Temple University)
Prime ideals and group actions in noncommutative algebra
Having originated from number theory, the notion of a prime ideal has become central in many different branches of algebra. This talk will focus on the role of prime ideals in the representation theory of noncommutative algebras and the use of group actions as an efficient tool in organizing the spectrum of all prime ideals of a given algebra. In many cases, the group in question is an affine algebraic group and geometric methods are essential.
Fri, March 30: Wilhelm Schalg (University of Chicago)
Invariant manifolds and dispersive Hamiltonian equations
We will review recent work on the role that center-stable manifolds play in the study of dispersive unstable evolution equations. More precisely, by means of the radial cubic nonlinear Klein-Gordon equation we shall exhibit a mechanism in which the ground state soliton generates a center-stable manifold which separates a region of data leading to finite time blowup from another where solutions scatter to a free wave in forward time. This is joint work with Kenji Nakanishi from Kyoto University, Japan.
Fri, April 13: Ricardo Cortez (Tulane)
Introduction to the method of regularized Stokeslets for fluid flow and applications to microorganism swimming
Biological flows, such as those surrounding swimming microorganisms, can be properly modeled using the Stokes equations for fluid motion with external forcing. The organism surfaces can be viewed as flexible interfaces imparting force or torque on the fluid. Interesting flows have been observed when the organism swims near a solid wall due to the hydrodynamic interaction of rotating flagella with a neighboring solid surface. I will introduce the method of regularized Stokeslets and some extensions of it that are used to compute these flows. The method is based on fundamental solutions of linear PDEs, leading to integral representations of the solution. I will present the idea of the method, some of the known results and applications to flows generated by swimming flagella.
Wed, April 18: Benedict Gross (Harvard)
The arithmetic of elliptic curves
The question of rational points on cubic curves has been of central interest in number theory for 350 years. About 50 years ago, Bryan Birch and Peter Swinnerton-Dyer formulated a precise conjecture for the rank of the group of rational points, in terms of the number of solutions to the equation (modulo p) for all primes p. I will review the progress that has been made on this conjecture, and will discuss a method, recently introduced by Manjul Bhargava, to bound the average rank.
Thur, April 19: Benedict Gross (Harvard)
Arithmetic invariant theory
David Mumford introduced geometric invariant theory to study the relation between the orbits of an algebraic group G on a linear representation V and the algebra of G-invariant polynomials on V. The geometric theory was developed over an algebraically closed ground field. We will consider the stable orbits in some simple representations over the rational numbers, and relate them to the arithmetic of hyperelliptic curves.
Fri, April 20: Bob Guralnick (USC)
Maps from the Generic Riemann surface to the Riemann sphere
Zariski, in his thesis, proved that any map from the generic Riemann surface of genus g > 6 could not be solvable. Using more sophisticated permutation group theory, we can prove a much stronger result: if f is an indecomposable map of degree n from the generic Riemann surface of genus g > 3 to the Riemann sphere, then the monodromy group of f is either the symmetric group of degree n with n > (g+2)/2 or the alternating group of degree n with n > 2g. We will discuss the main ideas used in solving this problem and some related problems. We will also discuss the analog of this problem in positive characteristic.
Fri, April 27: Gui-Qiang G. Chen (University of Oxford)
Nonlinear Partial Differential Equations of Mixed Type
Many nonlinear partial differential equations arising in mechanics and geometry naturally are of mixed hyperbolic-elliptic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear partial differential equations of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations), isometric embedding problems in in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear partial differential equations with these longstanding problems and will discuss some recent developments in the analysis of these nonlinear equations through the examples with emphasis on identifying/developing mathematical approaches, ideas, and techniques to deal with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed. This talk will be based mainly on the joint work correspondingly with Mikhail Feldman, Marshall Slemrod, as well as Myoungjean Bae and Dehua Wang.