General Spectral Flow Formula for Fixed Maximal Domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.Comment: 22 page

The spectral flow for Dirac operators on compact planar domains with local boundary conditions

Let $D_t$, $t \in [0,1]$ be an arbitrary 1-parameter family of Dirac type operators on a two-dimensional disk with $m-1$ holes. Suppose that all operators $D_t$ have the same symbol, and that $D_1$ is conjugate to $D_0$ by a scalar gauge transformation. Suppose that all operators $D_t$ are considered with the same locally elliptic boundary condition, given by a vector bundle over the boundary. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of the holes in the disk. This constant is calculated explicitly for the case of the annulus ($m=2$).Comment: 33 pages, 4 figures; section 9 adde

Unity and Disunity in Mathematics

The unity of mathematics has its power to compactify experiences in a form capable of being transferred and modified or adapted to new mathematical situations. Yet, we believe that the phrase "Unity of Mathematics" expresses a dream, an ideal that doesn't exist. We shall point to diachronic and cross cultural disunities, to semantic, semiotic and philosophic ambiguities and to the non-acceptance of certain mathematical texts by some practitioners of the subject.Comment: 4 pages. This is a condensed, sharpened and polemic outcome of our previous discussions on that topic, disseminated in arXiv:1204.514

Unity in Major Themes:Convergence vs. Arbitrariness in the Development of Mathematics

We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the development of mathematics as a goal-driven process as opposed to one that often seems to possess considerable arbitrariness.Comment: To the memory of Gian-Carlo Rota (April 27, 1932 - April 18, 1999), Contribution to the XI Oesterreichisches Symposion zur Geschichte der Mathematik, Organiser: Christa Binder, Topic: Der Blick aufs Ganze. Gibt es grosse Linien in der Entwicklung der Mathematik? Venue: Miesenbach (Austria), 22-28 April, 2012, 9 page

Missionaries, measles, and manuscripts: revisiting the Whitman tragedy

The missionaries Marcus Whitman, a doctor, and Narcissa Whitman, his wife, and twelve other members of the Waiilatpu Mission were murdered in November 1847 by a small contingent of the Cayuse Indians in the Oregon Territory. The murders became known as the “Whitman Massacre.” The authors examine the historical record, including archived correspondence held at the Yale University Libraries, for evidence of what motivated the killings and demonstrate that there were two valid perspectives, Cayuse and white. Hence, the event is better termed the “Whitman Tragedy.” The crucial component, a highly lethal measles epidemic, has been called the spark that lit the fuse of the tragedy

The topological meaning of Levinson's theorem, half-bound states included

We propose to interpret Levinson's theorem as an index theorem. This exhibits its topological nature. It furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds.Comment: 4 page