Vita for Robert L. Griess, Jr.Version 8 June, 2007; printed June 8, 2007.

Contents

1 Personal and Professional Data 21.1 Current Address . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Employment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Visiting Appointments for One Academic Term or Longer . . . 21.5 Fellowships, Prizes and Honors . . . . . . . . . . . . . . . . . 31.6 Selected List of Invited Lectures . . . . . . . . . . . . . . . . . 31.7 Recent visits, conference participation, seminar talks . . . . . 51.8 Doctoral Students at University of Michigan . . . . . . . . . . 91.9 Service to university and profession. . . . . . . . . . . . . . . 9

1.9.1 Major Committees in Mathematics Department, Uni-versity of Michigan. . . . . . . . . . . . . . . . . . . . . 9

1.9.2 Major Service to the Mathematics Department. . . . 101.9.3 Service to University of Michigan. . . . . . . . . . . . 101.9.4 Service to the American Mathematical Society. . . . . 101.9.5 Service to the mathematical community. . . . . . . . . 10

1.10 Outreach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.11 The Arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.11.1 Bodytalk . . . . . . . . . . . . . . . . . . . . . . . . . . 121.11.2 Benefit Performance . . . . . . . . . . . . . . . . . . . 131.11.3 Dedication . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Bibliography, Robert L. Griess, Jr. 132.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Published items . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Accepted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 In Preparation. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 References and publicity about my work 203.0.1 The announcement . . . . . . . . . . . . . . . . . . . . 223.0.2 Text of NSF press release. . . . . . . . . . . . . . . . . 22

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1 Personal and Professional Data

1.1 Current Address

Department of MathematicsUniversity of MichiganAnn Arbor, MI 48109-1109 USAOffice phone: 734-764-0361Eaddress: rlg@umich.edu

1.2 Education

B.S. University of Chicago, 1967M.S. University of Chicago, 1968Ph.D. University of Chicago, 1971

1.3 Employment

University of Michigan, T. H. Hildebrandt Research Instructor, 1971-73University of Michigan, Assistant Professor, 1973-76University of Michigan, Associate Professor, 1976-81University of Michigan, Professor, 1981-

1.4 Visiting Appointments for One Academic Term orLonger

Rutgers University, Visiting Assistant Professor, 1974-75Institute for Advanced Study, Visiting Member, 1979-80Institute for Advanced Study, Visiting Member, Winter, 1981Yale University, Visiting Professor, 1983-84Ecole Normale Superieure, Paris, 1986-87Institute for Advanced Study, Visiting Member, Fall 1994University of California, Santa Cruz, Visiting Professor, Winter, 2000.National Cheng Kung University, Distinguished Chair Professor in the

Department of Mathematics, fall term 2007; Tainan, Taiwan.Center for Mathematical Sciences, Zhejiang University, visiting professor,

winter term, 2008; Hongzhou, China.

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1.5 Fellowships, Prizes and Honors

Guggenheim Fellowship, 1981-82Hour Invited Address, American Mathematical Society summer meeting,

Pittsburgh, 1982.International Congress of Mathematicians, Invited Speaker, Warsaw 1983Maıtre de Recherche, CNRS, France for academic year 1986-87;

spent at Ecole Normale Superieure.Dozor Visiting Fellow, 1999, Ben-Gurion University of the Negev.Harold R. Johnson Diversity Service Award, University of Michigan, 2003.American Academy of Arts and Sciences membership, April 2007.

1.6 Selected List of Invited Lectures

This list does not include ordinary seminar or colloquium talks

Group Theory Conference, University of Minnesota, Duluth,August, 1977

Group Theory Conference, Santa Cruz, June-July 1979.Hour Invited Address, American Mathematical Society summer meeting,

Pittsburgh, 1982.Algebra Section, International Congress of Mathematicians, Warsaw, 1983Conference in honor of Guido Zappa, Florence, October, 1986Two Invited Lectures, College de France, Winter, 1987Conference in honor of Daniel Gorenstein, Rutgers University,

March, 1988Invited hour lecture, Lie theory section,

Canadian Mathematical Society winter meeting, December, 1991.Moonshine conferences, Ohio State University, May, 1993 and May, 1996Lecture series on finite simple groups, Elliptic Cohomology Conference,

July, 1995, Bellaterra, SpainLecture in Elliptic Cohomology Conference, University of Glasgow,

January, 1997.University of California Lie Theory Meeting, Santa Cruz, April, 1998Moonshine Conference, Montreal, May, 1999Amitsur Algebra Conference, Jerusalem, June, 1999Lie Theory meeting, Edmonton, June, 2000Vertex Operator Algebra Meeting, Santa Cruz, July, 2000

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Conference to honor Marshall Osborn, September, 2000Infinite dimensional Lie theory meeting, Toronto, October, 2000Invited one hour talk in Lie theory section, Canadian Mathematical

Society Summer Meeting, Saskatoon, 2-4 June, 2001.Invited Lecture, one-week Lie theory meeting, Beijing, June 2001.Invited Lecture series on finite groups and algebras, Universite de Lyon,

October, 2001.Invited Lecture, algebra meeting, RIMS, Kyoto, December, 2001.Invited Lecture, AMS meeting on group representations

to honor Jon Alperin, June, 2002.Plenary lecture at Brazilian XVII Algebra MeetingNovember, 2002: invited to be Distinguished Speaker in the 2002

Mathematics and Statistics Red Raider Mini-Symposium series“Contemporary Algebra and Algebraic Geometry”. Thelist of speakers was William Fulton, Robert Griess,James Lepowsky, Gregory Margulis and Efim Zelmanov.

Invited lecture: Conference on infinite dimensional Lie algebras atFields Institute, Toronto in July, 2003.

Special session on infinite dimensional Lie theory atBinghamton, N.Y. AMS meeting 11-12 October, 2003.

Invited lecture for conference to honor Walter Feit, New Haven,30 October-2 November.

Invited address at Moonshine Conference, Heriot-Watt University:Moonshine - the first quarter century and beyond A Workshop on theMoonshine Conjectures and Vertex Algebras Edinburgh,4-14 July 2004.

Invited address at ICRT III (International Conference on RepresentationTheory, III), Chengdu, China 30 July - 4 August, 2004.

Invitation to American Institute of Mathematics (Palo Alto, CA)conference on “Sphere packings, lattices, groups andinfinite dimensional algebra”; 16-20 August, 2004.

Invited hour lecture at 65th birthday conference for Koichiro Haradaat Hamamatsu City, Japan, 20-22 March, 2006: “Barnes-Wall lattices andrelatives”.

Four lectures on lattices in instructional conference “Sphere Packings:Exceptional Geometric Structures and Connections to other Fields”, 20-26November, 2005; Oberwolfach Mathematische Forschungsinstitut, http://www.mfo.de/;organizers:

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Henry Cohn cohn@microsoft.com,Robert Griess, rlg@umich.edu.Achill.Schuermann@Mathematik.Uni-Magdeburg.DE.

1.7 Recent visits, conference participation, seminar talks

Activities since January, 2000 (some overlap with above)

Official Appointments and Extended Visits (2000-2002).

Centre de Recerca Matematica Institut d’Estudis Catalans (CRM),Bellaterra, Spain; 26 April-20 May 2000.

Chinese Academy of Science (Beijing), last two weeks of June 2001;Tata Institute (Mumbai) three weeks August, 2001;Universite de Lyon 1, Mathematics Department (Lyon, France),

four weeks September-October, 2001;IHES, (Bures-sur-Yvette, France) two weeks, October-November, 2001Tsukuba University, (Tsukuba, Japan) Mathematics Department,

four weeks November-December 2001

Lectures, Visits, Publicity (2000-2002).Year 2000

Two lectures at University of Marakkech, Morocco, 16 April, 2000,Journee de Mathematiques.

26 April-20 May 2000, CRM, Bellaterra, SpainTwo Lectures, Universidad Publia de Navarra;

titles “The classification of finite simple groups” and “Pieces of Eight”.While in Pamplona, I was the subject of a press conference, thenfeatured in a half page Pamplona newspaper article a few days later:Noticias, “El padre del ‘Monstruo’ ”,Viernes, 19 de mayo, de 2000,ultima pagina (p. 80).

Lecture, Lie theory meeting, Edmonton, June, 2000.“Automorphism groups of VOAs and frame stabilizers.”

Lecture, Vertex Operator Algebra meeting, University of CaliforniaSanta Cruz

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Lectures, GTLT seminar University of Michigan:(1) The automorphism group of a finitely generated VOA is an algebraic

group (with Chongying Dong); fall, 2000.(2) general talk on ordinary and modular representations of finite groups;

winter 2000.Two lectures: University of Virginia, 30, 31 August, 2000.(1) Colloquium: “Vertex Operator Algebras and Automorphism Groups.”(2) Algebra Seminar: “Finite subgroups of Lie groups. ”University of Wisconsin, Madison. early September, 2000(1) Department Colloquium: “Pieces of Eight.”(2) Invited hour talk for the retirement conference of J. Marshall Osborn:

“Finite simple groups and nonassociative objects.”Infinite Dimensional Lie Theory, Fields Institute Toronto,

September 2000. One week.Colloquium, Columbia University, 11 October, 2000.

”Groups, Nonassociative algebras and vertex operator algebras.”Invited hour talk at conference, Vertex Operator Algebras,

Fields Institute Toronto, October, 2000. One week.Year 2001

Lecture, University of Michigan Undergraduate Math Club: ThursdayFebruary 15, 2001: “Big dimensions, big groups and big numbers:sometimes a big deal. ”

Basic Notions Seminar University of Michigan, 30 March, 2001. The workof John Griggs Thompson, National Medal of Science, November 2000.

Canadian Mathematical Society Meeting, June, Saskatoon;Lie theory meeting, Chinese Academy of Sciences (Beijing), June;2 at Tata Institute, August;Bangalore Institute of Statistics, August;4 at Universite de Lyon, September-October;2 at Institut Henri Poincare in Paris, October-November;4 at University of Tsukuba, November-December;RIMS Kyoto, December;University of Tokyo, December;Osaka University, December.

Year 2002Algebra Seminar, Notre Dame University, 26 March;GTLT at University of Michigan, 1 April;June, 2002, invited lecture at an AMS conference on

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group representations to honor Jon Alperin.August, 2002: Plenary lecture at XVII Brazilian Algebra Conference and

lecture tour of Brazilian universities:Universidade Federal do Rio de Janeiro,University of Campinas, University of Sao Paolo andIMPA (mathematics institute in Rio).

November, 2002: invited to be Distinguished Speaker in the 2002Mathematics and Statistics Red Raider Mini-Symposium series“Contemporary Algebra and Algebraic Geometry”.The list of speakers was William Fulton, Robert Griess,James Lepowsky, Gregory Margulis and Efim Zelmanov.

Columbia University, Colloquium, 4 December, 2002Year 2003

University of Virginia, Colloquium and algebra seminar,30 and 31 January, 2003

Penn State University, Number theory seminar, 21 March, 2003.University of California Santa Cruz, algebra seminar, 6 June, 2003.Special session on Lie algebras and mathematical physics at AMS meeting

in Bloomington, Indiana, 3-5 April, 2003.Algebra seminar, “New uniqueness proof for unimodular lattices of

rank at most 8”, University of California, Santa Cruz, 6 June, 2003.Conference on infinite dimensional Lie algebras at Fields Institute,

Toronto in July, 2003.Special session on infinite dimensional Lie theory at Binghamton, N.Y.

AMS meeting 11-12 October, 2003.GTLT seminar University of Michigan, 27 October: “ Uniqueness and

existence theorems for lattices in dimensions a power of 2”.Invited lecture for conference to honor Walter Feit, New Haven,

30 October-2 November.Year 2004

Universitat Freiburg, Algebra Seminar 27 February, 2004:Finite Subgroups of Lie groups.

GTLT seminar University of Michigan, 15 March “The Barnes-Wall andYpsilanti lattices”.

Michigan State University, Department Colloquium, 25 March, 2004:Automorphisms of Vertex Operator Algebras.

Dicken Elementary School, Ann Arbor. Half hour presentation aboutsequences and integers at Math-Science Night.

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Algebra seminar, “Finite subgroups of Lie groups”, China University ofScience and Technology, Hefei, China; 26 May, 2004.

Algebra seminar, Fudan University, Shanghai, China. “GNAVOA”; 1June, 2004.

Algebra seminar, Jiantong University, Shanghai, China. “GNAVOA”; 2June, 2004.

Two invited addresses at Moonshine Conference, Heriot-Watt University,4-13 July, 2004 (“ Moonshine - the first quarter century and beyond: AWorkshop on the Moonshine Conjectures and Vertex Algebras”) : (1) A his-tory of the monster and its construction; (2) VOAs and their automorphismgroups.

Invited address at ICRT III (International Conference on RepresentationTheory, III), Chengdu, China 30 July - 4 August, 2004. Topic: VOAs

and their automorphism groups.Invitation to American Institute of Mathematics (Palo Alto, CA)

conference on “Sphere packings, lattices, groups and infinitedimensional algebra”; 16-20 August, 2004. Topic: Uniqueness and

existence for Barnes-Wall and Ypsilanti lattices.GTLT seminar, September 2004, A survey of uniqueness proofs for the

E8 lattice.Visit Microsoft, 17-23 October 2004, to collaborate with Henry Cohn.

Year 2005Algebra seminars (2 hours each), invited by Prof. Ali Nesin, Bilgi Uni-

versity, Istanbul, Turkey. These were expository talks. 25 February 2005:The finite simple groups. 4 March, 2005: Finite subgroups of Lie groups.

Algebra seminar, Wayne State University, 25 April, 2005. “Uniquenesstheorems for lattices”.

Algebra seminar, Jiaotong University, 25 May, 2005. “ On the uniquenessof the moonshine vertex operator algebra”.

Algebra seminar, East China Normal University, 7 June, 2005. “Auto-morphisms of Vertex Operator Algebras”.

Algebra seminar, Peiking University, 13 June, 2005. “Finite Subgroupsof Lie Groups.”

Special session on Lie theory, joint AMS-Taiwan Math Society meeting,14-18 December, 2005. “Results for uniqueness of the moonshine module”

Four lectures on lattices in instructional conference “Sphere Packings:Exceptional Geometric Structures and Connections to other Fields”, 20-26

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November, 2005; Oberwolfach Mathematische Forschungsinstitut, http://www.mfo.de/;organizers:

Henry Cohn cohn@microsoft.com,Robert Griess, rlg@umich.edu.Achill.Schuermann@Mathematik.Uni-Magdeburg.DE.Special session on Lie theory, joint AMS-Taiwan Math Society meeting,

14-18 December, 2005. Lecture on a uniqueness result for the moonshinemodule, about joint work with Chongying Dong and Ching Hung Lam.

Year 2006Group Theory/Lie Theory Seminar at University of Michigan, Thursday,

2 February, 2006 “Spherical codes and Barnes-Wall lattices”Invited hour lecture at 65th birthday conference for Koichiro Harada

at Hamamatsu City, Japan, 20-22 March, 2006: “Barnes-Wall lattices andrelatives”.

Colloquium, National Cheng Kung University, Tainan, Taiwan, May,2006.

Algebra seminar, Fudan University, Shanghai, China, 23 May, 2006.

1.8 Doctoral Students at University of Michigan

Arnold Feldman, 1975: “Fitting Height of Solvable Groups AdmittingFixed Point Free Automorphism Groups”.

Manfred Steeg, 1976: “Finite p-Groups with Trivial Schur Multiplier”.

Thomas Richardson, 1991: “Local Subgroups of the Monster and OddCode Loops”.

Darrin Frey, 1995: “Conjugacy of Alternating Groups of Degree 5 andSL(2, 5) Subgroups of the Complex Lie Group of Type E8. ”

Michael Kantor, 1997: “SL and PSL Subgroups of E8 and their Actionson a Maximal Torus”.

Current doctoral student: Ivan Middleton.

1.9 Service to university and profession.

1.9.1 Major Committees in Mathematics Department, Univer-sity of Michigan.

Five year term on Long Range Planning Committee, many personnel com-mittees, several terms each on Doctoral Committee, Admissions Committee,

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Qualifying Review Examination Committee, Minority Outreach Committee,Area leaders (algebra).

1.9.2 Major Service to the Mathematics Department.

In 1978, Professor James Kister and I rewrote the graduate programdescription.

In 1984-86, I took major responsibility for setting up the foundations ofthe mathematics department computer system. My recommendations wereaccepted by my department and our setup was a model for other mathematicsdepartments in the mid to late eighties.

In the nineties decade, I took major initiative in the Mathematics Depart-ment minority outreach, mainly in boosting our department’s committmentto hosting over four hundred middle school kids annually in the University ofMichigan’s King-Chavez-Parks visitation program. This is the highest levelacademic unit participation at the University of Michigan. We hope thatsuch sustained effort will promote increased diversity in mathematics classesand careers. My involvement continues to the present.

1.9.3 Service to University of Michigan.

Several grant application review committees for the Rackham GraduateSchool, University Senate Rules Committee.

1.9.4 Service to the American Mathematical Society.

Consultant for Steele Prize award to Daniel Gorenstein, 1989.Central Section Program Committee, two calendar years 1990-91

(Chair, 1991).

1.9.5 Service to the mathematical community.

Editor of Journal of Group Theory, 1996-2001.Editor for Electronic Research Announcements of the AMS (ERA-AMS),

2006-.Consultant for the National Medal of Science award given to John G.

Thompson in November, 2000.

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In July, 2001, I served on a grant review panel for the Foundation forScience and Technology of Portugal.

Selection committee for Algebra, International Congress of Mathemati-cians, 2006.

1.10 Outreach

In May, 2003, I received the Harold R. Johnson Diversity Service Award.Below is the report published in the Unversity Record.

Harold R. Johnson Diversity Service Award winners announcedBy the Office of the Provost and Executive Vice President for Academic

AffairsFive U-M faculty members have been named winners of the 2003 Harold

R. Johnson Diversity Service Award, and one has been named a specialhonoree. Established in 1996, the award was named in recognition of HaroldJohnson, dean emeritus of the School of Social Work. Award recipients arerecognized for their commitment to the development of a more culturally andethnically diverse campus community. Awardees receive $5,000 to furthertheir personal research, teaching and scholarship activities.

Robert Griess, Jr.Robert Griess Jr. is a professor in the Department of Mathematics. Pe-

ter Hinman, professor and associate chair of mathematics, notes that for morethan 10 years Griess has led the department’s effort in the King/Chavez/ParksCollege Day visitation program. Griess is single-handedly responsible for thefact that the department every year hosts 30-50 groups of 7th graders. Griessbelieves one positive contact for these children with a professional mathemati-cian has the potential to change their thinking about mathematics, Hinmansays. Griess is committed to seeing more students from underrepresentedminority groups stay in the pipeline leading to undergraduate and advanceddegrees in mathematics-based disciplines. Besides recruiting for the program,he has spent much time developing support materials for it. He has video-taped many of the best presentations over the years and scheduled showingsof the tapes for potential presenters who are unfamiliar with teaching middleschool students.

When U-M was selected as the site for the Fifth Conference for AfricanAmerican Researchers in the Mathematical Sciences (CAARMS), held in1999, Griess immediately saw the value of putting undergraduates from un-derrepresented minority groups into contact with the 100 or more African

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American mathematics professionals and graduate students who would at-tend the conference. With no external funding, Griess organized a “shadowconference” for these undergraduates that ran parallel with CAARMS, bor-rowed speakers from the main conference, and capitalized upon opportunitiessuch as the CAARMS social events to put the students into direct contactwith the people at the main conference who could serve as role models.

Griess also has taken a personal, direct interest in students from under-represented minority backgrounds who enter the University’s mathematicsgraduate program and has a demonstrated commitment to academic excel-lence and diversity.

1.11 The Arts

1.11.1 Bodytalk

On 2 July, 2002, I performed a role in a multimedia concert event at thePower Center in Ann Arbor.

The concert was by Peter Sparling, and was called “Bodytalk: A Vaudevillefor Dancing Man at Middle Age.” It was mainly a dance concert but involvedfilms, slides, poems, monologue, video, and music. In my part, Peter Sparlingand Janet Lilly danced a duet, to a recorded monologue by Charles Baxter,as I stood onstage and drew at a light table. My rapid, evolving drawingswere instantly projected to a large screen in the background. The originaldark lines on light background were reversed on screen to white lines on darkbackground.

The Ann Arbor News review by critic Susan Isaacs Nisbett was quitefavorable to the concert. The excerpt which applies to me is below. (Theentire review is roughly twelve column inches.)

“... Who does not recognize the questions of boundaries and borders, selfand others, posed by Charles Baxter’s poem ’The System’ or recognized howtwo people can be two or one as Sparling and Janet Lilly dance the poemand Bob Griess draws them? ”

Background:Peter Sparling is on the University of Michigan dance faculty. He directs

the Peter Sparling Dance Company in Ann Arbor. He danced with the JoseLimon company and was principal dancer in the Martha Graham Company.http://tenaya.physics.lsa.umich.edu/

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˜dwchin/DanceGallery/Company/sparling bio.ht mlCharles Baxter was at that time on the faculty of the University of Michi-

gan and is a writer. He recently received a National Book Award for his novelThe Feast of Love. http://www.charlesbaxter.com/

1.11.2 Benefit Performance

On 17 May, 2003, I drew on stage at a Dance Gallery performance in AnnArbor. Immediately after, the drawings were offered for sale to the audience,to benefit the Dance Gallery.

1.11.3 Dedication

A piece of music was dedicated to me. It is a work for orchestra and elec-tronics titled “Monsterology.” The composer is

Lawrence FrittsAssociate Professor of Composition and TheorySchool of MusicUniversity of Iowa

Larry Fritts told me in March, 2005, that the University of Iowa ChamberOrchestra will perform its world premiere on 1 May, 2005 and he expects aCD to be out in 2006. He says “The piece sounds absolutely gorgeous and Iwas truly inspired by your work”.

2 Bibliography, Robert L. Griess, Jr.

2.1 Overview

My early work (seventies decade) was mainly on Schur multipliers, groupextensions and cohomology, and the properties of and classification of finitesimple groups. Main accomplishments were the determination of Schur mul-tipliers of many finite simple groups, collaboration on odd standard form [23]and initial investigations on the monster (1973), first reported in [9].

During the eighties, I became involved with the monster’s constructionand consequences [21],[22],[26], attempts to build a general theory of sporadicgroups (e.g., groups and various nonassociative objects) and efforts to launchthe classification of finite subgroups of exceptional Lie groups. The latterprogram achieved a level of resolution by the turn of the millenium [54],[60].

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The nineties decade saw my growing involvement in vertex operator alge-bras and their automorphisms, and in lattices [57]. The 196884-dimensionalalgebra associated to the monster has taken on a life of its own, being knownas the Griess algebra in literature of physics, Lie theory, nonassociative alge-bras and elsewhere. I recently gave an elementary proof of a classic theoremabout lattices [64] and have started a program of existence and uniquenessfor high dimensional unimodular lattices.

2.2 Published items

[1] Schur Multipliers of the Known Finite Simple Groups, Ph.D. Thesis,University of Chicago, 1971.

[2] Schur multipliers of the known finite simple groups, Bull. Amer. Math.Soc., 78, 68-71, 1972.

[3] Schur multipliers of finite simple groups of Lie type, Trans. Amer. Math.Soc., 183, 355-421, 1973.

[4] A characterization of U3(2n) by its Sylow 2-subgroup, Trans. Amer. Math.

Soc., 175, 181-186, 1973.

[5] Automorphisms of extra special groups and nonvanishing degree 2 coho-mology, Pacific J. Math., 48, 403-422, 1973.

[6] Automorphisms of extra special groups and nonvanishing degree 2 coho-mology (research announcement for [5]), in Finite Groups 1972: Proceedingsof the Gainesville Conference on Finite Groups, (T. Gagen, M. P. Hale andE. E. Shult, eds.), North Holland Publishing Co., Amsterdam, 68-73, 1973.

[7] Schur multipliers of some sporadic simple groups, J. Algebra, 32, 445-466,1974.

[8] The automorphism group of the Tits simple group 2F4(2)′ (with R. Lyons),

Proc. Amer. Math. Soc., 52, 75-78, 1975.

[9] The structure of the “ Monster” simple group, Proceedings of the Con-ference on Finite Groups, Park City, Utah, February 1975, Academic Press,Inc.,1976, 113-118.

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[10] On a subgroup of order 215|GL(5, 2)| in E8(C) , the Dempwolff groupand Aut(D8 ◦ D8 ◦ D8) , J. Algebra, 40, 1976, 271-279.

[11] The splitting of extensions of SL(3, 3) by the vector space F33, Pacific

J. Math., 63,1976, 405-409.

[12] Maximal subgroups and field automorphisms of finite Chevalley groups(research announcement for Maximal subgroups and field automorphisms offinite Chevalley groups), Proceedings of the Conference on Finite Groups,Park City, Utah, February 1975, Academic Press, Inc.,1976, 329-330.

[13] Maximal subgroups and field automorphisms of finite Chevalley groups(with N. Burgoyne and R. Lyons), Pacific J. Math., 71, 1977, 365-403.

[14] The Frattini module (with P. Schmid), Archiv der Mathematik, 30, 1978,256-266.

[15] Bender groups as standard subgroups (with D. R. Mason and G. M.Seitz), Trans. Amer. Math. Soc., 1978, 238, 179-211.

[16] A remark about groups of characteristic 2-type and p-type, Pacific J.Math., 74, 1978, 349-355.

[17] Finite groups whose involutions lie in the center, Quarterly J. Math., 29,1978, 241-247.

[18] Finite groups with unbalancing 2-components of {L3(4), He}-type (withR. Solomon), J. Algebra, 60, 1979, 95-125.

[19] Odd standard form problems, Proceedings of the Durham Conference(survey article), Academic Press, 1980, 199-206.

[20] Schur multipliers of the known finite simple groups, II, The Santa CruzConference on Finite Groups, Amer. Math. Soc., Providence, 1980, 279-282.

[21] A construction of F1 as automorphisms of a 196,883-dimensional algebra,Proc. Natl. Acad. Sci., USA, 78, 1981, 689-691.

[22] The friendly giant, Invent. Math., 69, 1982, 1-102.

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[23] Finite groups with standard components of Lie type over fields of char-acteristic two (with R. H. Gilman), J. Algebra, 80, 1983, 383-516.

[24] Quotients of infinite reflection groups, Math. Ann., 263, 1983, 267-278.

[25] Schur multipliers of the known finite simple groups, III, in Proceedingsof the Rutgers Groups Theory Year, 1983-1984, 1984, Cambridge UniversityPress, 69-80.

[26] The sporadic simple groups and construction of the monster, in Pro-ceedings of the International Conference of Math., Warsaw, Polish ScientificPublishers, Warszawa; North Holland, Amsterdam, 1984, 369-384.

[27] A brief introduction to the finite simple groups, in Vertex Operators inMathematics and Physics, Mathematical Sciences Research Publications 3,(J. Lepowsky, S. Mandelstam, I. Singer, eds.), Springer-Verlag, New York,1985,217-230.

[28] A remark about representations of ·1, Comm. Algebra, 13, 1985, 835-844.

[29] The monster and its nonassociative algebra, in Proceedings of the Mon-treal Conference on Finite Groups, Contemporary Mathematics, 45, 121-157,1985, American Mathematical Society, Providence, RI.

[30] Code loops, J. Algebra, 100, 1986, 224-234.

[31] A structure constant problem of Goddard and Olive, dedicated to Profes-sor Bertram Huppert on the occasion of his sixtieth birthday, in Proceedingsof the Montreal Conference on Infinite Dimensional Lie Algebras and theirApplications, World Scientific (ed. S. Kass) , 1986

[32] Sporadic groups, code loops and nonvanishing cohomology, J. Pure Appl.Algebra, 44, 1987, 191-214.

[33] The Schur multiplier of McLaughlin’s simple group, Arch. Math. Vol.48, 1987, 31.

[34] On finite simple subgroups of the complex Lie group of type E8 (withA. M. Cohen), Proc. Symp. Pure Math., 47, 1988, 367-405.

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[35] Code loops and a large finite group containing triality for D4, Proc. AttiDel Convegno Internazionale di Teoria Dei Gruppi E Geometria Combinato-ria, Firenze, 23-25 October 1986, Serie II, 19, 1988, pg.79-98.

[36] A uniqueness proof for the monster (with U. Meierfrankenfeld and Y.Segev), Annals, 130,1989, 567-602.

[37] A Moufang loop, the exceptional Jordan algebra and a cubic form in 27variables, J. Algebra, Vol. 131,1990, No. 1, 281-293.

[38] Elementary Abelian p-subgroups of algebraic groups, Geom. Dedicata39, 1991, 253-305.

[39] Nonlocal Lie primitive subgroups of Lie groups, Canadian Journal ofMath., 1992, 88-103 (with Arjeh Cohen).

[40] Automorphisms of Modular Lie Algebras, Nova Journal of Algebra andGeometry, Vol. 1, No. 4, 1992, 339-345.

[41] Non-local Lie primitive subgroups of Lie groups, (with A. Cohen), Cana-dian Journal of Mathematics, 45, No. 1, 1993, 88-103.

[42] The group L(2, 61) embeds in the Lie group of type E8, (with A.Cohenand B. Lisser), Comm. Algebra, 21(6), 1993, 1889-1907.

[43] Embeddings of U3(8), Sz(8) and the Rudvalis group in algebraic groupsof type E7 , (with A. Ryba), Inventiones Mathematicae 116, 1994, 1215-241(special issue dedicated to Armand Borel).

[44] Minimal dimensions for modular representations of the Monster (withSteve Smith), Comm. Algebra, 22(15), 1994, 6279-6294.

[45] Basic Conjugacy Theorems for G2, Inventiones Math. 121, 1995, 257-277.

[46] Codes, Loops and p-Locals, Groups, Difference Sets and the Monster,ed. Joe Ferrar and Koichiro Harada, 1995, 369-375.

[47] An infinite family of elementwise conjugate nonconjugate homomor-phisms, (with P. Fong), International Research Notes in Mathematics No.5, 1995, 249-252. (J. Arthur, ed.)

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[48] Twelve Sporadic Groups, Springer Monographs in Mathematics, 1998,Springer-Verlag.

[49] Framed vertex operator algebras, codes and the moonshine module, (withChongying Dong and Gerald Hohn), Comm. Math. Physics, 193, 1998, 407-448.

[50] The Conjugacy classes of elements in the Borovik group (with D. Frey),Journal of Algebra 203, 1998, 226-243.

[51] Embeddings of PSL(2, 32) and PGL(2, 31) in E8(C) (with A. Ryba),Duke Math. Journal, 94 (1), 1998, 181-211.

[52] Rank one lattice type vertex operator algebras and their automorphismgroups (with Chongying Dong), Journal of Algebra 208, 1998, 262-275. q-alg/9710017

[53] A vertex operator algebra related to E8 with automorphism group O+(10, 2),article in The Monster and Lie Algebras, ed. J. Ferrar and K. Harada, de-Gruyter, Berlin, 1998.

[54] Finite simple groups which projectively embed in an exceptional Liegroup are classified! (with A. Ryba), Bulletin Amer. Math. Soc. 36 (1),1999, 75-93.

[55] Rank one lattice type vertex operator algebras and their automorphismgroups, II: E-series (with Chongying Dong and A. Ryba), Journal of Algebra217, 1999, 701-710.

[56] Automorphisms of vertex operator algebras, a survey, Proceedings of theRaleigh Conference on affine algebras, quantum affine algebras and relatedtopics, 21-24 May, 1998. Contemporary Mathematics, volume 248, AmericanMathematical Society, 1999.

[57] Pieces of Eight, Advances in Mathematics, 148, 75-104 (1999).

[58] Embeddings of Sz(8) into exceptional Lie groups (with A. Ryba), J.reine angew. Math., 523 (2000), 55-68.

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[59] Embeddings of PSL(2, 41) and PSL(2, 49) in E8(C) (with A. Ryba)Computational algebra and number theory (Milwaukee, WI, 1996). J. Sym-bolic Comput. 31 (2001), no. 1-2, 211–227.

[60] Quasisimple finite subgroups of exceptional algebraic groups, with AlexRyba, Journal of Group Theory, 2002, 1-39.

[61] Embeddings of SL(2, 27) in algebraic groups of type E8, with Alex Ryba,Michigan Math Journal, 50 (2002), 89-99.

[62] Automorphism groups of finitely generated vertex operator algebras,with Chongying Dong, Michigan Math Journal, 50 (2002). 227-239. math.QA/0106051

[63] Frame stabilizers for the lattice vertex operator algebra of type E8, withGerald Hohn, J. reine angew. Math., 561 (2003), 1-37.

[64] Positive definite lattices of rank at most 8, Journal of Number Theory,103 (2003), 77-84.

[65] GNAVOA, I. (studies in groups, nonassociative algebras and vertex oper-ator algbras), (about 25 pages) article in Vertex Operator Algebras in Math-ematics and Physics, with S. Berman, Y. Billig and J. Lepowsky, FieldsInstitute Communications, Vol. 39, Amer. Math. Soc., Providence, 2003.

[66] Pieces of 2d: existence and uniqueness for Barnes-Wall and Ypsilantilattices. Advances in Mathematics, 196 (2005) 147-192. math.GR/0403480

[67] The rank two lattice type vertex operator algebras V +L and their auto-

morphism groups, with Chongying Dong, Michigan Math. J. 53 (2005), no.3, 691–715. math.QA/0409409

[68] Involutions on the the Barnes-Wall lattices and their fixed point sub-lattices, I. Pure and Applied Mathematics Quarterly, vol.1, no. 4, (SpecialIssue: In Memory of Armand Borel, Part 3 of 3) 989-1022, 2005.

[69] Review of the Mark Ronan book “Symmetry and the Monster, one of theGreatest Quests of Mathematics”, Notices AMS, February, 2007, p.234-239.

[70] Uniqueness results for the moonshine vertex operator algebra, about 24pages, with Chongying Dong and Ching Hung Lam, appeared in AmericanJournal of Mathematics.

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[CBWY] Corrections and additions to “ Pieces of 2d: existence anduniqueness for Barnes-Wall and Ypsilanti lattices. ” [Adv. Math. 196(2005) 147-192], Advances in Mathematics 211 (2007) 819-824.

2.3 In Preparation.

Several projects on groups, lattices and vertex operator algebras are in prepa-ration. Some are with collaborators.

3 References and publicity about my work

MathSciNet (available at www.ams.org) contains reviews and notes cita-tions of my work in regular research articles. In this category, some items ofrelatively general interest are these:

G. Glauberman, Mathematical Reviews 84m:20024J.-P. Serre, Sous-groupes finis des groupes de Lie. (French) [Finite sub-

groups of Lie groups] Seminaire Bourbaki, Vol. 1998/99. Asterisque No. 266(2000), Exp. No. 864, 5, 415–430.

J. Tits, (F-CDF) Le Monstre (d’apres R. Griess, B. Fischer et al.). (French)[The Monster (following R. Griess, B. Fischer et al.)] Seminare Bourbaki,Vol. 1983/84. Asterisque No. 121-122, (1985), 105–122.

D. Wales, Mathematical Reviews 99g:20085

Below is a partial list of references to my work, made outside regularscientific research journals. More can be obtained via internet search bycombining phrases like robert griess, monster, sporadic group, simple group,moonshine, physics, exceptional Lie group, griess algebra etc. (results mayvary with order of phrases and search engine).

National Science Foundation News, by Ralph Kazarian; March 14, 1980;NSF PR80-17; Scientist takes big step in solving major problem in mathe-matics; 2 page press release, reproduced below)

Scientific American, Science and the Citizen, May, 1980, p. 82.Scientific American, Martin Gardner Mathematical Games column in

June, 1980.

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Science News, 27 September, 1980, A monstrous piece of research, byLynn Arthur Steen, p.204-206

Encyclopedia Britannica Book of the Year, 1981 (events of 1980), p. 529(the article is flawed).

Mosaic, A Friendly Giant, by Henry T. Simmons, September-October1981, 23-36.

Jonathan Friendly, Ideas and Trends, “School of theorists works itself outof a job”, 22 June, 1980, New York Times.

Clive Cookson, “Mathematics: ‘The Monster’ unveiled”, London Times,31 May, 1980.

London Times Higher Education Supplement, “Michigan doctor claimsmonster maths breakthrough”, about May, 1980

Frances Buekenhout, Les Groupes Sporadiques, La Recherche, 131, Mars1982. 348-355.

Freeman Dyson, “Unfashionable Pursuits”, Mathematical Intelligencer,vol. 5, no. 3, 1983, 47-54.

Freeman Dyson, “Unfashionable Pursuits”, article in book “From Eros toGaia.”

The Timetables of Science, by Alexander Hellemans and Bryan Bunch,Simon and Schuster, New York, 1988; p. 297 and p.585.

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3.0.1 The announcement

Below is the text of the typed announcement which I photocopied and cir-culated by mail when I got an existence proof of the monster.

“ A CONSTRUCTION OF F1

I am extremely pleased to announce my recent construction of a finitegroup, G. There is no doubt that G is isomorphic to F1, a simple group oforder 246320597611213317·19·23·29·31·41·47·59·71 ∼ 8.080×1053 discovered in1973 by Bernd Fischer and me.

The constructionis direct, explicit and carried out entirely by hand. I likeit.

Best wishes for a Happy New Year!

Robert L. Griess, Jr.The Institute for Advanced StudyJanuary 14, 1980 ”

3.0.2 Text of NSF press release.

14 March, 1980A scientist has taken a major step toward the solution of a long-standing

problem in mathematics – an accomplishment that involved a structure withso many elements that he had to develop new ways of doing the computationsbecause even high speed computers could have taken many years of full-timecontinuous operation to do the job.

The segment of the problem he solved concerns a group called “The Mon-ster” by some mathematicians because of the enormous number of elementsinvolved – a number with 54 digits. This number is comparable to the massof the universe in grams.

The scientist – whose accomplishment has excited the mathematical com-munity – succeeded in constructing one of the two remiaining algebraic enti-ties in the list of what are called “sporadic” groups.

A group, the name given to one of the simplest and most useful of alge-braic structures, is a set of elements together with an operation (for example,addition or multiplication) that satisfies certain axioms. The elements of agroup can be thought of as any sort of object, such as numbers or physicalmotions.

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The scientist is Dr. Robert L. Griess, Jr., associate professor of mathe-matics at the University of Michigan and currently a member of the Institutefor Advanced Study in Princeton, new Jersey. He reported his accomplish-ment – which began more than six years ago – to the National SciencenFoundation’s (NSF) Division of Mathematical and Computer Sciences whichfunded his resesarch.

The group concept has found applications in many branches of mathemat-ics as well as in physics and chemistry. For example, the theory of groupshas applications in crystallography and in spectroscopy, the science of thestructure and properties of crystals and the study of colors, respectively.

Some groups have only a finite number of elements. Certain groups,called simple groups, can be thought of as the building blocks from whichall groups are constructed. For years, mathematicians knew of two largefamilies of finite simple groups. These families were extensively studied andare relatively well understood.

However, some finite groups do not fit in either of these two classes.These exceptional groups, the first of which was discovered in 1860, arecalled sporadic groups.

The long-standing problem mathematicians have been working on in-volves the classification of finite simple groups, assigning each of them to oneof the two principal families of groups or to one of the 26 sporadic groups.Once this is done mathematicians will have solved one of the principal prob-lems of the theory of finite simple groups. Solution of the problem of classi-fying all finite simple groups can perhaps be compared to the completion ofthe table of elements in chemistry.

The problem was studied without major dramatic success until, in 1963,two mathematicians proved that the number of elements of essentially allfinite simple groups was even. At this point it was thought that the majorobstacle in solving the main problem of finite groups was past, and that asolution was in sight.

Two years later, in 1965, however, the mathematical community wasstartled by the discovery of a new sporadic group, the first since 1860. Since1965, several new sporadic groups have been discovered until today 24 areknown and two others are believed to exist.

Dr. Griess’ work has brought this goal a large step closer because nowit appears that only one more sporadic group is left for mathematicians toconstruct. When the final sporadic group has been constructed, a major gapin the solution of the classsification problem will have been filled.

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