James R. Conant
- Courses5
- Reviews20
- School: University of California San Diego
- Campus:
- Department: Mathematics
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Location:
9500 Gilman Dr
La Jolla, CA - 92093 - Dates at University of California San Diego: November 2016 - April 2017
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Biography
James R. Conant is a/an Visiting Professor in the University Of California department at University Of California
University of California San Diego - Mathematics
Experience
UC San Diego
Visiting Professor
Teach large lecture calculus courses. Engage in pure mathematics research.
UC San Diego
Visiting Professor
Teaching large lecture classes. Mathematics research.
University of Tennessee
Professor
Teach classes, conduct research. As Undergraduate Director, administer undergraduate program.
University of Tennessee
Professor
Teaching, research, service.
Art of Problem Solving
Campus Director, AoPS Academy-Carmel Valley
James worked at Art of Problem Solving as a Campus Director, AoPS Academy-Carmel Valley
Milarepa Osel Cho Dzong
Board Member
Helped in initiating and building a Buddhist retreat center in the mountains of Tennessee.
Education
Rutgers University-New Brunswick
Bachelor's degree
Mathematics
University of California, San Diego
Doctor of Philosophy (Ph.D.)
Mathematics
Art of Problem Solving
Campus Director, AoPS Academy-Carmel Valley
Publications
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Higher-order intersections in low-dimensional topology
Proc. Natl. Acad. Sci. USA
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato–Levine, and Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Higher-order intersections in low-dimensional topology
Proc. Natl. Acad. Sci. USA
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato–Levine, and Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
Whitney tower concordance of classical links
Geometry and Topology
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato–Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4–ball bounded by a link in the 3–sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants.
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Higher-order intersections in low-dimensional topology
Proc. Natl. Acad. Sci. USA
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato–Levine, and Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
Whitney tower concordance of classical links
Geometry and Topology
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato–Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4–ball bounded by a link in the 3–sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants.
On the number of tilings of a square by rectangles.
Annals of Combinatorics
We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n−1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Higher-order intersections in low-dimensional topology
Proc. Natl. Acad. Sci. USA
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato–Levine, and Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
Whitney tower concordance of classical links
Geometry and Topology
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato–Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4–ball bounded by a link in the 3–sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants.
On the number of tilings of a square by rectangles.
Annals of Combinatorics
We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n−1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.
Hairy graphs and the unstable homology of Mod(g,s), Out(F_n) and Aut(F_n)
Journal of Topology
We study a family of Lie algebras 𝔥𝒪 which are defined for cyclic operads 𝒪. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper, we introduce a hairy graph homology theory for 𝒪. We show that the homology of 𝔥𝒪 embeds in hairy graph homology via a trace map that generalizes the trace map defined by Morita. For the Lie operad, we use the trace map to find large new summands of the abelianization of 𝔥𝒪 which are related to classical modular forms for SL2(ℤ). Using cusp forms, we construct new cycles for the unstable homology of Out(Fn), and using Eisenstein series, we find new cycles for Aut(Fn). For the associative operad, we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of 𝔥𝒪 in the associative case.
Tree homology and a conjecture of Levine
Geometry and Topology
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism η′: T → D′ is an isomorphism. Both T and D′ are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of Out(Fn). In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain T and range D′ of Levine’s map. The isomorphism η′ is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.
Higher-order intersections in low-dimensional topology
Proc. Natl. Acad. Sci. USA
We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato–Levine, and Arf invariants. We also define higher-order Sato–Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
Whitney tower concordance of classical links
Geometry and Topology
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato–Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4–ball bounded by a link in the 3–sphere. Applications include computation of the grope filtration and new geometric characterizations of Milnor’s link invariants.
On the number of tilings of a square by rectangles.
Annals of Combinatorics
We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n−1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.
Hairy graphs and the unstable homology of Mod(g,s), Out(F_n) and Aut(F_n)
Journal of Topology
We study a family of Lie algebras 𝔥𝒪 which are defined for cyclic operads 𝒪. Using his graph homology theory, Kontsevich identified the homology of two of these Lie algebras (corresponding to the Lie and associative operads) with the cohomology of outer automorphism groups of free groups and mapping class groups of punctured surfaces, respectively. In this paper, we introduce a hairy graph homology theory for 𝒪. We show that the homology of 𝔥𝒪 embeds in hairy graph homology via a trace map that generalizes the trace map defined by Morita. For the Lie operad, we use the trace map to find large new summands of the abelianization of 𝔥𝒪 which are related to classical modular forms for SL2(ℤ). Using cusp forms, we construct new cycles for the unstable homology of Out(Fn), and using Eisenstein series, we find new cycles for Aut(Fn). For the associative operad, we compute the first homology of the hairy graph complex by adapting an argument of Morita, Sakasai and Suzuki, who determined the complete abelianization of 𝔥𝒪 in the associative case.
Homotopy approximations to the space of knots, Feynman diagrams, and a conjecture of Scannell and Sinha
American Journal of Mathematics
Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in R^n, for n ≥ 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E2 stage precisely give the space of irreducible Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of long knots in R^3 are in one-to-one correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree. Our proof yields a sequence of new presentations for the space of irreducible Feynman diagrams
