Bruno Harris, professor of mathematics, works on algebraic topology and algebraic geometry areas with many applications to physics, including String Theory. Topology studies qualitative features, as distinguished from quantitative ones, of problems thus topology decides whether an equation has solutions, and how many solutions, but does not calculate these solutions exactly. Algebraic geometry deals with equations in many variables involving relatively simple functions-polynomials.

A few of Professor Harris's noteworthy results:

1."On the Homotopy Groups of the Classical Groups," Annals of Math (1961), and "Suspensions and Characteristic Maps for Symmetric Spaces," Annals of Math (1962). The Classical compact simple Lie groups come in four infinite series: SU(n), SO(2n), SO(2n+1), SP(n). In the mid 1950s, Bott had calculated their homotopy groups in dimensions i (Bott Periodicity), but only for i less than 2n. These groups are completely determined by their Dynkin diagrams (just dots and lines) which contain a complete set of assembly instructions, just like a chromosome. Two of the series, SO(2n+1) and Sp(n), have diagrams which are identical except for one wrong-way arrow. Based just on this observation, Serre made the surprising conjecture that their homotopy groups in all dimensions should be identical except for the prime 2 (the conjecture is mentioned at the end of a survey article "Topology of Lie Groups and Characteristic Classes " by A. Borel, Bulletin AMS (mid 1950s). This striking conjecture led many topologists to attempt a proof, but without success, until in the first paper above I provided a very short and simple-minded proof using Symmetric space concepts, at the same time going beyond the conjecture and including SU(n). Even after this proof, other topologists tried doing it again with their own methods, but still unsuccessfully. The second paper above brings in suspension and further symmetric spaces. Incidentally, Bott periodicity itself is given a very short and simple proof in my later paper, "Bott periodicity via Simplicial Spaces," Journal of Algebra (1980): If U is the infinite unitary group and BU its classifying space, the periodicity statement is that B(BU)=U. I show that this fact is exactly the Spectral theorem of elementary linear algebra for unitary matrices. Here is the Math Reviews review by Harold Hastings: "The author gives a beautiful well-motivated proof of the complex Bott periodicity theorem using only two essential properties of the complex numbers (unitary matrices can be diagonalized; the Stiefel manifolds of k-frames in C^N are highly connected for large n), and the group completion theorem."

2. "K-Groups of Rings of Algebraic Integers" (with G.Segal), Annals of Math , vol. 101, 1975. The theorem here gives a partial result on Quillen's higher K-groups of any ring of algebraic integers, in every odd dimension, namely a cyclic direct summand is constructed, but in certain cases its order is determined only up to a factor of 2 or 1/2. It turned out that Lichtenbaum conjectured at about the same time that this was the entire group (without the extra factor ambiguity). Later the complete result was more or less proved by several people, and in many of these papers the methods of this paper continued to be used. Even the ambiguous factor turned out to be necessary and has only recently been settled by Weibel. The tools used in this paper include the Cebotarev density theorem and a simple transfer procedure from linear groups over a finite field to those over a ring of algebraic integers. Segal originally envisaged a greater use and development of the Transfer, but this only came to full fruition in his later work.

3. "Harmonic Volumes," Acta Mathematica , Vol. 150 (1983), and "Homological versus Algebraic Equivalence in a Jacobian," Proceedings of the National Academy of Sciences of the United States of America , Vol. 80 ((1983). The first paper uses Chen's Iterated Integrals on Riemann surfaces to give an explicit formula for the Abel-Jacobi image of a certain cycle on the Jacobian (the image being in a higher intermediate Jacobian). Previously Griffiths had given a general theorem for the derivative with respect to parameters of the Abel-Jacobi images of cycles and so was able to show the existence in parametrized families of cycles with certain properties (homologous to zero but not algebraically equivalent to zero), but his method could not give explicit examples, e.g., defined over integers. The method of the first paper above can be used to give such explicit examples, in fact very simple ones that lend themselves to arithmetic computation (and such arithmetic computation was next done by S. Bloch and his students, C. Schoen and D. Zelinsky). The first paper also gives "generic" results in the sense of Griffiths for the cycle obtained from a Riemann surface in its Jacobian. Later results of Hain give further significance to this cycle for the moduli space of curves. Here is the Math Reviews review by David Lieberman of the second paper: "The existence of cycles which are homologous to 0, but not algebraically equivalent to 0, was first shown (non-constuctively) by P. Griffiths ( Annals of Math , 1969). In the present article the author produces the first explicit example of this phenomenon. If C denotes the degree 4 Fermat curve, J its Jacobian and C^- the 'inverse' of C on J, then X= C-(C^-) povides the example. To achieve this, the author gives formulae for calculating the Weil-Abel-Jacobi mapping, W, of X to the intermediate Jacobian of J and for checking that W(X) is not zero. G. Ceresa ( Annals , 1983) and the author ( Acta Math , 150 (1983) have also shown for generic J, C that W(X) is non-zero, employing arguments in the spirit of Griffiths's earlier work."

4."Height Pairings and the Heat Equation," Topology , Vol. 32, 1993, showed that the kernel of the Heat operator (on differential forms) on a compact Kahler manifold, and a related "linking kernel" were convenient for calculating height pairings of complex cycles. An analogous kernel on a real Riemannian manifold can be used for linking numbers of (real) cycles and further for defining a real-valued generalized linking number which reduces modulo the integers to the Abel-Jacobi image. This relation of the Archimedean Height pairing to the Heat Kernel aroused the interest of Lang and Jorgenson, and they included part of this paper in their survey article "The Ubiquitous Heat Kernel" (p.655-682 of Mathematics Unlimited: 2001 and Beyond , edited by B. Engquist and W. Schmid, Springer, 2001).

5. Bin Wang's Ph.D. thesis (Brown, 1994) and further work with Bin Wang: Previous to Bin's work, the Archimedean height pairing (of complex cycles) required disjointness of the supports of the cycles. In a very original thesis, Bin Wang obtained a formula for the asymptotics of the height pairing as the cycles move in a 1-parameter family towards a position where they intersect - the leading term in the asymptotics involved as coeffient an integer which he showed to be a Chern number describing the intersection. (Previously Hain had used Hodge theory to obtain an integer coefficient but left its actual value unspecified.) Barry Mazur then became interested in Bin Wang's result and formulated a conjectural "Incidence Divisor" on the Chow variety to describe pairs of cycles of "linking dimension" which intersect. Bin later proved these conjectures of Mazur and began applying the to important problems in Algebraic Geometry.

5. My book Iterated Integrals and Cycles on Algebraic Manifolds ( Nankai Tracts in Mathematics , Vol. 7, World Scientific Publishing, 2004) describes some of the work in #3 and #4 (above) with further developments. The book is based on a course I gave at the Nankai Mathematical Institute during fall semester 2001. I will always be most grateful to Professor S.S. Chern for inviting me to give a lecture at the Nankai Conference in 2000 on the work of W.L. Chow and K.T. Chen, to give this course the following year, and to publish the monograph based on it.

6. Current work: What may be called "Linking Integrals" occur in various parts of mathematics, ranging from arithmetic geometry to V. Arnold's work on "Asymptotic Linking" of streamlines in Fluid Mechanics (see the book Topological Methods in Hydrodynamics , by Arnold and Khesin, 1998), and I am exploring and trying to develop such ideas.

1."On the Homotopy Groups of the Classical Groups," Annals of Math (1961), and "Suspensions and Characteristic Maps for Symmetric Spaces," Annals of Math (1962). The Classical compact simple Lie groups come in four infinite series: SU(n), SO(2n), SO(2n+1), SP(n). In the mid 1950s, Bott had calculated their homotopy groups in dimensions i (Bott Periodicity), but only for i less than 2n. These groups are completely determined by their Dynkin diagrams (just dots and lines) which contain a complete set of assembly instructions, just like a chromosome. Two of the series, SO(2n+1) and Sp(n), have diagrams which are identical except for one wrong-way arrow. Based just on this observation, Serre made the surprising conjecture that their homotopy groups in all dimensions should be identical except for the prime 2 (the conjecture is mentioned at the end of a survey article "Topology of Lie Groups and Characteristic Classes " by A. Borel, Bulletin AMS (mid 1950s). This striking conjecture led many topologists to attempt a proof, but without success, until in the first paper above I provided a very short and simple-minded proof using Symmetric space concepts, at the same time going beyond the conjecture and including SU(n). Even after this proof, other topologists tried doing it again with their own methods, but still unsuccessfully. The second paper above brings in suspension and further symmetric spaces. Incidentally, Bott periodicity itself is given a very short and simple proof in my later paper, "Bott periodicity via Simplicial Spaces," Journal of Algebra (1980): If U is the infinite unitary group and BU its classifying space, the periodicity statement is that B(BU)=U. I show that this fact is exactly the Spectral theorem of elementary linear algebra for unitary matrices. Here is the Math Reviews review by Harold Hastings: "The author gives a beautiful well-motivated proof of the complex Bott periodicity theorem using only two essential properties of the complex numbers (unitary matrices can be diagonalized; the Stiefel manifolds of k-frames in C^N are highly connected for large n), and the group completion theorem."

2. "K-Groups of Rings of Algebraic Integers" (with G.Segal), Annals of Math , vol. 101, 1975. The theorem here gives a partial result on Quillen's higher K-groups of any ring of algebraic integers, in every odd dimension, namely a cyclic direct summand is constructed, but in certain cases its order is determined only up to a factor of 2 or 1/2. It turned out that Lichtenbaum conjectured at about the same time that this was the entire group (without the extra factor ambiguity). Later the complete result was more or less proved by several people, and in many of these papers the methods of this paper continued to be used. Even the ambiguous factor turned out to be necessary and has only recently been settled by Weibel. The tools used in this paper include the Cebotarev density theorem and a simple transfer procedure from linear groups over a finite field to those over a ring of algebraic integers. Segal originally envisaged a greater use and development of the Transfer, but this only came to full fruition in his later work.

3. "Harmonic Volumes," Acta Mathematica , Vol. 150 (1983), and "Homological versus Algebraic Equivalence in a Jacobian," Proceedings of the National Academy of Sciences of the United States of America , Vol. 80 ((1983). The first paper uses Chen's Iterated Integrals on Riemann surfaces to give an explicit formula for the Abel-Jacobi image of a certain cycle on the Jacobian (the image being in a higher intermediate Jacobian). Previously Griffiths had given a general theorem for the derivative with respect to parameters of the Abel-Jacobi images of cycles and so was able to show the existence in parametrized families of cycles with certain properties (homologous to zero but not algebraically equivalent to zero), but his method could not give explicit examples, e.g., defined over integers. The method of the first paper above can be used to give such explicit examples, in fact very simple ones that lend themselves to arithmetic computation (and such arithmetic computation was next done by S. Bloch and his students, C. Schoen and D. Zelinsky). The first paper also gives "generic" results in the sense of Griffiths for the cycle obtained from a Riemann surface in its Jacobian. Later results of Hain give further significance to this cycle for the moduli space of curves. Here is the Math Reviews review by David Lieberman of the second paper: "The existence of cycles which are homologous to 0, but not algebraically equivalent to 0, was first shown (non-constuctively) by P. Griffiths ( Annals of Math , 1969). In the present article the author produces the first explicit example of this phenomenon. If C denotes the degree 4 Fermat curve, J its Jacobian and C^- the 'inverse' of C on J, then X= C-(C^-) povides the example. To achieve this, the author gives formulae for calculating the Weil-Abel-Jacobi mapping, W, of X to the intermediate Jacobian of J and for checking that W(X) is not zero. G. Ceresa ( Annals , 1983) and the author ( Acta Math , 150 (1983) have also shown for generic J, C that W(X) is non-zero, employing arguments in the spirit of Griffiths's earlier work."

4."Height Pairings and the Heat Equation," Topology , Vol. 32, 1993, showed that the kernel of the Heat operator (on differential forms) on a compact Kahler manifold, and a related "linking kernel" were convenient for calculating height pairings of complex cycles. An analogous kernel on a real Riemannian manifold can be used for linking numbers of (real) cycles and further for defining a real-valued generalized linking number which reduces modulo the integers to the Abel-Jacobi image. This relation of the Archimedean Height pairing to the Heat Kernel aroused the interest of Lang and Jorgenson, and they included part of this paper in their survey article "The Ubiquitous Heat Kernel" (p.655-682 of Mathematics Unlimited: 2001 and Beyond , edited by B. Engquist and W. Schmid, Springer, 2001).

5. Bin Wang's Ph.D. thesis (Brown, 1994) and further work with Bin Wang: Previous to Bin's work, the Archimedean height pairing (of complex cycles) required disjointness of the supports of the cycles. In a very original thesis, Bin Wang obtained a formula for the asymptotics of the height pairing as the cycles move in a 1-parameter family towards a position where they intersect - the leading term in the asymptotics involved as coeffient an integer which he showed to be a Chern number describing the intersection. (Previously Hain had used Hodge theory to obtain an integer coefficient but left its actual value unspecified.) Barry Mazur then became interested in Bin Wang's result and formulated a conjectural "Incidence Divisor" on the Chow variety to describe pairs of cycles of "linking dimension" which intersect. Bin later proved these conjectures of Mazur and began applying the to important problems in Algebraic Geometry.

5. My book Iterated Integrals and Cycles on Algebraic Manifolds ( Nankai Tracts in Mathematics , Vol. 7, World Scientific Publishing, 2004) describes some of the work in #3 and #4 (above) with further developments. The book is based on a course I gave at the Nankai Mathematical Institute during fall semester 2001. I will always be most grateful to Professor S.S. Chern for inviting me to give a lecture at the Nankai Conference in 2000 on the work of W.L. Chow and K.T. Chen, to give this course the following year, and to publish the monograph based on it.

6. Current work: What may be called "Linking Integrals" occur in various parts of mathematics, ranging from arithmetic geometry to V. Arnold's work on "Asymptotic Linking" of streamlines in Fluid Mechanics (see the book Topological Methods in Hydrodynamics , by Arnold and Khesin, 1998), and I am exploring and trying to develop such ideas.

Here is a detailed review from 1995:

American Mathematical Society

Post Office Box 6248, Providence, RI 02940-6248

USA Location: 201 Charles Street, Providence, RI 02904

Telephone: 401-455-4000, 1-800-321-4AMS (4267)

FAX: 401-455-4004, Telex: 797192

Samuel M. Rankin III

Internet: smr@math.ams.org

Associate Executive Director

July, 1995

Professor Bruno Harris

Department of Mathematics

Brown University

Box 1917

Providence, RI 02912

Dear Professor Harris:

The American Mathematical Society assists the National Security Agency in evaluating the scientific merit of research proposals submitted to the NSA under their program for sponsoring mathematical research at nonprofit institutions. The Society performs this service by convening an advisory panel which, through a combination of peer review and its own expertise, makes recommendations to the NA for funding. The final decision on awarding a grant is made solely by the NSA.

As is customary, I am sending you copies of the reviews which were used by the advisory panel in evaluating the scientific merit of your proposal. In the case of multiple principal investigators, these are only being sent to the PI listed first on the proposal. These copies have been edited to remove any information which may identify the reviewer. The process of peer review, of course, depends on this anonymity to protect its value.

The advisory panel used the information in these reviews to provide consensus recommendations to the NSA. In addition, your proposal as studied by one or more panelist whose own expertise was instrumental in formulating the final recommendation of the panel. A always, reviews, and in particular overall ratings, are open to interpretation and several factors were considered in evaluating the reviews. There was adequate provision for complete discussion of each proposal and the reviews at the panel's meeting.

We hope that these reviews will provide useful information to you.

Sincerely,

Samuel M. Rankin, III

SMR/edf

Cc: National Security Agency

National Security Agency

American Mathematical Society

Mathematical Sciences Program

PO Box 6248, Providence, RI 02940

Proposal No.: 95S-035 Principal Investigator: Bruno Harris

Title: Height Pairing of Algebraic Cycles

Please evaluate this proposal using the criteria presented on the enclosed information sheet. Continue on additional sheets and necessary. Please type your review if possible.

The Bruno Harris proposal is very strong. The reviewers' comments were uniformly positive and enthusiastic. One that stands out was the remark: "Speaking personally, I have been interested in the phenomena which is the main subject-matter of the present proposal (for applications to the theory of heights in certain number theoretic situations) for a while now but I feel that I only came to understand whatever concrete things I understand in this subject by studying the joint work of Harris and Wang".

Overall rating:

[X] Excellent [ ] Very good [ ] Good [ ] Fair [ ] Poor

Heat Kernels and Cycles, in "The Ubiquitous Heat Kernel", edited by J. Jorgenson & L. Walling, Contemporary Mathematics AMS, 2006

"Iterated Integrals and Cycles on Algebraic Manifolds" Nankai Tracts in Mathematics Vol.7, World Scientific,2004

Height pairings asymptotics and Bott-Chern forms. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 93--113, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000. w/Wang, Bin

Archimedean height pairing of intersecting cycles. Internat. Math. Res. Notices 1993, no. 4, 107--111. w/Wang, B.

Cycle pairings and the heat equation. Topology 32 (1993), no. 2, 225--238.

Iterated integrals and Epstein zeta functions with harmonic rational function coefficients. Illinois J. Math. 34 (1990), no. 2, 325--336.

An analytic function and iterated integrals. Compositio Math. 65 (1988), no. 2, 209--221.

Differential characters and the Abel-Jacobi map. Algebraic $K$-theory: connections with geometry and topology (Lake Louise, AB, 1987), 69--86, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 279, Kluwer Acad. Publ., Dordrecht, 1989.

A triple product for automorphic forms. Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 133, 67--75.

Harmonic volumes. Acta Math. 150 (1983), no. 1-2, 91--123.

Homological versus algebraic equivalence in a Jacobian. Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 4 i., 1157--1158.

Triple products, modular forms, and harmonic volumes. Algebraists' homage: papers in ring theory and related topics (New Haven, Conn., 1981), pp. 287--295, Contemp. Math., 13, Amer. Math. Soc., Providence, R.I., 1982.

Bott periodicity via simplicial spaces. J. Algebra 62 (1980), no. 2, 450--454.

Group cohomology classes with differential form coefficients. Algebraic K-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), pp. 278--282. Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976.

K_{i} groups of rings of algebraic integers. Ann. of Math. (2) 101 (1975), 20--33. w/Segal, G.

Suspension, automorphisms, and division algebras. Algebraic $K$-theory, II: "Classical" algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Seattle Res. Center, Seattle, Wash., 1972), pp. 337--346. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973 18F25 w/Stasheff, J.

Suspension, automorphisms, and division algebras. Algebraic $K$-theory, II: "Classical" algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Seattle Res. Center, Seattle, Wash., 1972), pp. 337--346. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973 w/Stasheff, J.

The Adams $e$-homomorphism and non-stable homotopy groups. Topology 8 1969 337--344.

$J$-homomorphisms and cobordism groups. Invent. Math. 7 1969 313--320.

The $K$-theory of a class of homogeneous spaces. Trans. Amer. Math. Soc. 131 1968 323--332.

Torsion in Lie groups and related spaces. Topology 5 1966 347--354.

Some calculations of homotopy groups of symmetric spaces. Trans. Amer. Math. Soc. 106 1963 174--184.

Suspensions and characteristic maps for symmetric spaces. Ann. of Math. (2) 76 1962 295--305.

Cohomology of Lie triple systems and Lie algebras with involution. Trans. Amer. Math. Soc. 98 1961 148--162.

On the homotopy groups of the classical groups. Ann. of Math. (2) 74 1961 407--413.

A generalization of $H$-spheres. Bull. Amer. Math. Soc. 66 1960 503--505.

Derivations of Jordan algebras. Pacific J. Math. 9 1959 495--512.

Commutators in division rings. Proc. Amer. Math. Soc. 9 1958 628--630.

Centralizers in Jordan algebras. Pacific J. Math. 8 1958 757--790.

Year | Degree | Institution |
---|---|---|

1956 | PhD | Yale University |

1954 | MA | Yale University |

1952 | BS | California Institute of Technology |

NSF Postdoctoral Fellow, Yale, 1956-57

ONR Fellow, Princeton/IAS, 1960-61

ARDC Fellow, Princeton/IAS, 1964-65

ONR Fellow, Princeton/IAS, 1960-61

ARDC Fellow, Princeton/IAS, 1964-65

American Mathematical Society

Although my research interests are in Topology and Algebraic Geometry, I have taught almost every undergraduate and graduate course in the Mathematics department. In addition I have introduced several new courses in the Brown Mathematics Curriculum, all of which are currently being taught:these are Honors Linear Algebra (Math 54), Calculus for Engineering and Physical Sciences (Math 19,20), Foundations of Analysis (Math 101), Undergraduate Topology (math 141), on the graduate level co-sponsored "Manifolds". Curently I am teaching a new course, "Geometry and Physics" (Math 272).