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Kirsten Graham Wickelgren

Professor of Mathematics
Mathematics

Selected Publications


On quadratically enriched excess and residual intersections

Journal Article Journal 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 text Cite

EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal Article Journal 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 text Cite

The Galois action on the lower central series of the fundamental group of the Fermat curve

Journal Article Israel 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 text Cite

An explicit self-duality

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

Compactly supported A1-Euler characteristic and the Hochschild complex

Journal Article Topology 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 text Cite

Applications to A1 -enumerative geometry of the A1 -degree

Journal Article Research 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 text Cite

An arithmetic count of the lines meeting four lines in P3

Journal Article Transactions 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 text Cite

An arithmetic count of the lines on a smooth cubic surface

Journal Article Compositio 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 text Cite

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Journal Article Linear Algebra and Its Applications · June 15, 2020 © 2020 Elsevier Inc. Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions ess ... Full text Open Access Cite

Examples of wild ramification in an enriched riemann–hurwitz formula

Journal Article Contemporary Mathematics · January 1, 2020 © 2020 American Mathematical Society. M. Levine proved an enrichment of the classical Riemann–Hurwitz formula to an equality in the Grothendieck–Witt group of quadratic forms. In its strongest form, Levine’s theorem includes a technical hypothesis on ramif ... Full text Cite

The class of Eisenbud-Khimshiashvili-Levine is the local A 1 -Brouwer degree

Journal Article Duke 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 text Cite

Classification of problematic subgroups of U(n)

Journal Article Transactions 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 text Cite

The simplicial EHP sequence in A1–algebraic topology

Journal Article Geometry and Topology · January 1, 2019 © 2019, Mathematical Sciences Publishers. All rights reserved. We give a tool for understanding simplicial desuspension in A1–algebraic topology: We show that X→Ω(S1∧ X) → Ω(S1∧ X∧ X) is a fiber sequence up to homotopy in 2–localized A1 algebraic topology ... Full text Cite

Unstable Motivic Homotopy Theory

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 item Cite

The Galois action and cohomology of a relative homology group of Fermat curves

Journal Article Journal 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 text Cite

An Étale realization which does NOT exist

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 text Cite

Massey products 〈y,x,x,…,x,x,y〉 in Galois cohomology via rational points

Journal Article Journal 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 text Cite

On quadratically enriched excess and residual intersections

Journal Article Journal 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 text Cite

EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal Article Journal 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 text Cite

The Galois action on the lower central series of the fundamental group of the Fermat curve

Journal Article Israel 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 text Cite

An explicit self-duality

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

Compactly supported A1-Euler characteristic and the Hochschild complex

Journal Article Topology 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 text Cite

Applications to A1 -enumerative geometry of the A1 -degree

Journal Article Research 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 text Cite

An arithmetic count of the lines meeting four lines in P3

Journal Article Transactions 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 text Cite

An arithmetic count of the lines on a smooth cubic surface

Journal Article Compositio 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 text Cite

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Journal Article Linear Algebra and Its Applications · June 15, 2020 © 2020 Elsevier Inc. Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions ess ... Full text Open Access Cite

Examples of wild ramification in an enriched riemann–hurwitz formula

Journal Article Contemporary Mathematics · January 1, 2020 © 2020 American Mathematical Society. M. Levine proved an enrichment of the classical Riemann–Hurwitz formula to an equality in the Grothendieck–Witt group of quadratic forms. In its strongest form, Levine’s theorem includes a technical hypothesis on ramif ... Full text Cite

The class of Eisenbud-Khimshiashvili-Levine is the local A 1 -Brouwer degree

Journal Article Duke 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 text Cite

Classification of problematic subgroups of U(n)

Journal Article Transactions 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 text Cite

The simplicial EHP sequence in A1–algebraic topology

Journal Article Geometry and Topology · January 1, 2019 © 2019, Mathematical Sciences Publishers. All rights reserved. We give a tool for understanding simplicial desuspension in A1–algebraic topology: We show that X→Ω(S1∧ X) → Ω(S1∧ X∧ X) is a fiber sequence up to homotopy in 2–localized A1 algebraic topology ... Full text Cite

Unstable Motivic Homotopy Theory

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 item Cite

The Galois action and cohomology of a relative homology group of Fermat curves

Journal Article Journal 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 text Cite

An Étale realization which does NOT exist

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 text Cite

Massey products 〈y,x,x,…,x,x,y〉 in Galois cohomology via rational points

Journal Article Journal 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 text Cite

The simplicial suspension sequence in A1-homotopy

Journal Article Geometry 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 ... Full text Cite

Desuspensions of S 1 Λ (P1/Q - {0, 1, ∞ })

Conference International 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,∞}. ... Full text Cite

What is… an anabelian scheme?

Journal Article Notices of the American Mathematical Society · March 1, 2016 Full text Cite

Galois Action on the Homology of Fermat Curves

Conference · January 1, 2016 In his paper titled “Torsion points on Fermat Jacobians, roots of circular units and relative singular homology,” Anderson determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group (Forumala pre ... Full text Cite

Women in Topology

Journal Article · May 21, 2015 Full text Open Access Cite

Splitting varieties for triple Massey products

Journal Article Journal 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 ... Full text Cite

An Abel map to the compactified Picard scheme realizes Poincaré duality

Journal Article Algebraic 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 ... Full text Cite

2-Nilpotent real section conjecture

Journal Article Mathematische 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 ... Full text Cite

On 3-nilpotent obstructions to π1 sections for ℙ1-{0,1,∞}

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 ... Full text Cite

n-nilpotent obstructions to pi(1)sections of P-1 - {0, 1, infinity} and Massey products

Conference GALOIS-TEICHMUELLER THEORY AND ARITHMETIC GEOMETRY · January 1, 2012 Link to item Cite

3-nilpotent obstructions to pi_1 sections for P^1_Q - {0,1,infty}

Journal Article The 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 ... Link to item Cite

Universal covering spaces and fundamental groups in algebraic geometry as schemes

Journal Article Journal 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 ... Full text Cite

On weakly mixing and doubly ergodic nonsingular actions

Journal Article Colloquium 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 ... Full text Cite