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Dive into the research topics where Richard E. Borcherds is active.

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Featured researches published by Richard E. Borcherds.


Inventiones Mathematicae | 1998

Automorphic forms with singularities on Grassmannians

Richard E. Borcherds

We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moores extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.


Duke Mathematical Journal | 1999

The Gross-Kohnen-Zagier theorem in higher dimensions

Richard E. Borcherds

The Gross-Kohnen-Zagier theorem describes Heegner points on a modular curve in terms of coefficients of modular forms. We give another proof of this theorem which generalizes to higher dimensions.


Advances in Mathematics | 1990

The monster Lie algebra

Richard E. Borcherds

Abstract We calculate the multiplicities of all the roots of the “monster Lie algebra.” This gives an example of a Lie algebra all of whose simple roots and root multiplicities are known and which is not finite dimensional or an affine Kac-Moody algebra. There seem to be several similar infinite dimensional Lie algebras, which to a sympathetic eye appear to correspond to some of the sporadic simple groups.


Topology | 1996

The moduli space of Enriques surfaces and the fake monster Lie superalgebra

Richard E. Borcherds

Abstract We show that the moduli space of complex Enriques surfaces is an affine variety with a copy of the affine line removed. We do this by using the denominator function of a generalized Kac-Moody superalgebra (associated with superstrings on a 10-dimensional torus) to construct a non-vanishing section of an ample line bundle on the moduli space.


Duke Mathematical Journal | 2000

Reflection groups of Lorentzian lattices

Richard E. Borcherds

The aim of this paper is to provide evidence for the following new principle: interesting reflection groups of Lorentzian lattices are controlled by certain modular forms with poles at cusps. We use this principle to explain many of the known examples of such reflection groups, and to find several new examples of reflection groups of Lorentzian lattices, including one whose fundamental domain has 960 faces.


Journal of Algebra | 1991

Central extensions of generalized Kac-Moody algebras

Richard E. Borcherds

The main result of Borcherds [1] states that graded Lie algebras with an “almost positive definite” contravariant bilinear form are essentially the same as central extensions of generalized Kac-Moody algebras. In this paper we calculate these central extensions. Ordinary Kac-Moody algebras have nontrivial centers when the Cartan matrix is singular; generalized Kac-Moody algebras turn out to have some “extra” center in their universal central extensions whenever they have simple roots of multiplicity greater than 1, and in particular the dimension of the Cartan subalgebra can be larger than the number of rows of the Cartan matrix.


Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | 1985

The Leech lattice

Richard E. Borcherds

New proofs of several known results about the Leech lattice are given. In particular I prove its existence and uniqueness and prove that its covering radius is the square root of 2. I also give a uniform proof that the 23 ‘holy constructions’ of the Leech lattice all work.


Archive | 1994

Sporadic Groups and String Theory

Richard E. Borcherds

I will start by giving some well known product identities. The first is


Journal of Algebra | 1990

Lattices like the Leech lattice

Richard E. Borcherds


Algebra & Number Theory | 2011

Renormalization and quantum field theory

Richard E. Borcherds

{\sum\limits_{n \in {\mathbf{Z}}} {{{\left( { - 1} \right)}^n}q} ^{3{{\left( {n + 1/6} \right)}^2}/2}} = \,{q^{1/24}}\prod\limits_{n > 0} {\left( {1\, - \,{q^n}} \right)}

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L. Queen

University of Cambridge

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N. I. Shepherd-Barron

Massachusetts Institute of Technology

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Tony Pantev

University of California

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