The Experts below are selected from a list of 240 Experts worldwide ranked by ideXlab platform
Liqun Tao - One of the best experts on this subject based on the ideXlab platform.
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homology stability for the special linear group of a field and milnor witt k theory
Documenta Mathematica, 2010Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Let F be a field of characteristic zero and let ft;n be the stabilization homomorphism from the nth integral homology of SLt(F) to the nth integral homology of SLt+1(F). We prove the following results: For all n, ft;n is an isomorphism if tn + 1 and is surjective for t = n, confirming a conjecture of C-H. Sah. fn;n is an isomorphism when n is odd and when n is even the kernel is isomorphic to the (n + 1)st power of the fundamental ideal of the Witt Ring of F. When n is even the Cokernel of fn−1;n is isomorphic to the nth Milnor-Witt K-theory group of F. When n is odd, the Cokernel of fn−1;n is isomorphic to the square of the nth Milnor K-group of F.
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the third homology of the special linear group of a field
Journal of Pure and Applied Algebra, 2009Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Abstract We prove that for any infinite field F , the map H 3 ( SL n ( F ) , Z ) → H 3 ( SL n + 1 ( F ) , Z ) is an isomorphism for all n ≥ 3 . When n = 2 the Cokernel of this map is naturally isomorphic to 2 ⋅ K 3 M ( F ) , where K n M ( F ) is the n th Milnor K -group of F . We deduce that the natural homomorphism from H 3 ( SL 2 ( F ) , Z ) to the indecomposable K 3 of F , K 3 ( F ) ind , is surjective for any infinite field F .
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homology stability for the special linear group of a field and milnor witt k theory
arXiv: K-Theory and Homology, 2008Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism and when n is even the kernel of f(n,n) is the (n+1)st power of the fundamental ideal of the Witt Ring of the field.. If n is even, then the Cokernel of f(n-1,n) is naturally isomorphic to the n-th Milnor-Witt K-group of F, MWK(n,F) and when n>2 is odd the Cokernel of f(n-1,n) is the square of the nth Milnor K-group of F.
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the third homology of the special linear group of a field
arXiv: K-Theory and Homology, 2008Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the Cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.
Melanie Matchett Wood - One of the best experts on this subject based on the ideXlab platform.
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random integral matrices universality of surjectivity and the Cokernel
arXiv: Probability, 2018Co-Authors: Hoi H Nguyen, Melanie Matchett WoodAbstract:For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the Cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n^{-1+epsilon}.
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on a cohen lenstra heuristic for jacobians of random graphs
Journal of Algebraic Combinatorics, 2015Co-Authors: Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, Melanie Matchett WoodAbstract:In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen---Lenstra-type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over $${\mathbb {Z}}_p$$Zp, distributed according to Haar measure, have Cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.
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on a cohen lenstra heuristic for jacobians of random graphs
arXiv: Combinatorics, 2014Co-Authors: Julien Clancy, Nathan Kaplan, Timothy Leake, Sam Payne, Melanie Matchett WoodAbstract:In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over the p-adic integers, distributed according to Haar measure, have Cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.
Kevin Hutchinson - One of the best experts on this subject based on the ideXlab platform.
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homology stability for the special linear group of a field and milnor witt k theory
Documenta Mathematica, 2010Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Let F be a field of characteristic zero and let ft;n be the stabilization homomorphism from the nth integral homology of SLt(F) to the nth integral homology of SLt+1(F). We prove the following results: For all n, ft;n is an isomorphism if tn + 1 and is surjective for t = n, confirming a conjecture of C-H. Sah. fn;n is an isomorphism when n is odd and when n is even the kernel is isomorphic to the (n + 1)st power of the fundamental ideal of the Witt Ring of F. When n is even the Cokernel of fn−1;n is isomorphic to the nth Milnor-Witt K-theory group of F. When n is odd, the Cokernel of fn−1;n is isomorphic to the square of the nth Milnor K-group of F.
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the third homology of the special linear group of a field
Journal of Pure and Applied Algebra, 2009Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Abstract We prove that for any infinite field F , the map H 3 ( SL n ( F ) , Z ) → H 3 ( SL n + 1 ( F ) , Z ) is an isomorphism for all n ≥ 3 . When n = 2 the Cokernel of this map is naturally isomorphic to 2 ⋅ K 3 M ( F ) , where K n M ( F ) is the n th Milnor K -group of F . We deduce that the natural homomorphism from H 3 ( SL 2 ( F ) , Z ) to the indecomposable K 3 of F , K 3 ( F ) ind , is surjective for any infinite field F .
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homology stability for the special linear group of a field and milnor witt k theory
arXiv: K-Theory and Homology, 2008Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism and when n is even the kernel of f(n,n) is the (n+1)st power of the fundamental ideal of the Witt Ring of the field.. If n is even, then the Cokernel of f(n-1,n) is naturally isomorphic to the n-th Milnor-Witt K-group of F, MWK(n,F) and when n>2 is odd the Cokernel of f(n-1,n) is the square of the nth Milnor K-group of F.
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the third homology of the special linear group of a field
arXiv: K-Theory and Homology, 2008Co-Authors: Kevin Hutchinson, Liqun TaoAbstract:We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the Cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory.
Torsten Ehrhardt - One of the best experts on this subject based on the ideXlab platform.
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on the kernel and Cokernel of some toeplitz operators
2013Co-Authors: Torsten Ehrhardt, Ilya M SpitkovskyAbstract:We show that the kernel and/or Cokernel of a block Toeplitz operator T (G) are trivial if its matrix-valued symbol G satisfies the condition \(G(t^{-1})G(t)^*\;=\;I_N\). As a consequence, the Wiener–Hopf factorization of G (provided it exists) must be canonical. Our setting is that of weighted Hardy spaces on the unit circle. We extend our result to Toeplitz operators on weighted Hardy spaces on the real line, and also Toeplitz operators on weighted sequence spaces.
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invertibility theory for toeplitz plus hankel operators and singular integral operators with flip
Journal of Functional Analysis, 2004Co-Authors: Torsten EhrhardtAbstract:Abstract It is well known that a Toeplitz operator is invertible if and only if its symbols admits a canonical Wiener–Hopf factorization, where the factors satisfy certain conditions. A similar result holds also for singular integral operators. More generally, the dimension of the kernel and Cokernel of Toeplitz or singular integral operators which and Fredholm operators can be expressed in terms of the partial indices ϰ 1 ,…,ϰ N ∈ Z of an associated Wiener–Hopf factorization problem. In this paper we establish corresponding results for Toeplitz plus Hankel operators and singular integral operators with flip under the assumption that the generating functions are sufficiently smooth (e.g., Holder continuous). We are led to a slightly different factorization problem, in which pairs (ϱ 1 ,ϰ 1 ),…,(ϱ N ,ϰ N )∈{−1,1}× Z , instead of the partial indices appear. These pairs provide the relevant information about the dimension of the kernel and Cokernel and thus answer the invertibility problem.
James Conant - One of the best experts on this subject based on the ideXlab platform.
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addendum to the johnson Cokernel and the enomoto satoh invariant the es trace detects all top level partitions
arXiv: Quantum Algebra, 2016Co-Authors: James ConantAbstract:The degree $d$ part of the Cokernel $\mathsf C_d$ of the Johnson homomorphism decomposes into irreducible $\mathrm{SP}$-modules indexed by partitions of $d-2r$ for $r\geq 0$: $$\mathsf C_d\cong \mathsf C_d(d)\oplus \mathsf C_d(d-2)\oplus\cdots.$$ In this addendum we calculate $\mathsf{C}_d(d)$ precisely: it is isomorphic to the $\mathrm{GL}(V)$-decomposition of a space of coinvariants $(V^{\otimes d})_{D_{2d}}$, and the isomorphism is induced by Enomoto and Satoh's trace map. This establishes Conjecture 7.2 of the paper "The Johnson Cokernel and the Enomoto-Satoh invariant."
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hopf algebras and invariants of the johnson Cokernel
Algebraic & Geometric Topology, 2016Co-Authors: James Conant, Martin KassabovAbstract:We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the Cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson Cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of [J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of Topology, Geom. Dedicata 176 (2015), 345--374.].
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the johnson Cokernel and the enomoto satoh invariant
Algebraic & Geometric Topology, 2015Co-Authors: James ConantAbstract:We study the Cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto and Satoh and Conant, Kassabov and Vogtmann is given, and using technology from the author’s work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto and Satoh. The Enomoto‐Satoh trace is the rank-1 part of the new trace, and it is here that the new series of representations is found. The rank-2 part is also investigated, though a fuller investigation of the higher-rank case is deferred to another paper. 17B40; 20C15, 20F28
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the johnson Cokernel and the enomoto satoh invariant
arXiv: Quantum Algebra, 2013Co-Authors: James ConantAbstract:We study the Cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto-Satoh and Conant-Kassabov-Vogtmann is given, and using technology from the author's work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto-Satoh. The Enomoto-Satoh trace is the rank 1 part of the new trace. The rank 2 part is also investigated.