Web: http://www.ima.umn.edu | Email: ima-staff@ima.umn.edu | Telephone: (612) 624-6066 | Fax: (612) 626-7370
Additional newsletters available at http://www.ima.umn.edu/newsletters

IMA Newsletter #357

July 2006

News and Notes

Recent and upcoming IMA programs

The 2005–2006 IMA thematic program on Imaging drew to a close at the end of June. The very exciting ten month program brought in over 700 visitors, including over 70 who visited for at least a month. Bytes, Camera, Action at IMA Workshop, which appeared in the June edition of SIAM News, reports on one of the Imaging workshops.

The 2006–2007 thematic program on Applications of Algebraic Geometry is proving tremendously popular, with 95 confirmed visitors of a month for more, about a third of whom will be at the IMA for the duration of the program. But first come the summer 2006 IMA programs: the recently concluded New Directions short course on Biophysical Fluid Dynamics, the 2006 summer program on Symmetries and Overdetermined Systems of Partial Differential Equations, the IMA PI Summer Program for Graduate Students on Topology and its Applications, and the Mathematical Modeling in Industry X workshop for graduate students.

Blackwell–Tapia Conference

The IMA is now accepting participant applications for the fourth Blackwell–Tapia Conference, to be held at the IMA on November 3–4, 2006. The one and a half day meeting honors David Blackwell and Richard Tapia, who inspired a generation of African-American, Native American and Latino/Latina students to pursue careers in mathematics.

The conference will include a mix of activities, including talks by William Massey and Ehran Cinlar of Princeton, Illya Hicks of Texas A&M, and Mark Lewis of Cornell; a session of poster presentations; panel discussions about career opportunities in mathematics, and about recruitment and retention of a diverse mathematics workforce; and ample opportunities for discussion and interaction.

A high point of the meeting will be the awarding, during the conference banquet on Saturday evening, of 2006 Blackwell-Tapia prize to William Massey, in recognition of his outstanding achievements in queuing theory, stochastic networks, and the modeling of communications systems, and in increasing diversity in the mathematical sciences.

Limited funds are available for travel and local expenses, so apply soon.

IMA Events

PI Summer Graduate Program

Topology and its Applications

Mississippi State University, Starkville

July 10-28, 2006

Organizer: Kevin Knudsen (Mississippi State University)

Topological issues arise in a number of diverse areas. In molecular biology, for example, the geometric features of the surface of a molecule have been shown to influence certain protein docking processes. Knot theory is becoming increasingly important in the study of DNA. Computer scientists encounter topological problems in attempts to reconstruct surfaces from sampled data. Topology in phase space can help overcome the inherent sensitivity in longtime simulations of dynamical systems. During this three-week meeting there will be week-long courses on: A typical day's schedule will consist of two lectures in the morning, one after lunch, and informal problem sessions in the late afternoon. This program is open only to graduate students from IMA Participating Institutions.

Symmetries and Overdetermined Systems of Partial Differential Equations

July 17 - August 4, 2006

Organizers: Michael Eastwood (University of Adelaide) and Willard Miller Jr. (University of Minnesota Twin Cities)

This summer program is dedicated to the memory of Thomas P. Branson, who played
a leading role in its conception and organization, but did not live to see its realization.

The symmetries to be studied in the this Summer Program naturally arise in several different ways. Firstly, there are the symmetries of a differential geometric structure. By definition, these are the vector fields that preserve the structure in question—the Killing fields of Riemannian differential geometry, for example. Secondly, the symmetries can be those of another differential operator. For example, the Riemannian Killing equation itself is projectively invariant whilst the ordinary Euclidean Laplacian gives rise to conformal symmetries. In addition, there are higher symmetries defined by higher order operators. Physics provides other natural sources of symmetries, especially through string theory and twistor theory.

These symmetries are usually highly constrained—viewed as differential operators, they themselves are overdetermined or have symbols that are subject to overdetermined differential equations. As a typical example, the symbol of a symmetry of the Laplacian must be a conformal Killing field (or a conformal Killing tensor for a higher order symmetry). The Summer Program will consider the consequences of overdeterminacy and partial differential equations of finite type.

Schedule

Monday, July 17

8:15a-9:10aRegistration and CoffeeEE/CS 3-176 SP7.17-8.4.06
9:10a-9:20aWelcome to the IMADouglas N. Arnold (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
9:20a-9:30aIntroduction by the OrganizersMichael Eastwood (University of Adelaide)
Willard Miller Jr. (University of Minnesota Twin Cities)
EE/CS 3-180 SP7.17-8.4.06
9:30a-10:30aExterior differential systems for ordinary differential equations Boris Doubrov (Belarus State University)EE/CS 3-180 SP7.17-8.4.06
10:30a-11:00aBreakEE/CS 3-176 SP7.17-8.4.06
11:00a-12:00pLinear wave equations in curved space-time, with emphasis on their separability and conservation laws Niky Kamran (McGill University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch Break SP7.17-8.4.06
2:00p-3:00pIntroduction to Moving Frames and Pseudo-Groups Peter J. Olver (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
3:00p-3:05pGroup Photo SP7.17-8.4.06
3:05p-3:30pBreakEE/CS 3-176 SP7.17-8.4.06
3:30p-4:30pProjective differential geometry without moving frames Michael Eastwood (University of Adelaide)EE/CS 3-180 SP7.17-8.4.06
5:00p-6:30pReception and Poster SessionLind Hall 400 SP7.17-8.4.06
Partially invariant solutions to ideal magnetohydrodynamicsSergey Golovin (Queen's University)

Tuesday, July 18

9:15a-9:30aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:30a-10:30aSuperintegrable classical and quantum systems Pavel Winternitz (University of Montreal)EE/CS 3-180 SP7.17-8.4.06
10:30a-11:00aBreakEE/CS 3-176 SP7.17-8.4.06
11:00a-12:00pExterior differential systems for ordinary differential equations (continued)Boris Doubrov (Belarus State University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch Break SP7.17-8.4.06
2:00p-3:00pLinear wave equations in curved space-time, with emphasis on their separability and conservation laws (continued)Niky Kamran (McGill University)EE/CS 3-180 SP7.17-8.4.06
3:00p-3:30pBreakEE/CS 3-176 SP7.17-8.4.06
3:30p-4:30pPseudogroups, moving frames, invariants, and symmetries (continued)Peter J. Olver (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06

Wednesday, July 19

9:15a-9:30aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:30a-10:30aAmbient metric constructions in CR and conformal geometry Kengo Hirachi (University of Tokyo)EE/CS 3-180 SP7.17-8.4.06
10:30a-11:00aBreakEE/CS 3-176 SP7.17-8.4.06
11:00a-12:00pProjective differential geometry without moving frames (continued)Michael Eastwood (University of Adelaide)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch Break SP7.17-8.4.06
2:00p-3:00pOverdetermined systems, conformal differential geometry, and the BGG complex Andreas Cap (University of Vienna)EE/CS 3-180 SP7.17-8.4.06
3:00p-3:30pBreakEE/CS 3-176 SP7.17-8.4.06
3:30p-4:30pProjective differential geometry with moving frames Joseph Landsberg (Texas A & M University)EE/CS 3-180 SP7.17-8.4.06
6:30p-8:30pWorkshop Dinner SP7.17-8.4.06

Thursday, July 20

9:15a-9:30aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:30a-10:30aPseudogroups, moving frames, invariants, and symmetries (continued)Peter J. Olver (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
10:30a-11:00aBreakEE/CS 3-176 SP7.17-8.4.06
11:00a-12:00pGeometry of linear differential systems—towards "contact geometry of second order" Keizo Yamaguchi (Hokkaido University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch Break SP7.17-8.4.06
2:00p-3:00pSuperintegrable classical and quantum systems (continued)Pavel Winternitz (University of Montreal)EE/CS 3-180 SP7.17-8.4.06
3:00p-3:30pBreakEE/CS 3-176 SP7.17-8.4.06
3:30p-4:30pAmbient metric constructions in CR and conformal geometry (continued)Kengo Hirachi (University of Tokyo)EE/CS 3-180 SP7.17-8.4.06

Friday, July 21

Monday, July 24

9:15a-9:30aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:30a-10:30aOverdetermined systems, conformal differential geometry, and the BGG complex (continued)Andreas Cap (University of Vienna)EE/CS 3-180 SP7.17-8.4.06
10:30a-11:00aBreakEE/CS 3-176 SP7.17-8.4.06
11:00a-12:00pProjective differential geometry with moving frames (continued)Joseph Landsberg (Texas A & M University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch Break SP7.17-8.4.06
2:00p-3:00pGeometry of linear differential systems—towards "contact geometry of second order" (continued)Keizo Yamaguchi (Hokkaido University)EE/CS 3-180 SP7.17-8.4.06
3:00p-3:30pBreakEE/CS 3-176 SP7.17-8.4.06
3:30p-4:30pSuperintegrable classical and quantum systems (continued)Pavel Winternitz (University of Montreal)EE/CS 3-180 SP7.17-8.4.06
8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aSymplectification procedure for the equivalence problem of vector distributionsIgor Zelenko (International School for Advanced Studies (SISSA/ISAS))EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aThe Structure of Continuous PseudogroupsJuha Pohjanpelto (Oregon State University)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pTBADaniel Fox (Georgia Institute of Technology)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pAnalogues of the Dolbeault complex and the separation of variablesVladimir Soucek (Karlovy (Charles) University)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pGroup-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces and Lie groupsStephen Anco (Brock University)EE/CS 3-180 SP7.17-8.4.06
4:30p-6:00pReception and Poster Session SP7.17-8.4.06
Differential invariants of Lie pseudogroups in mechanics of fluidsSergey Golovin (Queen's University)

Tuesday, July 25

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aThe ambient metric to all orders in even dimensionsRobin Graham (University of Washington)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aRigidity of the adjoint varietiesColleen Robles (University of Rochester)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pSecond order superintegrable systems in two and three dimensions. (Solving a system in multiple ways)Willard Miller Jr. (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pGalilean vector fields: tensor products and invariants with using moving frames approachAnatoly Nikitin (National Academy of Sciences of Ukraine)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pExterior differential systems with symmetryMark Fels (Utah State University)EE/CS 3-180 SP7.17-8.4.06

Wednesday, July 26

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aExact and quasi-exact solvability superintegrability in Euclidean spaceGeorge Pogosyan (Yerevan State University)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aQuadratic algebras and superintegrable systemsJonathan Kress (University of New South Wales)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pMultiplication of solutions for systems of partial differential equationsJens Jonasson (Linköping University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pA geometric approach to Kalnins and Miller non-regular separationClaudia Chanu (Università di Torino)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pDiscrete symmetries and Lie algebra automorphismsPeter Hydon (University of Surrey)EE/CS 3-180 SP7.17-8.4.06

Thursday, July 27

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aSeparation of variables for systems of first-order Partial Differential Equations: the Dirac equation in two-dimensional manifoldsGiovanni Rastelli (Università di Torino)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aClassification of 3-dimensional scalar discrete integrable equationsThomas Wolf (Brock University)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pInvolutive differential systems and tableaux over Lie algebrasLorenzo Nicolodi (Università di Parma)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pProjective-type differential invariants and geometric evolutions of KdV-typeGloria Mari Beffa (University of Wisconsin)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pTBAEE/CS 3-180 SP7.17-8.4.06

Friday, July 28

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aSeparation of variables theory for the Hamilton-Jacobi equation from the perspective of the invariant theory of Killing tensorsRay McLenaghan (University of Waterloo)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aGeometry "a la Cartan" revisited: Hamilton-Jacobi theory in moving framesRoman Smirnov (Dalhousie University)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pTwistor forms on Riemannian manifoldsUwe Semmelmann (Universität zu Köln)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pGlobally hyperbolic Lorentzian manifolds with special holonomyHelga Baum (Humboldt-Universität)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pOn the integrability of multi-dimensional quasilinear systemsEugene Ferapontov (Loughborough University)EE/CS 3-180 SP7.17-8.4.06

Monday, July 31

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aFinite element exterior calculus and its applicationsDouglas N. Arnold (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aA simple criterion for involutivityElizabeth L. Mansfield (University of Kent at Canterbury)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pDifferential and variational calculus in invariant framesIrina Kogan (North Carolina State University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pAlgorithmic Symmetry Analysis and Overdetermined Systems of PDEGreg Reid (University of Western Ontario)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pRational and Algebraic Invariants of a Group ActionEvelyne Hubert (Institut National de Recherche en Informatique Automatique (INRIA))EE/CS 3-180 SP7.17-8.4.06

Tuesday, August 1

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aOverdetermined elliptic boundary value problemsJukka Tuomela (University of Joensuu)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreak EE/CS 3-180 SP7.17-8.4.06
10:00a-10:45aGeometric integration and controlDebra Lewis (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pOverdetermined systems, invariant connections, and short detour complexesA. Rod Gover (University of Auckland)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pHigher spin gauge theories and unfolded dynamicsMikhail Vasiliev (P. N. Lebedev Physics Institute)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pSpecial polynomials associated with rational solutions of the Painleve equations and applications to soliton equationsPeter A. Clarkson (University of Kent at Canterbury)EE/CS 3-180 SP7.17-8.4.06

Wednesday, August 2

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aCR-manifolds, differential equations and multicontact structures (tentative) Gerd Schmalz (University of New England)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aWünsch's calculus for parabolic geometriesJan Slovak (Masaryk University)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pDifferential equations and conformal structuresPawel Nurowski (University of Warsaw)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pSymmetry algebras for even number of vector fields and for linearly perturbed complex structuresChong-Kyu Han (Seoul National University)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pProjective transformations and integrable systemsVladimir S. Matveev (Katholieke Universiteit Leuven)EE/CS 3-180 SP7.17-8.4.06

Thursday, August 3

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aThe Work of Thomas P. Branson
(Michael Eastwood, moderator)
Michael Eastwood (University of Adelaide)EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aGeometric analysis in parabolic geometriesBent Orsted (Aarhus University)EE/CS 3-180 SP7.17-8.4.06
10:45a-11:15aBreakEE/CS 3-176 SP7.17-8.4.06
11:15a-12:00pEquivariant differential operators, classical invariant theory, unitary representations, and Macdonald polynomialsSiddhartha Sahi (Rutgers University)EE/CS 3-180 SP7.17-8.4.06
12:00p-2:00pLunch SP7.17-8.4.06
2:00p-2:45pRepresentation theory of SU(n,1) and Sp(n,1)Michael Cowling (University of New South Wales)EE/CS 3-180 SP7.17-8.4.06
2:45p-3:15pBreakEE/CS 3-176 SP7.17-8.4.06
3:15p-4:00pTBAPaul-Andi Nagy (Humboldt-Universität)EE/CS 3-180 SP7.17-8.4.06

Friday, August 4

8:45a-9:00aCoffeeEE/CS 3-176 SP7.17-8.4.06
9:00a-9:45aThe Uniqueness of the Joseph Ideal for the Classical GroupsPetr Somberg (Karlovy (Charles) University (UK))EE/CS 3-180 SP7.17-8.4.06
9:45a-10:00aBreakEE/CS 3-176 SP7.17-8.4.06
10:00a-10:45aFinal Discussion Group
Willard Miller Jr., moderator
Willard Miller Jr. (University of Minnesota Twin Cities)EE/CS 3-180 SP7.17-8.4.06
Abstracts
Stephen Anco (Brock University) Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces and Lie groups
Abstract: This talk will present a broad, unified generalization of much recent work on the geometric derivation of soliton equations and their bi-Hamiltonian integrability structure from curve flows in various kinds of Riemannian geometries and Klein geometries. In particular, it will be shown that for each symmetric space G/H there is a hierarchy of geometric curve flows in which the components of the principal normal vector along the curve satisfy a group-invariant bi-Hamiltonian soliton equation. The derivation is based on a natural construction of moving parallel frames and moving covariantly-constant frames in such spaces. Examples of symmetric spaces include, among others, constant curvature manifolds e.g. Sn, flat conformal manifolds, Kahler and quaternion manifolds e.g. CPn, QPn, compact Lie group manifolds e.g. SU(n), Sp(n). These examples lead to a wide class of bi-Hamiltonian soliton equations describing multicomponent group-invariant systems of modified KdV type, nonlinear Schrodinger type, and sine-Gordon type. The corresponding curve flows are related to geometric variants of mKdV maps, Schrodinger maps, and wave maps on G/H.
Douglas N. Arnold (University of Minnesota Twin Cities) Finite element exterior calculus and its applications
Abstract: Finite element exterior calculus is a theoretical approach to the design and understanding of discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These spaces connect to each other in discrete subcomplexes of elliptic differential complexes, which are themselves connected to the continuous elliptic complex through projections which commute with the complex differential. This structure relates directly to the stability of discretization methods based on the finite element spaces. Applications include elliptic systems, electromagnetism, elasticity, elliptic eigenvalue problems, and preconditioners.
Helga Baum (Humboldt-Universität) Globally hyperbolic Lorentzian manifolds with special holonomy
Abstract: The (connected) holonomy groups of Riemanninan manifolds are well known for a long time. The (connected) holonomy groups of Lorentzian manifolds were classified only recently by Thomas Leistner. Anton Galaev finished this classification by describing local analytic metrics for all of these Lorentzian holomomy groups (including the still missing coupled types). The next step in this program is to describe Lorentzian metrics with special holonomy and precribed global properties (geodesically complete, globally hyperbolic,...). In the talk I will explain a method to construct globally hyperbolic Lorentzian manifolds with special holonomy using Riemannian spin manifolds with Codazzi spinors. This is a joint work with Olaf Müller.
Gloria Mari Beffa (University of Wisconsin) Projective-type differential invariants and geometric evolutions of KdV-type
Abstract: In this talk we will consider curves in any flat homogeneous manifolds G/H with G semisimple. We will define projective-type differential invariants for these curves and we will prove that there exist curve evolutions invariant under G such that, when written in terms of the differential invariant of the curves, they become completely integrable equations of KdV-type if appropriate initial conditions are chosen. We will also describe the background Poisson Geometry that causes this relation.
Andreas Cap (University of Vienna) Overdetermined systems, conformal differential geometry, and the BGG complex
Abstract: The starting point of my lectures will be a way to rewrite certain overdetermined systems on Riemannian manifolds in closed form. The method is based on including the orthogonal group O(n) into the pseudo-orthogonal group O(n+1,1) and analyzing the standard representation of O(n+1,1) from the point of view of this subgroup. Next, I will indicate how, replacing direct observations by tools from representation theory, this method can be generalized to a large class of systems. Then I will explain how the inclusion of O(n) into O(n+1,1) that we started from is related the passage from Riemannian to conformal geometry. Refining the methods slightly, one obtains a construction for a large family of conformally invariant differential operators. In the end, I want to sketch how the ideas generalize further to a large class of geometric structures called parabolic geometries.
Claudia Chanu (Università di Torino) A geometric approach to Kalnins and Miller non-regular separation
Abstract: A geometric interpretation of non-regular additive separation for a PDE, as described by Kalnins and Miller is provided. This general picture contains as special cases both fixed energy separation and constrained separation of Helmoltz equation (not necesarily orthogonal). Moreover, the geometrical approach to non-regular separation allows you to explain why there are some coordinates in Euclidean 3-space in which a R-separable solutions of Helmoltz equation exist (depending on a fewer number of parameters than in the regular case) but which are apparently not related to classical Staeckel form. The differential equations that characterize this kind of non-regular R-separation on a general Riemannian manifold are given. Moreover, for the Euclidean 3-dimensional space general conditions on the form of the metric tensor in these coordinates are provided.
Peter A. Clarkson (University of Kent at Canterbury) Special polynomials associated with rational solutions of the Painleve equations and applications to soliton equations
Abstract: In this talk I shall discuss special polynomials associated with rational solutions for the Painleve equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, modified Korteweg-de Vries, classical Boussinesq and nonlinear Schrodinger equations. The Painleve equations (PI-PVI) are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For PII these polynomials are known as the Yablonskii-Vorob'ev polynomials, first derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of PIV are described and it is shown that their roots also have a highly regular structure. It is well known that soliton equations have symmetry reductions which reduce them to the Painleve equations. Hence rational solutions of soliton equations arising from symmetry reductions of the Painleve equation can be expressed in terms of the aforementioned special polynomials. Also the motion of the poles of the rational solutions of the Korteweg-de Vries equation is described by a constrained Calogero-Moser system describes the motion of the poles of rational solutions of the Korteweg-de Vries equation, as shown by Airault, McKean, and Moser in 1977. The motion of the poles of more general rational solutions of equations in the Korteweg-de Vries, modified Korteweg-de Vries and classical Boussinesq equations, and the motion of zeroes and poles of rational and new rational-oscillatory solutions of the nonlinear Schrodinger equation will be discussed.
Michael Cowling (University of New South Wales) Representation theory of SU(n,1) and Sp(n,1)
Abstract: The groups SU(n,1) and Sp(n,1) act on spheres with Carnot-- Caratheodory structures. The analysis of these actions leads to information about the unitary and uniformly bounded representations of these groups. This talk surveys recent work of Francesca Astengo, the speaker and Bianca Di Blasio on the representations of the real rank one semisimple Lie groups.
Boris Doubrov (Belarus State University) Exterior differential systems for ordinary differential equations
Abstract: We consider geometric structures associated with systems of ordinary differential equations. In particular, we explore various exterior differential systems defined by ODE's and show how to construct associated absolute parallelisms and Cartan connections in a natural way. The whole theory is split into a series of examples from the simplest ones to the general case of arbitrary order ODEs. We also present practical algorithms suitable for explicit calculation of invariants of ordinary differential equations. In the case of a single ODE we list the generators in the algebra of all contact invariants. As an application, we discuss the classes of trivializable equations and Elie Cartan's C-classe equations, which can be solved without any integration at all.
Michael Eastwood (University of Adelaide) The Work of Thomas P. Branson
(Michael Eastwood, moderator)
Abstract: Tom Branson played a leading role in the conception and organization of this Summer Program. Tragically, he passed away in March this year and the Summer Program is now dedicated to his memory. This session will be devoted to a discussion of his work. The format will be decided in consultation with others during the earlier part of the Program and anyone wishing to present material is asked to contact the moderator.
Mark Fels (Utah State University) Exterior differential systems with symmetry
Abstract: Given a symmetry group of an exterior differential system, we'll investigate ways to use the group to simplify finding integral manifolds.
Eugene Ferapontov (Loughborough University) On the integrability of multi-dimensional quasilinear systems
Abstract: In this talk I will give an overview of the method of hydrodynamic reductions applied to the classification of various particularly interesting classes of multi-dimensional PDEs, including hydrodynamic type systems and equations of the dispersionless Hirota type. The integrability conditions constitute overdetermined systems for coefficients of the equations, which are in involution. Differential-geometric interpretation of these conditions will be discussed.
Sergey Golovin (Queen's University) Differential invariants of Lie pseudogroups in mechanics of fluids
Abstract: We present bases of differential invariants for Lie pseudogroups admitted by the main models of fluid mechanics. Among them infinite-dimensional parts of symmetry groups of Navier-Stokes equations in general (V.O. Bytev, 1972) and rotationally-symmetric (L.V. Kapitanskij, 1979) cases; stationary gas dynamics equations (M. Munk, R. Prim, 1947); stationary incompressible ideal magnetohydrodynamics (O.I. Bogoyavlenskij, 2000). Applications of the obtained bases to construction of differentially-invariant solutions and group foliations of the differential equations are demonstrated.
A. Rod Gover (University of Auckland) Overdetermined systems, invariant connections, and short detour complexes
Abstract: With mild restrictions, each overdetermined differential operator is equivalent to a (tractor-type) connection on a prolonged system, and this connection depends only on the operator concerned. On the other hand in Riemannian geometry (for example), natural conformally invariant overdetermined operators may, given suitable curvature restrictions, be extended to an elliptic conformally invariant complex that we term a short detour complex. (These complexes yield an approach to studying deformations of various structures, and these complexes and their hyperbolic variants also have a role in gauge theory.) These constructions are intimately related.
Robin Graham (University of Washington) The ambient metric to all orders in even dimensions
Abstract: The ambient metric associated to a conformal manifold is an important object in conformal geometry. However, the basic construction is obstructed at finite order in even dimensions. This talk will describe how to complete the construction to all orders in even dimensions. One obtains a family of smooth ambient metrics determined up to smooth diffeomorphism. These ambient metrics arise as an invariantly defined smooth part of inhomogeneous Ricci-flat metrics with asymptotic expansions involving log terms. This is joint work with Kengo Hirachi.
Chong-Kyu Han (Seoul National University) Symmetry algebras for even number of vector fields and for linearly perturbed complex structures
Abstract: We discuss the existence of solutions and the dimension of the solution spaces for infinitesimal symmeries of the following two cases: firstly, even number (2n) of vector fields in a manifold of dimension 2n+1, and secondly, almost complex manifold with linearly perturbed structure. We use the method of complete prolongation for thses overdetermined linear pde systems of first order and checking the integrability of the associated Pfaffian systems.
Kengo Hirachi (University of Tokyo) Ambient metric constructions in CR and conformal geometry
Abstract: Ambient metric is a basic tool in CR and conformal geometry. It was first introduced by Fefferman in an attempt to write down the asymptotic expansion of the Bergman kernel and later was generalized to the case of conformal geometry by Fefferman-Graham. In these talks, I will start with the construction of the ambient metric and describe its applications, including the construction of local CR/ conformal invariants (with an application to the Bergman kernel), invariant differential operators and Branson's Q-curvature.
Evelyne Hubert (Institut National de Recherche en Informatique Automatique (INRIA)) Rational and Algebraic Invariants of a Group Action
Abstract: We consider a rational group action on the affine space and propose a construction of a finite set of rational invariants and a simple algorithm to rewrite any rational invariant in terms of those generators. The construction is shown to be an algebraic analogue of the moving frame construction of local invariants [Fels & Olver 1999]. We introduce a finite set of replacement invariants that are algebraic functions of the rational invariants. They are the algebraic analogues of the Cartan's normalized invariants and give rise to a trivial rewriting. This is joint work with Irina Kogan, North Carolina State University.
Peter Hydon (University of Surrey) Discrete symmetries and Lie algebra automorphisms
Abstract: This talk reviews a method that enables one to construct all discrete point symmetries of any given differential equation that has nontrivial Lie point symmetries. The method extends to other classes of symmetries, including contact, internal and generalized symmetries. It is based on the observation that the adjoint action of an arbitrary symmetry induces a Lie algebra automorphism. By classifying all such automorphisms, it becomes possible to find discrete symmetries with little more effort than it takes to determine the Lie symmetries. To this end, we present a fairly concise classification of all automorphisms of real Lie algebras of dimension five or less.
Jens Jonasson (Linköping University) Multiplication of solutions for systems of partial differential equations
Abstract: The Cauchy-Riemann equations is an example of a system of partial differential equations that is equipped with a multiplication (a bi-linear operation) on its solution set. This multiplication is an immediate consequence of the multiplication of holomorphic functions in one complex variable. Another, more sophisticated, example is the multiplication of cofactor pair systems, that provides a method for generating large families of dynamical systems that can be solved through the method of separation of variables.
Niky Kamran (McGill University) Linear wave equations in curved space-time, with emphasis on their separability and conservation laws
Abstract: In the first lecture, we shall review the construction of the basic first-order linear field equations for fields of spin s in curve space-time, and show how separable second-order wave equations arise from these in the important setting provided by the Kerr geometry. We will also describe the construction of conserved energy densities for these wave equations with suitable positivity properties outside the ergosphere. In the second lecture, we will show how these ingredients can be used to study the long-time behavior of solutions to the Cauchy problem, and in particular how one can prove decay as t tends to infinity for spin 1/2 fields and scalar waves. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau.
Irina Kogan (North Carolina State University) Differential and variational calculus in invariant frames
Abstract: The talk is devoted to theoretical and computational aspects of performing differential and variational calculus relative to a group-invariant frame on a jet bundle. Many important systems of differential equations and variational problems, arising in geometry and physics, admit a group of symmetries. S. Lie recognized that symmetric problems can be expressed in terms of group-invariant objects: differential invariants, invariant differential forms, and invariant differential operators. It is desirable from both computational and theoretical points of view to use a group-invariant basis of differential operators (invariant frame) and the dual basis of differential forms (invariant coframe) to perform further computations with symmetric systems. Complexity of the structure equations for a non-standard coframe and non-commutativity of differential operators present, however, both theoretical and computational challenges. I will present formulas and symbolic algorithms for vector fields prolongation, integration by parts, Euler-Lagrange and Helmholtz operators, all relative to an invariant frame. Applications to the problem of finding group-invariant conservation laws and solving invariant inverse problem of calculus of variations will be considered. This talk is based on joint work with P. Olver and I. Anderson.
Jonathan Kress (University of New South Wales) Quadratic algebras and superintegrable systems
Abstract: Commutators of the symmetries of a superintegrable system do not necessarily close to form a finite-dimensional Lie algebra, but instead may be quadratic in a basic set symmetries. This is known to be a generic property of non-degenerate superintegrable systems. This talk will discuss the structure of these algebras and their uses, for example, in the classification of superintegrable systems.
Joseph Landsberg (Texas A & M University) Projective differential geometry with moving frames
Abstract: I will describe the projective differential invariants of submanifolds of projective space and give several examples of their uses. Applications will include: Griffiths-Harris rigidity of homogeneous varieties and studying the spaces of lines on a projective variety. I will also explain relations with the study of G-structures. These lectures will also serve as an elementary introduction to moving frames.
Debra Lewis (University of Minnesota Twin Cities) Geometric integration and control (tentative title)
Abstract: The global trivializations of the tangent and cotangent bundles of Lie groups significantly simplifies the analysis of variational problems, including Lagrangian mechanics and optimal control problems, and Hamiltonian systems. In numerical simulations of such systems, these trivializations and the exponential map or its analogs (e.g. the Cayley transform) provide natural mechanisms for translating traditional algorithms into geometric methods respecting the nonlinear structure of the groups and bundles. The interaction of some elementary aspects of geometric mechanics (e.g. non-commutativity and isotropy) with traditional methods for vector spaces yields new and potentially valuable results.
Elizabeth L. Mansfield (University of Kent at Canterbury) A simple criterion for involutivity
Abstract: One of the ways overdetermined systems have been studied is via Spencer cohomology of the symbol of the system. This machinery can seem rather forbidding but nevertheless intriguing as to what it might offer, as it is intrinsically co-ordinate independent. In this talk we "deconstruct" the key definitions and prove a relationship between a system being a characteristic set and being involutive. In fact, we turn the concepts around so that we can use the now familiar concepts of syzygies (a.k.a. compatibility conditions) to investigate involutivity.
Vladimir S. Matveev (Katholieke Universiteit Leuven) Projective transformations and integrable systems
Abstract: I will explain the proof of the following two theorems Thm1 (Lichnerowicz-Obata-Conjecture): Let (M,g) be a complete Riemannian manifold (of dimension >1) of nonconstant curvature. Suppose a connected Lie group acts on (M,g) by projective transformations. Then, it acts by affine transformations. Thm2 (Geodesic Rigidity Theorem): Let M be a closed manifold. Suppose there exists two nonproportional Riemannian metrics on it such that they are projectively equivalent. Then, the manifold admits a reducible metric, or is diffeomorphic to a reducible space form. The main new instrument of the proof is a connection between projectively equivalent metrics and quadratic in velocity integrals.
Ray McLenaghan (University of Waterloo) Separation of variables theory for the Hamilton-Jacobi equation from the perspective of the invariant theory of Killing tensors
Abstract: The theory of algebraic invariants of Killing tensors defined on pseudo-Riemannian spaces of constant curvature under the action of the isometry group is described. The theory is illustrated by the computation of bases for the invariants and reduced invariants on three dimensional Euclidean and Minkowski spaces. The invariants are employed to characterize the orthogonally separable coordinate webs for the Hamilton-Jacobi equation for the geodesics and the Laplace and wave equations.
Willard Miller Jr. (University of Minnesota Twin Cities) Final Discussion Group
Willard Miller Jr., moderator
Abstract: A primary aim of this Summer Program is to promote fruitful interaction between various research groups and individuals currently working, perhaps unwittingly, on overlapping themes. This session will be devoted to a public discussion of problems and possible directions for future research and collaboration. The format will be decided in consultation with others during the earlier part of the Program and anyone wishing to present material is asked to contact the moderator.
Lorenzo Nicolodi (Università di Parma) Involutive differential systems and tableaux over Lie algebras
Abstract: I will outline some recent work (joint with E. Musso) on the construction of involutive differential systems based on the concept of a tableau over a Lie algebra. Particular cases of this scheme lead to differential systems describing various familiar classes of submanifolds in homogeneous spaces which constitute integrable systems. This offers another perspective for better understanding the geometry of these submanifolds.
Anatoly Nikitin (National Academy of Sciences of Ukraine) Galilean vector fields: tensor products and invariants with using moving frames approach
Abstract: All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. Tensor products and joint invariants for such representations are found with using moving frames approach.
Pawel Nurowski (University of Warsaw) Differential equations and conformal structures
Abstract: We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature \$(3,2)\$. The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group \$G_2\$. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.
Peter J. Olver (University of Minnesota Twin Cities) Introduction to Moving Frames and Pseudo-Groups
Abstract: I will present the basics of the equivariant method of moving frames. First, the relevant constructions for finite-dimensional Lie group actions will be presented. Applications include the classification of differential invariants, invariant differential equations and variational problems, symmetry and equivalence problems, and the design of invariant numerical algorithms. Then I will introduce infinite-dimensional Lie pseudo-groups and discuss how to extend the moving frame methods. The lectures will include a self-contained introduction to the variational bicomplex.
Bent Orsted (Aarhus University) Geometric analysis in parabolic geometries
Abstract: Many aspects of parabolic geometries are by now well understood, especially those related to differential geometry and the symmetries of natural differential operators associated with these geometries. In this talk we shall see how some aspects of geometric analysis may be generalized from the best-known cases, namely Riemannian and conformal geometry, resp. CR geometry, to more general geometries. In particular we shall give results about Sobolev spaces and inequalities, and also mention results about unitary representations of the natural symmetry groups.
George Pogosyan (Yerevan State University) Exact and quasi-exact solvability superintegrability in Euclidean space
Abstract: We show that separation of variables for second-order superintegrable systems in two- and three-dimensional Euclidean space generates both exactly solvable and quasi-exactly solvable problems in quantum mechanics. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of quasi-exactly solvable problems via partial differential equations have only incorporated separability, not multiseparability. We also propose another definition of exactly and quasi-exactly solvability. The quantum mechanical problem is called exactly solvable if the solution of Schroedinger equation, can be expressed in terms of hypergeometrical functions and is quasi-exactly solvable if the Schroedinger equation admit polynomial solutions with the coefficients necessarily satisfying the three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is quasi-exactly solvable in one set of separable coordinates, but is not exactly solvable in any other separable coordinates. The work done with colloboration with E.Kalnins and W.Miller Jr.
Juha Pohjanpelto (Oregon State University) The Structure of Continuous Pseudogroups
Abstract: I will report on my ongoing joint work with Peter Olver on developing systematic and constructive algorithms for analyzing the structure of continuous pseudogroups and identifying various invariants for their action. Unlike in the finite dimensional case, there is no generally accepted abstract object to play the role of an infinite dimensional pseudogroup. In our approach we employ the bundle of jets of group transformations to parametrize a pseudogroup, and we realize Maurer-Cartan forms for the pseudogroup as suitably invariant forms on this pseudogroup jet bundle. Remarkably, the structure equations for the Maurer-Cartan forms can then be derived from the determining equations for the infinitesimal generators of the pseudogroup action solely by means of linear algebra. A moving frames for general pseudogroup actions is defined as equivariant mappings from the space of jets of submanifolds into the pseudogroup jet bundle. The existence of a moving frame requires local freeness of the action in a suitable sense and, as in the finite dimensional case, moving frames can be used to systematically produce complete sets of differential invariants and invariant coframes for the pseudogroup action and to effectively analyze their algebraic structure. Our constructions are equally applicable to finite dimensional Lie group actions and provide a slight generalization of the classical moving frame methods in this case.
Giovanni Rastelli (Università di Torino) Separation of variables for systems of first-order Partial Differential Equations: the Dirac equation in two-dimensional manifolds
Abstract: The problem of solving Dirac equation on two-dimensional manifolds is approached from the separation of variables point of vue, with the aim of setting the basis for the analysis in higher dimensions. Beginning from a sound definition of multiplicative separation for systems of two first-order PDE of "eigenvalue problem"-type and the characterization of those systems admitting multiplicatively separated solutions in some arbitrarily given coordinate system, more structure is step by step added to the problem by requiring the separation constants are associated with differential operators and commuting differential operators. Finally, the requirement that the original system coincides with the Dirac equation on a two-dimensional manifold allows the characterization of those metric tensors admitting separation of variables for the same Dirac equation and of the symmetries associated with the separated coordinates. The research is done in collaboration with R.G. McLenaghan.
Greg Reid (University of Western Ontario) Algorithmic Symmetry Analysis and Overdetermined Systems of PDE
Abstract: Topics covered in this talk include 1. Characterization of separation of variables by higher order symmetries (80's) 2. Point and non-local symmetries (80's) 3. Computation of the structure of finite and infinite Lie pseudo-groups of symmetries (90's) 4. Numerical Jet Geometry and Numerical Algebraic Geometry (00's) 5. New problems in deformation of PDE systems (10's?) The talk will have a retrospective feel, while looking forward to new problems and describing some links with a forth-coming special year on Algebraic Geometry and its Applications at the IMA (06-07). From my earliest work, on the connection between symmetries and separation of variables with Kalnins and Miller, I focused on the extraction of structural information using computer algebra. After leaving separation of variables, I developed the algorithmic analysis and associated theory of overdetermined systems of PDE. Linear systems are always shadowed by non-linear ones. Poor underlying complexity means that modern tools such as Numerical Algebraic Geometry (essentially computing with generic points on the jet components of over-determined systems) are needed. Unifying analysis and algebraic techniques via deformations of PDE pose intriguing open problems.
Colleen Robles (University of Rochester) Rigidity of the adjoint varieties
Abstract: I will discuss recent results on the (extrinsic) rigidity of the adjoint varieties. Consider the adjoint action of a simple Lie group G on its Lie algebra g. This induces an action on the projective space P(g). The action of G on P(g) has a unique closed orbit (the orbit of a highest weight space), and this orbit is an algebraic variety X. For example, when G=Sp(2n) is symplectic, X is the Veronese embedding of projective n-space. When G=SL(n+1) is the special linear group, X is the space of trace-free, rank=1 matrices. In order to study the rigidity of a projective variety Y, we look at the set C(k,y) of lines having contact to order k at y in Y. Note that C(1,y) is just the (projectivized) tangent space. So we say any two varieties (of the same dimension) are identical to first-order. The contact sets C(k,y) arise as the zero sets of ideals I(k,y) generated by differential invariants. In general, we say two varieties X and Y agree to order k at x and y if (1) I(j,x) = I(j,y) for all j = 1, 2, ..., k. We say X is rigid to order k if this condition forces Y to be (projectively equivalent to) X.
Siddhartha Sahi (Rutgers University) Equivariant differential operators, classical invariant theory, unitary representations, and Macdonald polynomials
Abstract: The various subjects in the title are connected by a common strand! In my talk, which is introductory in nature, I will give an overview of the subjects, and describe this fascinating connection.
Gerd Schmalz (University of New England) CR-manifolds, differential equations and multicontact structures (tentative)
Abstract: Cartan's method of moving frames has been successfully applied to the study of CR-manifolds, their mappings and invariants. For some types of CR-manifolds there is a close relation to the point-wise or contact geometry of differential equations. This can be used to find CR-manifolds with special symmetries. The recently introduced notion of multicontact structures provides a general framework comprising certain geometries of differential equations and CR-manifolds which in turn give examples with many symmetries.
Uwe Semmelmann (Universität zu Köln) Twistor forms on Riemannian manifolds
Abstract: Twistor forms, also called conformal Killing forms, are differential forms in the kernel of a certain conformally invariant first order differential operator. They are defined similar to twistor spinors and can be considered as a natural generalization of conformal vector fields. Twistor forms have many interesting properties and appear in different areas of differential geometry. Since they define quadratic first integrals of the geodesic equation they were also intensivly studied in physics. Aim of the talk is to define twistor forms, to give examples and to describe their basic properties. Moreover we want to give a survey on more recent results and developments, in particular in relation with special holonomies.
Jan Slovak (Masaryk University) Wünsch's calculus for parabolic geometries
Abstract: The conformally invariant objects were always understood as affine invariants of the underlying Riemannian connections which did not depend on the choice within the conformal class. Although this definition is so easy to understand, the description of such invariants is a difficult task and many mathematicians devoted deep papers to this problem in the last 80 years. The classical approach coined already by Veblen and Schouten was to elaborate special tensorial objects out of the curvatures, designed to eliminate the transformation rules of the Riemannian connections under conformal rescaling. The most complete treatment of such a procedure was given in a series of papers by Günther and Wünsch in 1986. They provide a version of calculus which allows to list all invariants in low homogeneities explicitly. The aim of this talk is to present a concise version of a similar calculus for all parabolic geometries, relying on the canonical normal Cartan connections.
Roman Smirnov (Dalhousie University) Geometry "a la Cartan" revisited: Hamilton-Jacobi theory in moving frames
Abstract: I will review the Hamilton-Jacobi theory of orthogonal separation of variables in the context of the Cartan geometry, in particular, its most valuable asset, - the method of moving frames. The central concept in this setting is that of frames of eigenvectors (eigenforms) of Killing two-tensors which provides a natural presentation of the theory in terms of principal fiber bundles. Eisenhart (implicitly) employed this idea in 1934 to study orthogonal separation of variables in Euclidean 3-space for geodesic Hamiltonians. I will show how the corresponding problem for natural Hamiltonians can be solved with the aid of a more general version of the moving frames method than the one used by Eisenhart (joint work with J.T. Horwood and R.G. McLenaghan). As an application, the approach outlined above together with symmetry methods will be used to determine a new class of maximally superintegrable and multi-separable potentials in Euclidean 3-space. These potentials given by a formula depending on an arbitrary function do not appear in Evans' classification of 1990. A particular example of such a potential is the potential of the Calogero-Moser system (joint work with P. Winternitz).
Petr Somberg (Karlovy (Charles) University) The Uniqueness of the Joseph Ideal for the Classical Groups
Abstract: The Joseph ideal is a unique ideal in the universal enveloping algebra of a simple Lie algebra attached to the minimal coadjoint orbit. For the classical groups, its uniqueness - in a sense of the non-commutative graded deformation theory - is equivalent to the existence of tensors with special properties. The existence of these tensors is usually concluded abstractly via algebraic geometry, but we present explicit formulae. This allows a rather direct computation of a special value of the parameter in the family of ideals used to determine the Joseph ideal.
Vladimir Soucek (Karlovy (Charles) University) Analogues of the Dolbeault complex and the separation of variables
Abstract: The Dirac equation is an analogue of the Cauchy-Riemann equations in higher dimensions. An analogues of the del-bar operator in the theory of several complex variables in higher dimensions is the Dirac operator D in several vector variables. It is possible to construct a resolution starting with the operator D, which is an analogue of the Dolbeault complex. A suitable tool for study of the properties of the complex is the separation of variables for spinor valued fields in several vector variables and the corresponding Howe dual pair.
Jukka Tuomela (University of Joensuu) Overdetermined elliptic boundary value problems
Abstract: I will first report on some recent work on generalising Shapiro-Lopatinski condition to overdetermined problems. The technical difficulty in this extension is that the parametrices are no longer pseudodifferential operators, but Boutet de Monvel operators. Then I discuss some numerical work related to these issues, and present one possibility to treat overdetermined problems numerically. In this approach there is no need to worry about inf-sup condition: for example one can stably compute the solution of the Stokes problem with P1/P1 formulation.
Mikhail Vasiliev (P. N. Lebedev Physics Institute) Higher spin gauge theories and unfolded dynamics
Abstract: I will discuss nonlinear equations of motion of higher spin gauge fields. The driving idea is to study most symmetric field theories, assuming that whatever theory of fundamental interactions is it should be very symmetric. The formulation is based on the unfolded dynamics formalism which is an overdetermined multidimensional covariant extension of the one-dimensional Hamiltonian dynamics. General properties of the unfolded dynamics formulation will be discussed in some detail with the emphasize on symmetries and coordinate independence.
Pavel Winternitz (University of Montreal) Superintegrable classical and quantum systems
Abstract: These lectures will cover the following topics
1. Definition and basic properties.
2. Lie symmetries and higher order symmetries.
3. Quadratic superintegrability and the separation of variables in spaces of constant and variable curvature.
4. Superintegrability and exact solvability.
5. Superintegrability without separation of variables. Third order integrals of motion. Velocity dependent forces.
6. Integrable and superintegrable systems involving particles with spin.
Thomas Wolf (Brock University) Classification of 3-dimensional scalar discrete integrable equations
Abstract: Joint work with S. Tsarev and A. Bobenko. A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. Recent progress in discrete differential geometry, reported in a review by A.Bobenko and Yu. Suris (see www.arxiv.org, math.DG/0504358) has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. In particular it was discovered that classical transformations of remarkable classes of smooth surfaces (Baecklund transfromations, Ribaucour transformations etc.) after discretization of the respective classes of surfaces become just their extension with an "extra discrete dimension" in an absolutely symmetric way. The requirement of consistency of the original difference systems with the operation of adding such "extra discrete dimension" gives an BIG overdetermined system of equations for the coefficients of the original difference equation describing the discrete system in question. This reqirement was considered in the review by A.Bobenko and Yu. Suris as the fundamental property giving a criterion of "discrete integrability". In this talk we describe our recent results on complete classification of a class of 3-dimensional scalar discrete integrable equations.
Keizo Yamaguchi (Hokkaido University) Geometry of linear differential systems—towards "contact geometry of second order"
Abstract: Starting from the geometric construction of jet spaces, defining the symbol algebra of these canonical (contact) systems, the goal of these lectures is to formulate submanifolds of 2-jet spaces as PD-manifolds (R,D1,D2), i.e. D1 and D2 are a pair of subbundles of the tangent bundle of R. This will also serve as a preparation to symmetries of p.d.e. and to parabolic geometry associated with various p.d.e.s, which will be discussed in later weeks.
Igor Zelenko (International School for Advanced Studies (SISSA/ISAS)) Symplectification procedure for the equivalence problem of vector distributions
Abstract: My talk is devoted to the equivalence problem of non-holonomic vector distributions and it is based on the joint work with Boris Doubrov. The problem was originated by E. Cartan in the beginning of twenty century, who treated the first nontrivial case of the fields of planes in a five-dimensional ambient space with his method of equivalence. In our talk we would like to describe a new rather effective approach to this problem, which we call the symplectification procedure. The starting point of this procedure is to lift the distribution to a special submanifold of the cotangent bundle, foliated by the characteristic curves. The invariants of the distributions can be obtained from the study of the dynamics of this lifting along the characteristic curves. The case of rank two distributions (fields of planes) will be discussed in more detail. In this case we succeeded to construct the canonical frame and to find the most symmetric models for the arbitrary dimension of the ambient manifold, generalizing the mentioned work of Cartan. The new effects in the case of distributions of rank greater than two will be discussed as well.
Visitors in Residence
Silas Alben Harvard University 6/27/2006 - 7/1/2006
Jung-Ha An University of Minnesota Twin Cities 9/1/2005 - 8/31/2007
Stephen Anco Brock University 7/15/2006 - 8/4/2006
Douglas N. Arnold University of Minnesota Twin Cities 7/15/2001 - 8/31/2007
Donald G. Aronson University of Minnesota Twin Cities 9/1/2002 - 8/31/2007
Evgeniy Bart University of Minnesota Twin Cities 9/1/2005 - 8/31/2007
Helga Baum Humboldt-Universität 7/16/2006 - 8/5/2006
Gloria Mari Beffa University of Wisconsin 7/23/2006 - 8/4/2006
Paul Bendich Duke University 7/10/2006 - 7/28/2006
Melisande Fortin Boisvert McGill University 7/16/2006 - 8/4/2006
Andreas Cap University of Vienna 7/16/2006 - 8/4/2006
Mark Chanachowicz University of Waterloo 7/16/2006 - 8/5/2006
Qu Changzheng Northwest (Xibei) University 7/24/2006 - 7/31/2006
Claudia Chanu Università di Torino 7/14/2006 - 8/2/2006
Jeongoo Cheh University of St. Thomas 7/17/2006 - 8/4/2006
Qianyong Chen University of Minnesota Twin Cities 9/1/2004 - 8/31/2006
Peter A. Clarkson University of Kent at Canterbury 7/16/2006 - 8/4/2006
Michael Cowling University of New South Wales 7/16/2006 - 8/4/2006
Luca Degiovanni Università di Torino 7/14/2006 - 8/1/2006
Brian DiDonna University of Minnesota Twin Cities 9/1/2004 - 8/31/2006
Hongjie Dong University of Chicago 7/16/2006 - 7/30/2006
Boris Doubrov Belarus State University 7/16/2006 - 8/5/2006
Michael Eastwood University of Adelaide 7/15/2006 - 8/5/2006
Mark Fels Utah State University 7/23/2006 - 7/26/2006
Eugene Ferapontov Loughborough University 7/16/2006 - 7/29/2006
Daniel Fox Georgia Institute of Technology 7/16/2006 - 7/29/2006
Peter Franek Karlovy (Charles) University 7/16/2006 - 8/4/2006
Michal Godlinski University of Warsaw 7/17/2006 - 8/4/2006
Vladislav Goldberg New Jersey Institute of Technology 7/15/2006 - 7/22/2006
Hubert Goldschmidt Columbia University 7/15/2006 - 7/22/2006
Sergey Golovin Queen's University 7/16/2006 - 7/29/2006
A. Rod Gover University of Auckland 7/25/2006 - 8/2/2006
Robin Graham University of Washington 7/23/2006 - 7/28/2006
Jooyoung Hahn Korea Advanced Institute of Science and Technology (KAIST) 8/26/2005 - 7/31/2006
Hazem Hamdan University of Minnesota Twin Cities 7/17/2006 - 8/4/2006
Chong-Kyu Han Seoul National University 7/16/2006 - 8/4/2006
Gloria Haro Ortega University of Minnesota Twin Cities 9/1/2005 - 8/31/2007
Frank Hausser Caesar Research Center 6/18/2006 - 7/1/2006
Kengo Hirachi University of Tokyo 7/16/2006 - 8/5/2006
Doojin Hong Eduard Cech Center 7/17/2006 - 7/22/2006
Moritaka Hosotubo National Institute of Science and Technology Policy 7/6/2006 - 7/6/2006
Evelyne Hubert Institut National de Recherche en Informatique Automatique (INRIA) 7/15/2006 - 8/5/2006
Peter Hydon University of Surrey 7/14/2006 - 8/4/2006
Yuko Itoh National Institute of Science and Technology Policy 7/6/2006 - 7/6/2006
Jens Jonasson Linköping University 7/16/2006 - 8/4/2006
Sookyung Joo University of Minnesota Twin Cities 9/1/2004 - 8/31/2006
Andreas Juhl Uppsala University 7/16/2006 - 7/23/2006
Ernie Kalnins University of Waikato 7/13/2006 - 7/24/2006
Niky Kamran McGill University 7/16/2006 - 7/21/2006
Chiu Yen Kao University of Minnesota Twin Cities 9/1/2004 - 8/31/2006
Joseph Kenney University of Minnesota Twin Cities 7/17/2006 - 8/4/2006
Artemiy Vladimirovich Kiselev Ivanovo State University 7/17/2006 - 7/31/2006
Mary Elizabeth Kloc University of Wisconsin 7/9/2006 - 7/29/2006
Irina Kogan North Carolina State University 7/16/2006 - 8/5/2006
Jonathan Kress University of New South Wales 7/14/2006 - 8/4/2006
Svatopluk Krysl Karlovy (Charles) University 7/16/2006 - 8/5/2006
Song-Hwa Kwon University of Minnesota Twin Cities 8/30/2005 - 8/31/2007
Joseph Landsberg Texas A & M University 7/16/2006 - 7/28/2006
Chang-Ock Lee Korea Advanced Institute of Science and Technology (KAIST) 8/1/2005 - 7/31/2006
Guang-Tsai Lei GTG Research 7/17/2006 - 8/4/2006
Thomas Leistner University of Adelaide 7/15/2006 - 8/5/2006
Felipe Leitner Universität Stuttgart 7/16/2006 - 8/5/2006
Debra Lewis University of Minnesota Twin Cities 7/15/2004 - 8/31/2006
Xiantao Li Pennsylvania State University 7/8/2006 - 7/16/2006
Hstau Liao University of Minnesota Twin Cities 9/2/2005 - 8/31/2007
Xiaolong Liu University of Iowa 7/16/2006 - 8/4/2006
Alison Malcolm University of Minnesota Twin Cities 9/1/2005 - 8/31/2006
Elizabeth L. Mansfield University of Kent at Canterbury 7/16/2006 - 8/4/2006
Ian Marquette University of Montreal 7/16/2006 - 7/28/2006
Vladimir S. Matveev Katholieke Universiteit Leuven 7/17/2006 - 8/4/2006
Bonnie McAdoo Clemson University 7/16/2006 - 8/7/2006
Ray McLenaghan University of Waterloo 7/16/2006 - 8/5/2006
Willard Miller Jr. University of Minnesota Twin Cities 7/17/2006 - 8/4/2006
Emilio Musso Università di L'Aquila 7/15/2006 - 7/31/2006
Paul-Andi Nagy Humboldt-Universität 7/16/2006 - 8/4/2006
Zoltan Neufeld University College Dublin 6/18/2006 - 7/1/2006
Lorenzo Nicolodi Università di Parma 7/15/2006 - 7/31/2006
Anatoly Nikitin National Academy of Sciences of Ukraine 7/16/2006 - 8/4/2006
Pawel Nurowski University of Warsaw 7/16/2006 - 8/5/2006
Luke Oeding Texas A & M University 7/16/2006 - 8/4/2006
Peter J. Olver University of Minnesota Twin Cities 7/17/2006 - 8/4/2006
Kaoru Ono Hokkaido University 7/6/2006 - 7/6/2006
Bent Orsted Aarhus University 7/28/2006 - 8/4/2006
Saadet S. Ozer Yeditepe University 7/16/2006 - 8/4/2006
Teoman Ozer Istanbul Technical University 7/15/2006 - 8/8/2006
Lawrence J. Peterson University of North Dakota 7/23/2006 - 7/29/2006
Peter Philip University of Minnesota Twin Cities 8/22/2004 - 8/18/2006
George Pogosyan Yerevan State University 7/16/2006 - 7/29/2006
Juha Pohjanpelto Oregon State University 7/16/2006 - 7/25/2006
Gregory J. Randall University of the Republic 8/18/2005 - 7/31/2006
Arvind Satya Rao University of Iowa 7/16/2006 - 7/27/2006
Giovanni Rastelli Università di Torino 7/14/2006 - 8/2/2006
Greg Reid University of Western Ontario 7/17/2006 - 8/4/2006
Chan Roath Ministry of Education, Youth and Sport 7/15/2006 - 8/15/2006
Colleen Robles University of Rochester 7/16/2006 - 7/29/2006
Vincent G. J. Rodgers University of Iowa 7/23/2006 - 7/28/2006
Siddhartha Sahi Rutgers University 7/23/2006 - 8/4/2006
Arnd Scheel University of Minnesota Twin Cities 7/15/2004 - 8/31/2007
Gerd Schmalz University of New England 7/28/2006 - 8/4/2006
Uwe Semmelmann Universität zu Köln 7/16/2006 - 7/30/2006
Neil Seshadri University of Tokyo 7/16/2006 - 8/5/2006
Michael J. Shelley New York University 6/17/2006 - 7/1/2006
Astri Sjoberg University of Johannesburg 7/16/2006 - 8/4/2006
Jan Slovak Masaryk University 7/17/2006 - 8/4/2006
Dalibor Smid Karlovy (Charles) University 7/16/2006 - 8/5/2006
Roman Smirnov Dalhousie University 7/16/2006 - 8/5/2006
Tatiana Soleski University of Minnesota Twin Cities 9/1/2005 - 8/31/2007
Petr Somberg Karlovy (Charles) University 7/16/2006 - 8/5/2006
Vladimir Soucek Karlovy (Charles) University 7/16/2006 - 8/5/2006
Adam Szereszewski University of New South Wales 7/16/2006 - 8/4/2006
Dennis The McGill University 7/16/2006 - 8/5/2006
Carl Toews University of Minnesota Twin Cities 9/1/2005 - 8/31/2007
Yoshihiro Tonegawa Hokkaido University 7/6/2006 - 7/6/2006
Frédérick Tremblay University of Montreal 7/19/2006 - 7/31/2006
Jukka Tuomela University of Joensuu 7/23/2006 - 8/4/2006
Francis Valiquette University of Minnesota Twin Cities 7/17/2006 - 8/4/2006
Nicolas Vandenberghe Université d'Aix-Marseille II (Université de la Méditerranée) 6/18/2006 - 7/3/2006
Mikhail Vasiliev P. N. Lebedev Physics Institute 7/16/2006 - 8/5/2006
Alfredo Villanueva University of Iowa 7/16/2006 - 8/4/2006
Sung Ho Wang Korea Institute for Advanced Study (KIAS) 7/24/2006 - 7/28/2006
Xiaoqiang Wang University of Minnesota Twin Cities 9/1/2005 - 8/4/2006
Ben Warhurst University of New South Wales 7/16/2006 - 8/5/2006
Pavel Winternitz University of Montreal 7/16/2006 - 7/22/2006
Thomas Wolf Brock University 7/16/2006 - 8/6/2006
Keizo Yamaguchi Hokkaido University 7/16/2006 - 8/5/2006
Jin Yue Dalhousie University 7/16/2006 - 8/5/2006
Ismet Yurdusen University of Montreal 7/16/2006 - 7/28/2006
Igor Zelenko International School for Advanced Studies (SISSA/ISAS) 7/16/2006 - 7/28/2006
Renat Zhdanov Bio-Key International 7/16/2006 - 8/5/2006
Legend: Postdoc or Industrial Postdoc Long-term Visitor

Participating Institutions: Air Force Research Laboratory, Carnegie-Mellon University, Consiglio Nazionale delle Ricerche, Georgia Institute of Technology, Indiana University, Iowa State University, Kent State University, Lawrence Livermore National Laboratory, Los Alamos National Laboratory, Michigan State University, Mississippi State University, Northern Illinois University, Ohio State University, Pennsylvania State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Seoul National University, Texas A & M University, University of Chicago, University of Cincinnati, University of Delaware, University of Houston, University of Illinois, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pittsburgh, University of Texas, University of Wisconsin, University of Wyoming, Wayne State University
Participating Corporations: 3M, Boeing, Corning, ExxonMobil, Ford, General Electric, General Motors, Honeywell, IBM, Johnson & Johnson, Lockheed Martin, Medtronic, Motorola, Schlumberger, Siemens, Telcordia