This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. Th... more This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers in number fields in terms of much more accessible "etale models". Suitable versions of the conjecture predict the cohomology of infinite general linear groups of rings of S-integers over suitable number fields; our survey focuses on an unstable version of this form of the conjecture.
For suitable rings of integers R, we show that the mod p group cohomology for GL n+3p-5 (R) comes... more For suitable rings of integers R, we show that the mod p group cohomology for GL n+3p-5 (R) comes from GL∞(R) when restricted to the diagonal matrices D n (R) for all ranks n > 2.
Topological data analysis studies point-clouds in high dimensional spaces by generating simplicia... more Topological data analysis studies point-clouds in high dimensional spaces by generating simplicial complexes and calculating their homology. For example, the subsets of place neurons that co-fire generate a simplicial complex having the topology of the environment. In the literature simplicial complexes are generated by listing all the faces of each simplex and by removing the duplicates. Our algorithm does not produce duplicates and can also generate $\Delta$-complexes and their boundary maps. To achieve this we make use of the divide and conquer paradigm and dynamic programming based on statistical hash tables.
Transactions of the American Mathematical Society, 2003
Conjecturally, for p p an odd prime and R R a certain ring of p p -integers, the stable general l... more Conjecturally, for p p an odd prime and R R a certain ring of p p -integers, the stable general linear group G L ( R ) GL(R) and the étale model for its classifying space have isomorphic mod p p cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if p p is regular and certain homology classes for S L 2 ( R ) SL_2(R) vanish. We check that this criterion is satisfied for p = 3 p=3 as evidence for the conjecture.
This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. Th... more This survey is mostly concerned with unstable analogues of the Lichtenbaum-Quillen Conjecture. The Lichtenbaum-Quillen Conjecture (now implied by the Voevodsky-Rost Theorem) attempts to describe the algebraic K-theory of rings of integers in number fields in terms of much more accessible "etale models". Suitable versions of the conjecture predict the cohomology of infinite general linear groups of rings of S-integers over suitable number fields; our survey focuses on an unstable version of this form of the conjecture.
For suitable rings of integers R, we show that the mod p group cohomology for GL n+3p-5 (R) comes... more For suitable rings of integers R, we show that the mod p group cohomology for GL n+3p-5 (R) comes from GL∞(R) when restricted to the diagonal matrices D n (R) for all ranks n > 2.
Topological data analysis studies point-clouds in high dimensional spaces by generating simplicia... more Topological data analysis studies point-clouds in high dimensional spaces by generating simplicial complexes and calculating their homology. For example, the subsets of place neurons that co-fire generate a simplicial complex having the topology of the environment. In the literature simplicial complexes are generated by listing all the faces of each simplex and by removing the duplicates. Our algorithm does not produce duplicates and can also generate $\Delta$-complexes and their boundary maps. To achieve this we make use of the divide and conquer paradigm and dynamic programming based on statistical hash tables.
Transactions of the American Mathematical Society, 2003
Conjecturally, for p p an odd prime and R R a certain ring of p p -integers, the stable general l... more Conjecturally, for p p an odd prime and R R a certain ring of p p -integers, the stable general linear group G L ( R ) GL(R) and the étale model for its classifying space have isomorphic mod p p cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if p p is regular and certain homology classes for S L 2 ( R ) SL_2(R) vanish. We check that this criterion is satisfied for p = 3 p=3 as evidence for the conjecture.
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