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Renner is known for his work on algebraic monoids,
a branch of algebra (MSC(2010):20M32) that he developed starting around 1979.
His creative academic background has forged a unique perspective combining geometry
and groups with convexity and semigroups. He and M.S. Putcha
(North Carolina State) have authored nearly two hundred papers
on algebraic monoids and related topics.
The theory of algebraic monoids is a natural synthesis of algebraic
group theory (Chevalley, Borel, Tits) and torus embeddings (Mumford, Kempf,
et al). The resulting theory is a significant part of the theory
of spherical embeddings (Brion, Luna, Vust). Horospherical varieties
(Popov, Vinberg) were a significant catalyst in the development.
Related work on symmetric varieties was motivated by classical enumerative
problems (De Concini, Procesi).
Renner's contribution contains results on the following issues.
Renner has authored a monograph entitled "Linear
Algebraic Monoids". The intention of this monograph is to convince
the reader that reductive monoids are among the darlings of mathematics. He does
this by systematically assembling many of the major known results, with many
proofs, examples, explanations, exercises and open problems.
Betti numbers of rationally smooth group embeddings (J. Alg. 319(2008), 360-376, J. of Alg. 332(2011), 159-186)
Descent systems for Bruhat posets (J. of Alg. Combinatorics, 29(2009), 413-435, Rocky Mountain Journal of Mathematics 41(2011), 1329-1359)
Blocks and representations of algebraic monoids (Fields Inst. Commun., v. 40, 2004, 125-143)
Cellular decompositions (analogous to Schubert cells) of
compactifications of G. (Can. Math. Bull. 46(2003), 140-148, J. Alg. 319(2008), 360-376,
J. Alg. 332(2011), 159-186)
Factorial hull (generalized Cox ring) for reductive monoids. (J. of
Pure and Appl. Alg. 138(1999), 279-296)
Finite analogues of reductive monoids. (in Semigroups, Formal Languages
and Groups, Kluwer Academic Publishers, 1995, pp. 381-390)
Combinatorial properties of orbits. (J. Alg. 116(1988), 385-399)
The monoid analogue of the Bruhat decomposition. (J. Alg. 101(1986),
The classification of normal reductive monoids in terms of discrete
data. (Trans. Amer. Math. Soc. 292(1985), 193-223)
Applications to topology and other areas of algebra. (J.P.A.A. 69(1990), 295-299;
J. Alg., 264(2003), 479-495)
LINEAR ALGEBRAIC MONOIDS
Encyclopedia of Mathematical Sciences 134
Invariant Theory V
Contents: *Background *Algebraic Monoids *Regularity Conditions
*Classification *Universal Constructions *Orbit Structure *Analogue of the Bruhat Decomposition
*Representations and Blocks *Monoids of Lie Type *Cellular Decompositions
*Conjugacy Classes *Centralizer of a Semisimple Element
*Combinatorics Related to Algebraic Monoids *Survey of Related Developments.
Key notions: Reductive group, regular monoid, diagonal idempotent,
blocks and representations, highest weight category, monoid Bruhat decomposition,
centralizer, divisor class group, adjoint quotient, flat deformation, generalized
Schubert cell, reductive monoid as spherical variety.
PhD THESES SUPERVISED
1. Marjorie Eileen Hull, ``The Centralizer of a Semisimple Element on a Reductive Algebraic Monoid", 1994. http://ir.lib.uwo.ca/digitizedtheses/2347/
2. Wenxue Huang, 1990-1995, ``Algebraic Monoids with Approaches to Linear Associative Algebras", 1995. http://ir.lib.uwo.ca/digitizedtheses/2494/
3. Zhuo Li, ``Orbit Structure of Finite and Reductive Monoids", 1997.
4. Zhenheng Li, ``The Renner Monoids and Cell Decompositions of the Classical Algebraic Monoids", 2001.
5. Letitia Golubitzky, ``Descent Systems, Eulerian Polynomials and Toric Varieties", 2011. http://ir.lib.uwo.ca/etd/134/
6. Richard Gonzales, ``GKM-theory of Rationally Smooth Group Embeddings", 2011. http://ir.lib.uwo.ca/etd/216/
7. Allen O'Hara, "A Study of Green's Relations on Algebraic Semigroups", 2015 (expected)
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