Formalized proof, computation, and the construction problem in algebraic geometry

An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory within a ZFC-like environment

Logic Programming and Logarithmic Space

We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation

A Note on the Morse Index Theorem for Geodesics between Submanifolds in semi-Riemannian Geometry

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse's proof. This approach reduces substantially the effort required in the proofs of the Theorem given in previous articles on the subject. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorems of Beem, Ehrlich and Kim.Comment: 12 pages, LaTeX2e, amsart style. To appear on the Journal of Mathematical Physic

The flat Grothendieck-Riemann-Roch theorem without adiabatic techniques

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.Comment: 21 pages. Comments are welcome. Final version. To appear in Journal of Geometry and Physic

Rough index theory on spaces of polynomial growth and contractibility

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.Comment: v4: final version, to appear in J. Noncommut. Geom. v3: added a computation of the homology of (a smooth subalgebra of) the uniform Roe algebra. v2: added as corollaries to the main theorem the multi-partitioned manifold index theorem and the higher-codimensional index obstructions against psc-metrics, added a proof of the strong Novikov conjecture for virtually nilpotent groups, changed the titl

Geometry Through Architectural Design

In her 1912 geometry book, Mabel Sykes surveys complex and beautiful architectural designs from around the world to inspire exercises on geometric proof, construction and computation. In over 1800 exercises, Sykes analyzes geometric patterns from ornamental and structural features found in tile mosaics, parquet floors, Gothic windows, trusses and arches. As Sykes\u27 writes, ``Geometry gives, as no other subject can give, an appreciation of form as it exists in the material world . We have chosen four examples to illustrate how her appealing designs and the accompanying exercises of this hidden gem can be incorporated into any geometry course