ABSTRACT:... (Bryant, Robert L.)

ABSTRACT: We find characterization of planar webs of maximum rank in terms of differential invariants. We prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature 2-form vanishes. We also give an invariant characterization of planar 4-webs admitting two or one abelian equations and show that in general such webs are not linearizable. (The talk will be based on a joint work of Goldberg, Vladislav with Lychagin, Valentin.)

ABSTRACT: Initial conditions violate, in general, the symmetry group of differential equations. Therefore, it seems that the group approach is quite useless in initial value problems. However, the so-called invariance principle together with an appropriate extension of the Lie group theory to differential equations involving distributions (generalized functions) provides a simple and effective method for tackling initial value problems.

ABSTRACT: Starting with important papers by L. Hormander, E. Nelson and others in the nineteen sixties, a new part of PDE theory has emerged, opening up promising directions in both theory and applications. Examples: Hypoellipticity, degenerate parabolic equations, spectral theory, symmetry based on representations, and Malliavin calculus, to mention just a few. Some of these are outlined where the author.s own work is closest. Earlier papers central to the theme are [Hul76] and [Jor75].

[Hul76] Hulanicki, A. The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Studia Math. 56 (1976), no. 2, 165--173.

[Jor75] Jørgensen, Palle E. T. Representations of differential operators on a Lie group. J. Functional Analysis 20 (1975), no. 2, 105--135.

In fact, in the 1970ties, and again more recently, there was and is a lot of interest in this subject.

One theme in our talk will be explicit formulas for partial Fourier transforms of various Lie theoretic heat kernels, and generators of non-commutative Brownian motion. This problem has a long history, including motivations from probability (P), and analysis (A). It is likely that the most interesting issue is the interplay between (A) and (P). History: The paper [Hul76; Theorem 2] gives an explicit formula for a partial Fourier transform of a certain Lie theoretic heat kernel, the case of the Heisenberg group. The partial Fourier transform of [Hul76] refers to the central scalar variable from Schrodinger.s representation of the Heisenberg group. Even before [Hul76], McKean had suggested that there should be probabilistic ways of getting explicit formulas, but that wasn.t followed up until much later.

Both the paper [Hul76] and McKean.s suggestion were motivated by a section in: Nelson, Edward; Analytic vectors.Ann. of Math. (2) 70 1959, 572--615.

And by the related paper: Nelson, Edward; and Stinespring, W. Forrest; Representation of elliptic operators in an enveloping algebra. Amer. J. Math. 81 1959, 547--560.

Some technical points: special coordinates in Lie groups; two ways of describing representations of Lie groups (differentiated and integrated), and a review of certain matrix Lie groups.

ABSTRACT: We will survey some recent results on the long-term behavior of the solutions of a number of relativistic wave equations in the Kerr metric of a rotating black hole in equilibrium. The problem of constructing conservation laws for higher spin equations, which is key to the analysis, will be highlighted. (The work presented is a collaboration of Kamran, Niky with Finster, Felix, Smoller, Joel and Yau, Shing-Tung .)

ABSTRACT: Spencer d-complex underlies many geometric constructions, which involve curvatures and other differential invariants. From the side of differential equations it is equally important: the corresponding cohomology contains information about generators of defining equations for a given system, obstructions to formal integrability etc.

In this talk I will review old and describe new methods of calculations of Spencer cohomology for general symbolic systems generated by equations of various orders. I will describe relations between involutivity, characteristics, restrictions and characteristisity, which generalize the classical theorems due to Quillen, Guillemin and Kuranishi. We will discuss a certain duality between involutive and Cohen-Macaulay systems. (The work presented is a cooperation between Kruglikov, Boris. and Lychagin, Valentin.)

ABSTRACT: We'' ll discuss recent advances in the theory of formal integrability. We use Buchsbaum-Rim complexes instead of Spencer d-complexes to get resolutions for a wide class of PDEs. This determines an exact form of compatibility conditions in terms of certain multibrackets. Some applications of the methods for classical geometrical problems will be discussed. (The work presented is a cooperation between Lychagin, Valentin and Kruglikov, Boris.)

ABSTRACT: ... (Muños-Masqué, Jaime)

ABSTRACT: New points of view were recently introduced in the geometric theory of PDE's, by adopting some algebraic topological approaches. In particular, integral (co)bordism groups are seen very useful to characterize global solutions. The methods developed by us, in the category of (non)commutative PDE's, in order to find integral bordism groups, allowed us to obtain, as a by-product, existence theorems for global solutions, in a pure geometric way. Another result that is directly related to the knowledge of integral bordism groups of PDE's, is the possibility to characterize PDE's by means of some important algebras, related to the conservation laws of these equations (Hopf algebras of PDE's). Objects of these algebras identify invariants of global solutions. Moreover, thanks to an algebraic characterization of PDE's, one has also a natural way to recognize quantized PDE's as objects of the category of quantum PDE's. These results have opened a new sector in Algebraic Topology, that we can formally define the algebraic topology of PDE's. Aim of the present talk is to report on some new issues in this direction, emphasizing the role just played by integral bordism groups and conservations laws. Applications to some important examples of the Mathematical Physics, as quantum black holes solutions of quantum PDE's, will be discussed too. (Prástaro, Agostino.)

ABSTRACT: We investigate the relation between the Korteweg - de Vries and modified Korteweg - de Vries equations (KdV and mKdV), and find a new algebro-analytic mechanism, similar to the Lax L-A pair, which involves a first-order operator Q instead of the third-order operator A. In our framework, eigenfunctions of the Schrödinger operator L, whose time-dependent potential solves the KdV equation, evolve according to a linear first-order partial differential equation, giving explicit control over their time evolution. As an application, we establish global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity. These classes may even include functions which tend to infinity with the space variable. (The talk will be based on a joint work of Kappeler, Thomas, Perry, Peter, Topalov, Peter and Shubin, Mikhail, see http://arxiv.org/pdf/math.AP/0601237)

ABSTRACT: This talk will discuss recent work on the Yang-Mills flow for a holomorphic vector bundle over a Kaehler manifold. In the case of stable vector bundles the flow converges to a Hermitian-Einstein metric. We will explain the behavior for unstable vector bundles. The relationship between the Harder-Narasimhan filtration, the singular set of the flow, and the limiting holomorphic structure will be explained. (Wentworth, Richard A.)

ABSTRACT: An asymptotic theory for a class of third - order differential equation was developed. We consider an Euler case for this class of differential equation ,as we shall see this Euler case represents a borderline between situations where two silutions are small compared to algebraic functions, and where two are small compared to exponential functions while the third solution has algebraic behavior in both situations. (Alhammadi, A.S.A.)(Visit also http://www.ams.org/mrlookup and search for Al-Hammadi in Author.)

ABSTRACT: Partial differential equations which are Darboux integrable admit explicit general solutions which can be written in terms of certian arbitrary functions and there derivatives. These formulas are suggestive that a superposition formula for the solutions can be written in terms of more basic differential equations. I'll pursue this idea by using the theory of quotients and exterior differential systems. ( Fels, Mark)

ABSTRACT: We show that the locality conditions for the cancellation of local anomalies in field theory admit a geometrical interpretation in terms of the equivariant cohomology of the variational bicomplex. Combining that result with the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gel'fand-Fuks cohomology, we obtain necessary and sufficient conditions for the cancellation of local gravitational and mixed anomalies. ( Ferreiro Pérez, Roberto)

ABSTRACT: A functional is a function from the space of functions to a number field, for example, f:{ a:(-¥,¥) ® (-¥,¥)} ® (-¥,¥). These three ¥'s are written as the same notation, but their original meanings are quite different. The purpose of this proceeding is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. For it we divide three ¥'s to three types of infinities. We extend the number field R to °(*R) under the base of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a *-finite lattice to a *-finite lattice with a double meaning. Considering a *-finite lattice with a double meaning, we find how to treat the domain for a functional in our theory of Fourier transformation, and calculate two typical examples. (The talk will be based on a joint work of Nitta, Takashi with Okada, Tomoko.)

ABSTRACT: A mass of fluid on a Riemannian manifold (M,g) is said to be self-gravitating if it is only subjected to its own gravitational force, which tends to collapse the fluid, and to its pressure, which tends to expand it. The equations modelling the equilibrium state of this kind of fluids define a difficult system of partial differential equations of free-boundary type. The literature on this subject is mainly focused on classifying the shapes of the fluid masses when they reach the equilibrium state. In this talk we review some results recently obtained by the author (Commun. Math. Phys. 2006). The main theorem shows that the boundary of a static self-gravitating fluid defines an isoparametric submanifold of M, a purely geometrical object introduced by Cartan, Levi-Civita and Segre in the nineteen thirties. The proof consists of the following three steps: