The Classification of the Finite Simple
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Boletín De La RSME
Boletín de la RSME Número 297, 16 de enero de 2012 Sumario Noticias de la RSME Noticias de la RSME Encuentro Conjunto RSME-SMM. Antonio Campillo. Torremolinos, Málaga, 17-20 de enero El programa científico consta de veinticuatro • Encuentro Conjunto RSME- La Real Sociedad Matemática Española y la sesiones especiales, ocho conferencias ple- SMM. Torremolinos, Málaga, Sociedad Matemática Mexicana celebran el narias y la conferencia de clausura del profe- 17-20 de enero Segundo Encuentro Conjunto RSME-SMM en sor Federico Mayor Zaragoza con el título preciso “La comunidad científica ante los de- • Noticias del CIMPA el Hotel Meliá Costa del Sol de Torremolinos (Málaga) los próximos días 17, 18, 19 y 20 de safíos presentes”. Como en la primera edi- • Acuerdo entre cuatro insti- enero. El Primer Encuentro Conjunto se ce- ción, el congreso ha programado alrededor tuciones para la creación del lebró en Oaxaca en julio de 2009 y la serie de doscientas conferencias invitadas, la mitad IEMath. Documento de In- continuará cada tres años a partir de 2014, de las cuales corresponden a la parte mexi- cana y la otra mitad a la española. vestigación de la RSME año en el que está prevista la tercera edición en México. El comité organizador está presi- Las conferencias plenarias están a cargo de • Ampliación del orden del dido por Daniel Girela, e integrado por José Samuel Gitler, José Luis Alías, José María día de la Junta General Ordi- Luis Flores, Cristóbal González, Francisco Pérez Izquierdo, Jorge Velasco, María Emilia naria de la RSME Javier Martín Reyes, María Lina Martínez, Caballero, Javier Fernández de Bobadilla, Francisco José Palma, José Ángel Peláez y • Noticia de la COSCE: Nue- Xavier Gómez-Mont y Eulalia Nulart. -
Lecture Notes in Mathematics
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann 1185 Group Theory, Beijing 1984 Proceedings of an International Symposium held in Beijing, Aug. 27-Sep. 8, 1984 Edited by Tuan Hsio-Fu Springer-Verlag Berlin Heidelberg New York Tokyo Editor TUAN Hsio-Fu Department of Mathematics, Peking University Beijing, The People's Republic of China Mathematics Subject Classification (1980): 05-xx, 12F-xx, 14Kxx, 17Bxx, 20-xx ISBN 3-540-16456-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16456-1 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 PREFACE From August 27 to September 8, 1984 there was held in Peking Uni• versity, Beijing an International Symposium on Group Theory. As well said by Hermann Wey1: "Symmetry is a vast subject signi• ficant in art and nature. Whenever you have to do with a structure endowed entity, try to determine the group of those transformations which leave all structural relations undisturbed." This passage underlies that the group concept is one of the most fundamental and most important in modern mathematics and its applications. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
The Thompson-Lyons Transfer Lemma for Fusion Systems
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 The Thompson-Lyons transfer lemma for fusion systems Justin Lynd Abstract A generalization of the Thompson transfer lemma and its various extensions, most recently due to Lyons, is proven in the context of saturated fusion systems. A strengthening of Alperin’s fusion theorem is also given in this setting, following Alperin’s own “up and down” fusion. The classical Thompson transfer lemma appeared as Lemma 5.38 in [12]; for a 2-perfect group G with S ∈ Syl2(G), it says that if T is a maximal subgroup of S and u is an involution in S − T , then the element u has a G-conjugate in T . Thompson’s lemma has since been generalized in a number of ways. Harada showed [9, Lemma 16] that the same conclusion holds provided one takes u to be of least order in S − T . Unpublished notes of Goldschmidt [6] extended this to show that one may find an G-conjugate of u in T which is extremal under the same conditions. An element t ∈ S is said to be extremal in S with respect to G if CS(t) is a Sylow subgroup of CG(t). In other words, hti is fully FS(G)-centralized, where FS(G) is the fusion system of G. Later, Thompson’s result and its extensions were generalized to all primes via an argument of Lyons [7, Proposition 15.15]. We prove here a common generalization of Lyons’ extension and his similar transfer result [8, Chapter 2, Lemma 3.1] (which relaxes the requirement that S/T be cyclic) in the context of saturated fusion systems. -
UT Austin Professor, Luis Caffarelli, Wins Prestigious Wolf Prize Porto Students Conduct Exploratory Visits
JANUARY//2012 NUMBER//41 UT Austin Professor, Luis Caffarelli, wins prestigious Wolf Prize Luis Caffarelli, a mathematician, Caffarelli’s research interests include Professor at the Institute for Compu- nonlinear analysis, partial differential tational Engineering and Sciences equations and their applications, (ICES) at the University of Texas at calculus of variations and optimiza- Austin and Director of CoLab tion. The ICES Professor is also widely Mathematics Program, has been recognized as the world’s leading named a winner of Israel’s prestigious specialist in free-boundary problems Wolf Prize, in the mathematics for nonlinear partial differential category. equations. This prize, which consists of a cer- Each year this prize is awarded to tificate and a monetary award of specialists in the fields of agriculture, $100,000, “is further evidence of chemistry, mathematics, physics Luis’ huge impact”, according to and the arts. Caffarelli will share the Alan Reid (Chair of the Department 2012 mathematics prize with of Mathematics). “I feel deeply Michael Aschbacher, a professor at honored”, states Caffarelli, who joins the California Institute of Technology. the UT Austin professors John Tate (Mathematics, 2002) and Allen Bard Link for Institute for Computational UT Austin Professor, Luis Caffarelli (Chemistry, 2008) as a Wolf Prize Engineering and Sciences: winner. http://www.ices.utexas.edu/ Porto Students Conduct Exploratory Visits UT Austin welcomed several visitors at the end of the Fall animation. Bastos felt he benefited greatly from the visit, 2011 semester, including Digital Media doctoral students commenting, “The time I spent in Austin and in College Pedro Bastos, Rui Dias, Filipe Lopes, and George Siorios of Station was everything I expected and more. -
A Sketch of the History and State of Automated Deduction
A Sketch of the History and State of Automated Deduction David A. Plaisted October 19, 2015 Introduction • Ed Reingold influenced me to do algorithms research early in my career, and this led to some good work. However, my main research area now is automatic theorem proving, so I will discuss this instead. • This talk will be a survey touching on some aspects of the state of the field of automated deduction at present. The content of the talk will be somewhat more general than the original title indicated; thus the change in title. This talk should be about 20 minutes long. I’m planning to put these notes on the internet soon. There is a lot to cover in 20 minutes, so it is best to leave most questions until the end. Proofs in Computer Science • There is a need for formal proofs in computer science, for example, to prove the correctness of software, or to prove theorems in a theoretical paper. • There is also a need for proofs in other areas such as mathematics. • It can be helpful to have programs that check proofs or even help to find the proofs. 1 Automated Deduction • This talk emphasizes systems that do a lot of proof search on their own, either to find the whole proof automatically or large parts of it, without the user specifying each application of each rule of inference. • There are also systems that permit the user to construct a proof and check that the rules of inference are properly applied. History of Automated Deduction Our current state of knowledge in automated deduction came to us by a long and complicated process of intellectual development. -
Daniel Gorenstein 1923-1992
Daniel Gorenstein 1923-1992 A Biographical Memoir by Michael Aschbacher ©2016 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. DANIEL GORENSTEIN January 3, 1923–August 26, 1992 Elected to the NAS, 1987 Daniel Gorenstein was one of the most influential figures in mathematics during the last few decades of the 20th century. In particular, he was a primary architect of the classification of the finite simple groups. During his career Gorenstein received many of the honors that the mathematical community reserves for its highest achievers. He was awarded the Steele Prize for mathemat- New Jersey University of Rutgers, The State of ical exposition by the American Mathematical Society in 1989; he delivered the plenary address at the International Congress of Mathematicians in Helsinki, Finland, in 1978; Photograph courtesy of of Photograph courtesy and he was the Colloquium Lecturer for the American Mathematical Society in 1984. He was also a member of the National Academy of Sciences and of the American By Michael Aschbacher Academy of Arts and Sciences. Gorenstein was the Jacqueline B. Lewis Professor of Mathematics at Rutgers University and the founding director of its Center for Discrete Mathematics and Theoretical Computer Science. He served as chairman of the universi- ty’s mathematics department from 1975 to 1982, and together with his predecessor, Ken Wolfson, he oversaw a dramatic improvement in the quality of mathematics at Rutgers. Born and raised in Boston, Gorenstein attended the Boston Latin School and went on to receive an A.B. -
Letter from the Chair Celebrating the Lives of John and Alicia Nash
Spring 2016 Issue 5 Department of Mathematics Princeton University Letter From the Chair Celebrating the Lives of John and Alicia Nash We cannot look back on the past Returning from one of the crowning year without first commenting on achievements of a long and storied the tragic loss of John and Alicia career, John Forbes Nash, Jr. and Nash, who died in a car accident on his wife Alicia were killed in a car their way home from the airport last accident on May 23, 2015, shock- May. They were returning from ing the department, the University, Norway, where John Nash was and making headlines around the awarded the 2015 Abel Prize from world. the Norwegian Academy of Sci- ence and Letters. As a 1994 Nobel Nash came to Princeton as a gradu- Prize winner and a senior research ate student in 1948. His Ph.D. thesis, mathematician in our department “Non-cooperative games” (Annals for many years, Nash maintained a of Mathematics, Vol 54, No. 2, 286- steady presence in Fine Hall, and he 95) became a seminal work in the and Alicia are greatly missed. Their then-fledgling field of game theory, life and work was celebrated during and laid the path for his 1994 Nobel a special event in October. Memorial Prize in Economics. After finishing his Ph.D. in 1950, Nash This has been a very busy and pro- held positions at the Massachusetts ductive year for our department, and Institute of Technology and the In- we have happily hosted conferences stitute for Advanced Study, where 1950s Nash began to suffer from and workshops that have attracted the breadth of his work increased. -
Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing. -
Representations of Finite Groups and Applications
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (241–266) REPRESENTATIONS OF FINITE GROUPS AND APPLICATIONS P H T Abstract We discuss some basic problems in representation theory of finite groups, and cur- rent approaches and recent progress on some of these problems. We will also outline some applications of these and other results in representation theory of finite groups to various problems in group theory, number theory, and algebraic geometry. 1 Introduction Let G be a finite group and F be a field. A (finite-dimensional) representation of G over F is a group homomorphism Φ: G GL(V ) for some finite-dimensional vector space ! V over F. Such a representation Φ is called irreducible if 0 and V are the only Φ(G)- f g invariant subspaces of V . Representation theory of finite groups started with the letter correspondence between Richard Dedekind and Ferdinand Georg Frobenius in April (12th, 17th, 26th) 1896. In the same year, Frobenius constructed the character table of PSL2(p), p any prime. Later on, the foundations of the complex representation theory (i.e. when F = C), were developed by Frobenius, Dedekind, Burnside, Schur, Noether, and others. Foundations of the modu- lar representation theory (that is, when p = char(F) > 0 and p divides G ) were (almost j j singlehandedly) laid out by Richard Brauer, started in 1935 and continued over the next few decades. A natural question arises: in a more-than-century-old theory such as the representa- tion theory of finite groups, what could still remain to be studied? By the Jordan–Hölder theorem, irreducible representations are the building blocks of any finite-dimensional rep- resentation of any finite group G. -
Fibrations and Idempotent Functors
Fibrations and idempotent functors MARTIN BLOMGREN Doctoral Thesis Stockholm, Sweden 2011 TRITA-MAT-11-MA-11 KTH ISSN 1401-2278 Institutionen för Matematik ISRN KTH/MAT/DA 11/04-SE 100 44 Stockholm ISBN 978-91-7501-235-3 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram- lägges till oentlig granskning för avläggande av teknologie doktorsexamen i matematik tisdagen den 31 januari 2012 klockan 13.00 i sal F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm. c Martin Blomgren, 2011 Tryck: Universitetsservice US AB iii Abstract This thesis consists of two articles. Both articles concern homotopi- cal algebra. In Paper I we study functors indexed by a small category into a model category whose value at each morphism is a weak equiv- alence. We show that the category of such functors can be understood as a certain mapping space. Specializing to topological spaces, this result is used to reprove a classical theorem that classies brations with a xed base and homotopy ber. In Paper II we study augmented idempotent functors, i.e., co-localizations, operating on the category of groups. We relate these functors to cellular coverings of groups and show that a number of properties, such as niteness, nilpotency etc., are preserved by such functors. Furthermore, we classify the values that such functors can take upon nite simple groups and give an ex- plicit construction of such values. iv Sammanfattning Föreliggande avhandling består av två artiklar som på olika sätt berör området algebraisk homotopiteori. I artikel I studerar vi funk- torer mellan en liten kategori och en modellkategori som ordnar en svag ekvivalens till varje mor. -
A Survey: Bob Griess's Work on Simple Groups and Their
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 13 (2018), No. 4, pp. 365-382 DOI: 10.21915/BIMAS.2018401 A SURVEY: BOB GRIESS’S WORK ON SIMPLE GROUPS AND THEIR CLASSIFICATION STEPHEN D. SMITH Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 S. Morgan, Chicago IL 60607-7045. E-mail: [email protected] Abstract This is a brief survey of the research accomplishments of Bob Griess, focusing on the work primarily related to simple groups and their classification. It does not attempt to also cover Bob’s many contributions to the theory of vertex operator algebras. (This is only because I am unqualified to survey that VOA material.) For background references on simple groups and their classification, I’ll mainly use the “outline” book [1] Over half of Bob Griess’s 85 papers on MathSciNet are more or less directly concerned with simple groups. Obviously I can only briefly describe the contents of so much work. (And I have left his work on vertex algebras etc to articles in this volume by more expert authors.) Background: Quasisimple components and the list of simple groups We first review some standard material from the early part of [1, Sec 0.3]. (More experienced readers can skip ahead to the subsequent subsection on the list of simple groups.) Received September 12, 2016. AMS Subject Classification: 20D05, 20D06, 20D08, 20E32, 20E42, 20J06. Key words and phrases: Simple groups, classification, Schur multiplier, standard type, Monster, code loops, exceptional Lie groups. 365 366 STEPHEN D. SMITH [December Components and the generalized Fitting subgroup The study of (nonabelian) simple groups leads naturally to consideration of groups L which are: quasisimple:namely,L/Z(L) is nonabelian simple; with L =[L, L].