FORTHCOMING LECTURES

Christian Bär

Universität Potsdam, Institut für Mathematik

BOUNDARY VALUE PROBLEMS FOR ELLIPTIC OPERATORS OF FIRST ORDER


Topic: We develop a systematic theory of boundary value problems for elliptic differential operators of first order on manifolds with compact boundary. This contains classical local elliptic boundary value conditions in the sense of Lopatinsky and Shapiro as well as the nonlocal boundary conditions introduced by Atiyah, Patodi, and Singer. We discuss regularity questions and index theory. In particular, we give a simple and natural proof of the relative index theorem due to Gromov and Lawson.
Wyk³ady:
2.XII, wtorek, 1515 (sala 1015, 60')
3.XII, ¶roda, 1415, (sala 1016, 90')
4.XII, czwartek, 1415, (sala 0094, 90')
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PAST LECTURES
BRUNO IOCHUM

Centre de Physique Theorique, Université Aix Marseille

 PHYSICS FROM SPECTRAL THEORY
(Wodzicki residues, Dixmier traces and heat expansion)

 Topics:
  • Quick review on pseudodifferential operators on manifolds.
  • Singularities of the kernel of pseudodifferential operators near the diagonal.
  • Wodzicki residue, Dixmier traces: Definition and properties and Connes' trace theorem.
  • Computation of residues for inverse powers of Dirac operators on manifolds.
  • Spectral triples and Spectral action.
  • Examples: Noncommutative torus.

> THOMAS SCHüCKER
Centre de Physique Theorique, Université Aix Marseille
Introduction to Einstein-Cartan theory and gravitational anomalies.
 Abstract: The first part intends to present the Einstein-Cartan theory in the language of differential forms valued in Lie algebras and of the Hodge star. The local theory (on open subsets of Rn) of this form calculus will be recalled and then applied to the first order variational formalism of general relativity. The second part intends to define infinitesimal anomalies of gauge groups induced by Weyl fermions and to compute these anomalies algebraically via the Wess-Zumino consistency condition.
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RAINALD FLUME
Rheinsche Friedrich-Wilhelms-Universitaet Bonn, RFN
On supersymmetric gauge theories and their relations to other fields of mathematical physics.
Content:
I. Basics of supersymmetry.
II. Algebraic renormalisation of SUSY gauge theories.
III. Some non-renormalisation theorems - holomorphicity a'la Seiberg.
IV. Instanton calculus.
V. SUSY Yang-Mills & Matrix Models.
VI. Double scaling limits in matrix models & critical Yang - Mills theories.
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NIALL O MURCHADHA
Physics Department, University College, Cork, Ireland.
General Relativity as a Dynamical System
Abstract: I begin with the ADM/Dirac analysis of the Einstein equations, showing how to transform them into a dynamical system. This naturally involves the split between the constraint equations and the dynamical equations. I will discuss the compact, without boundary, case but my main focus will be on the asymptotically flat case. I then show how the ADM energy-momentum-angular momentum naturally emerges. (ca. 4 lectures) I will discuss the various conformal methods of solving the constraints as an introduction to the Schoen-Yau proof of the positivity of the ADM energy. I will give a simple version of the first Schoen-Yau proof (the `riemannian' case). I will also outline Schoen's completion of the Yamabe proff, based on the positive energy proof. (ca. 5 lectures) I will then discuss the Hamiltonian based quasi-local energies of Brown-York and Kijowski-Liu-Yau. (ca. 3 lectures). Finally, I will discuss the evolution equations. In particular, I will focus on the amazing recent success of the `moving puncture' method in solving the binary black hole problem. (ca. 4 lectures)
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