Journal ArticleJournal fur die Reine und Angewandte Mathematik · September 1, 2023
We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We ...
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Journal ArticleJournal of the Institute of Mathematics of Jussieu · March 16, 2023
We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros ...
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Journal ArticleIsrael Journal of Mathematics · January 1, 2023
Information about the absolute Galois group GK of a number field K is encoded in how it acts on the étale fundamental group π of a curve X defined over K. In the case that K = ℚ(ζn) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderso ...
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Chapter · October 31, 2022
We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch. ...
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Journal ArticleTopology and its Applications · July 1, 2022
We show the A1-Euler characteristic of a smooth, projective scheme over a characteristic 0 field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported A1-Euler characteristic [Fo ...
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Journal ArticleResearch in Mathematical Sciences · June 1, 2021
These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel’s A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor ...
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Journal ArticleTransactions of the American Mathematical Society · May 1, 2021
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k, this enrichment counts the number of lines meeting four lines defined over k in P3k, with s ...
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Journal ArticleCompositio Mathematica · April 1, 2021
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field k, generalizing the counts that over C there are 27 lines, and over R the number of hyperbolic lines minus the number of elliptic lines is 3. In general, the lines a ...
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Journal ArticleDuke Mathematical Journal · February 15, 2019
Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud-Khimshiashvili-Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, t ...
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Journal ArticleTransactions of the American Mathematical Society · January 1, 2019
Let Ln denote the topological poset of decompositions of ℂn into mutually orthogonal subspaces. We classify p-toral subgroups of U(n) that can have noncontractible fixed points under the action of U(n) on Ln. ...
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Chapter · 2019
The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other ... ...
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Journal ArticleJournal of Algebra · July 1, 2018
For an odd prime p satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group GQ(ζp) on the homology of the degree p Fermat curve, building on work of Anderson. Further, we study the invariants and the first Gal ...
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Chapter · January 1, 2018
For a global field, local field, or finite field k with infinite Galois group, we show that there cannot exist a functor from the Morel-Voevodsky A1-homotopy category of schemes over k to a genuine Galois equivariant homotopy category satisfying a list of ...
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Journal ArticleJournal of Pure and Applied Algebra · July 1, 2017
For x an element of a field other than 0 or 1, we compute the order n Massey products 〈(1−x)−1,x−1,…,x−1,(1−x)−1〉 of n−2 factors of x−1 and two factors of (1−x)−1 by embedding P1−{0,1,∞} into its Picard variety and constructing Gal(ks/k) equivariant maps f ...
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Journal ArticleJournal fur die Reine und Angewandte Mathematik · September 1, 2023
We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We ...
Full textCite

Journal ArticleJournal of the Institute of Mathematics of Jussieu · March 16, 2023
We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros ...
Full textCite

Journal ArticleIsrael Journal of Mathematics · January 1, 2023
Information about the absolute Galois group GK of a number field K is encoded in how it acts on the étale fundamental group π of a curve X defined over K. In the case that K = ℚ(ζn) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderso ...
Full textCite

Chapter · October 31, 2022
We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch. ...
Cite

Journal ArticleTopology and its Applications · July 1, 2022
We show the A1-Euler characteristic of a smooth, projective scheme over a characteristic 0 field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported A1-Euler characteristic [Fo ...
Full textCite

Journal ArticleResearch in Mathematical Sciences · June 1, 2021
These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel’s A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor ...
Full textCite

Journal ArticleTransactions of the American Mathematical Society · May 1, 2021
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k, this enrichment counts the number of lines meeting four lines defined over k in P3k, with s ...
Full textCite

Journal ArticleCompositio Mathematica · April 1, 2021
We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field k, generalizing the counts that over C there are 27 lines, and over R the number of hyperbolic lines minus the number of elliptic lines is 3. In general, the lines a ...
Full textCite

Journal ArticleDuke Mathematical Journal · February 15, 2019
Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud-Khimshiashvili-Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, t ...
Full textCite

Journal ArticleTransactions of the American Mathematical Society · January 1, 2019
Let Ln denote the topological poset of decompositions of ℂn into mutually orthogonal subspaces. We classify p-toral subgroups of U(n) that can have noncontractible fixed points under the action of U(n) on Ln. ...
Full textCite

Chapter · 2019
The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other ... ...
Link to itemCite

Journal ArticleJournal of Algebra · July 1, 2018
For an odd prime p satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group GQ(ζp) on the homology of the degree p Fermat curve, building on work of Anderson. Further, we study the invariants and the first Gal ...
Full textCite

Chapter · January 1, 2018
For a global field, local field, or finite field k with infinite Galois group, we show that there cannot exist a functor from the Morel-Voevodsky A1-homotopy category of schemes over k to a genuine Galois equivariant homotopy category satisfying a list of ...
Full textCite

Journal ArticleJournal of Pure and Applied Algebra · July 1, 2017
For x an element of a field other than 0 or 1, we compute the order n Massey products 〈(1−x)−1,x−1,…,x−1,(1−x)−1〉 of n−2 factors of x−1 and two factors of (1−x)−1 by embedding P1−{0,1,∞} into its Picard variety and constructing Gal(ks/k) equivariant maps f ...
Full textCite

Journal ArticleGeometry and Topology · May 19, 2017
We study a version of the James model for the loop space of a suspension in unstable A1-homotopy theory. We use this model to establish an analog of G W Whitehead’s classical refinement of the Freudenthal suspension theorem in A1-homotopy theory: our resul ...
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ConferenceInternational Journal of Mathematics · June 1, 2016
We use the Galois action on Q1 -{0, 1,∞}) to show that the homotopy equivalence S1 Λ (Gm, Gm, S1 (1 -{0, 1,∞}) coming from purity, does not desuspend to a map Gm, Gm, 1 -{0, 1,∞}. ...
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Conference · January 1, 2016
In his paper titled â€œTorsion points on Fermat Jacobians, roots of circular units and relative singular homology,â€ Anderson determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group (Forumala pre ...
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Journal ArticleJournal of Pure and Applied Algebra · May 1, 2015
We construct splitting varieties for triple Massey products. For a, b, c∈F* the triple Massey product 〈a, b, c〉 of the corresponding elements of H1(F, μ2) contains 0 if and only if there are x∈F* and y∈F[a,c]* such that bx2=NF[a,c]/F(y), where NF[a,c]/F de ...
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Journal ArticleAlgebraic and Geometric Topology · March 23, 2015
For a smooth algebraic curve X over a field, applying H1 to the Abel map X→PicX∕∂X to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H1(X,Z∕ℓ)→H1(X∕∂X,Z∕ℓ(1))≅H1c(X,Z∕ℓ(1)). We show the analogous statement for the Abel ...
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Journal ArticleMathematische Annalen · February 1, 2014
We show a 2-nilpotent section conjecture over ℝ: for a geometrically connected curve X over ℝ such that each irreducible component of its normalization has ℝ-points, π0(X(ℝ)) is determined by the maximal 2-nilpotent quotient of the fundamental group with i ...
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Chapter · January 1, 2012
We study which rational points of the Jacobian of ℙ1-{0,1,∞} can be lifted to sections of geometrically 3-nilpotent quotients of étale π1 over the absolute Galois group. This is equivalent to evaluating certain triple Massey products of elements of k* ⊆ H1 ...
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Journal ArticleThe Arithmetic of Fundamental Groups - PIA 2010, editor J. Stix,
Contributions in Mathematical and Computational Sciences, Vol. 2,
Springer-Verlag Berlin Heidelberg, 2012 · January 1, 2012
We study which rational points of the Jacobian of P^1_K -{0,1,infty} can be
lifted to sections of geometrically 3 nilpotent quotients of etale pi_1 over
the absolute Galois group. This is equivalent to evaluating certain triple
Massey products of elements ...
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Journal ArticleJournal de Theorie des Nombres de Bordeaux · January 1, 2011
In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of w ...
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Journal ArticleColloquium Mathematicum · January 1, 2005
We study weak mixing and double ergodicity for nonsingular actions of locally compact Polish abelian groups. We show that if T is a nonsingular action of G, then T is weakly mixing if and only if for all cocompact subgroups A of G the action of T restricte ...
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