Lecture 2. Index problem for manifolds with a boundary. Index of the Dirac operator and anomalies. Lecture 3. Spectral asymmetry and Riemannian geometry. Heat equation and asymptotic heat kernel. The η-and ζ-functions. Lecture 4. Two-component spinor calculus. Dirac and Weyl equations in two-component spinor form. Weyl equation with spectral or local boundary conditions. Potentials for massless...

Let $X=\mathbb{R}\times M$ be the spacetime, where $M$ is a closed manifold equipped with Riemannian metric $g$, and we consider symmetric Klein-Gordon type operator $P$ on $X$, which asymptotically converges to $\partial_t^2-\triangle_g$ as $|t|\to\infty$, $\triangle_g$ Laplace-Beltrami $M$. We prove essential self-adjointness of $C_0^\infty(X)$. The idea proof closely related recent paper by ...

A scale of Sobolev spaces is introduced to measure the global hypoellipticity of the twisted Laplacian. The essential self-adjointness of the twisted Laplacian is established and the domain of the unique self-adjoint extension is determined in terms of the Sobolev spaces. The global hypoellipticity of the twisted Laplacian in the Gelfand–Shilov spaces is also proved. 1. The twisted Laplacian Le...

It is shown that a category-theoretic definition of chaotic system applies not only to the Smale horseshoe, a prototypical example, but also to Conway’s “Life.” Symbolic dynamics of the “Dining Philosophers” relational system is computed. A category composed of stochastic matrices is defined and its elementary properties are studied. A categorical variant of symbolic dynamics is applied to a fi...