Finiteness theorems for congruence reflection groups

This paper is a follow-up to the paper I. Agol, M. Belolipetsky, P. Storm, K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481?C498. The main purpose is to investigate the effective side of the method developed there and its possible application to the problem of classification of arithmetic hyperbolic reection groups.     

Finiteness theorems for submersions and souls

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose and tensors are both bounded in norm.

     

A matroid view of key theorems for edge-swapping algorithms

We demonstrate that two key theorems of Amaldi et?al. (Math Methods Oper Res 69:205?C223, 2009), which they presented with rather complicated proofs, can be more easily and cleanly established using a simple and classical property of binary matroids. Besides a simpler proof, we see that both of these key results are manifestations of the same essential property.     

Finiteness theorems for the Picard objects of an algebraic stack

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud?s relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron–Severi groups or of the Picard group itself. We give some examples and applications. In Appendix A, we prove the semicontinuity theorem for a (non-necessarily tame) algebraic stack.     

Combinatorial Laplacians of matroid complexes

We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid. The interpretation follows from a surprising orthogonal decomposition of the simplicial chain groups. This decomposition is in general finer than the spectral decomposition. As a consequence, the spectra are integral. One corollary to our combinatorial interpretation may be paraphrased as stating that one can ``hear" the characteristic polynomial of a matroid.

     

A matroid generalization of theorems of Lewin and Gallai

This paper presents a theorem concerning a matroid with the parity condition. The theorem provides matroid generalizations of graph-theoretic results of Lewin and Gallai.     

Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups

Supported by the National Science Foundation during 1986–1988 at the Mathematical Sciences Research Institute, Berkeley, and at the Institute for Advanced Study, Princeton.     

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH ⁱ (X) is finitely generated for i ≤ 4.