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This is Scott's BIBTEX file. References [1] I. R. Aitchison and J. H. Rubinstein. Fibered knots and involutions on homotopy spheres. In Four-manifold theory (Durham, N.H., 1982), volume 35 of Contemp. Math., pages 1{74. Amer. Math. Soc., Providence, RI, 1984. MR780575. [2] Selman Akbulut. The Dolgachev surface, 2008. arXiv:0805.1524. [3] Selman Akbulut. Cappell-Shaneson homotopy spheres are standard, 2009. arXiv:0907.0136. [4] Selman Akbulut and Robion Kirby. A potential smooth counterexample in dimension 4 to the Poincar´econjecture, the Schoenflies conjecture, and the Andrews-Curtis conjecture. Topology, 24(4):375{390, 1985. MR816520 DOI:10.1016/0040-9383(85)90010-2. [5] A. Alexeevski and S. Natanzon. Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves. Selecta Math. (N.S.), 12(3-4):307{377, 2006. MR2305607,arXiv:math.GT/0202164. [6] Henning Haahr Andersen. Tensor products of quantized tilting modules. Comm. Math. 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[29] Anna Beliakova and Christian Blanchet. Modular categories of types B, C and D. Com- ment. Math. Helv., 76(3):467{500, 2001. MR1854694 DOI:10.1007/PL00000385 arXiv:math. QA/0006227. [30] Anna Beliakova and Christian Blanchet. Skein construction of idempotents in Birman-Murakami-Wenzl algebras. Math. Ann., 321(2):347{373, 2001. MR1866492 DOI:10.1007/s002080100233 arXiv:math.QA/0006143. [31] Anna Beliakova and Stephan Wehrli. Categorification of the colored Jones polynomial and Rasmussen invariant of links. Canad. J. Math., 60(6):1240{1266, 2008. MR2462446 arXiv:math.QA/0510382. [32] Stephen Bigelow. Skein theory for the ADE planar algebras, 2009. arXiv:math.QA/0903. 0144. [33] Stephen Bigelow. Skein theory for the ADE planar algebras, 2009. arXiv:math.QA/0903. 0144. [34] Stephen Bigelow, Scott Morrison, Emily Peters, and Noah Snyder. Constructing the ex- tended haagerup planar algebra, 2009. arXiv:0909.4099, to appear Acta Mathematica. [35] Jocelyne Bion-Nadal. 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In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pages 775{785, Beijing, 2002. Higher Ed. Press. MR1957084 arXiv:math.OA/0304340. [40] Dietmar Bisch, Paramita Das, and Shamindra Kumar Ghosh. The planar algebra of group- type subfactors, 2008. arXiv:0807.4134. [41] Dietmar Bisch and Uffe Haagerup. Composition of subfactors: new examples of infinite depth subfactors. Ann. Sci. Ecole´ Norm. Sup. (4), 29(3):329{383, 1996. MR1386923. [42] Dietmar Bisch and Vaughan Jones. Algebras associated to intermediate subfactors. Invent. Math., 128(1):89{157, 1997. MR1437496. [43] Dietmar Bisch and Vaughan F. R. Jones. Singly generated planar algebras of small dimen- sion. Duke Math. J., 101(1):41{75, 2000. MR1733737. [44] Dietmar Bisch, Remus Nicoara, and Sorin Popa. Continuous families of hyperfinite subfac- tors with the same standard invariant. Internat. J. Math., 18(3):255{267, 2007. MR2314611 arXiv:math.OA/0604460 DOI:10.1142/S0129167X07004011. [45] Christian Blanchet. Hecke algebras, modular categories and 3-manifolds quantum invariants. Topology, 39(1):193{223, 2000. MR1710999 DOI:10.1016/S0040-9383(98)00066-4. [46] J. M. Boardman and R. M. Vogt. Homotopy-everything H-spaces. Bull. Amer. Math. Soc., 74:1117{1122, 1968. MR0236922 DOI:10.1090/S0002-9904-1968-12070-1. 3 [47] J. M. Boardman and R. M. Vogt. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, Vol. 347. Springer-Verlag, Berlin, 1973. MR0420609. [48] N. Bourbaki. El´ements´ de math´ematique. Fasc. XXXIV. Groupes et alg`ebres de Lie. Chapitre IV: Groupes de Coxeter et syst`emesde Tits. Chapitre V: Groupes engendr´espar des r´eflexions.Chapitre VI: syst`emesde racines. Actualit´esScientifiques et Industrielles, No. 1337. Hermann, Paris, 1968. MR0240238 (preview at google books). [49] Richard Brauer. On algebras which are connected with the semisimple continuous groups. Ann. of Math. (2), 38(4):857{872, 1937. 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  • A Brief History of an Important Classical Theorem

    A Brief History of an Important Classical Theorem

    Advances in Group Theory and Applications c 2016 AGTA - www.advgrouptheory.com/journal 2 (2016), pp. 121–124 ISSN: 2499-1287 DOI: 10.4399/97888548970148 A Brief History of an Important Classical Theorem L.A. Kurdachenko — I.Ya.Subbotin (Received Nov. 6, 2016 – Communicated by Francesco de Giovanni) Mathematics Subject Classification (2010): 20F14, 20F19 Keywords: Schur theorem; Baer theorem; Neumann theorem This note is a by-product of the authors’ investigation of relations between the lower and upper central series in a group. There are some famous theorems laid at the foundation of any research in this area. These theorems are so well-known that, by the established tra- dition, they usually were named in honor of the mathematicians who first formulated and proved them. However, this did not always hap- pened. Our intention here is to talk about one among the fundamen- tal results of infinite group theory which is linked to Issai Schur. Issai Schur (January 10, 1875 in Mogilev, Belarus — January 10, 1941 in Tel Aviv, Israel) was a very famous mathematician who proved many important and interesting results, first in algebra and number theory. There is a biography sketch of I. Schur wonderfully written by J.J. O’Connor and E.F. Robertson [6]. The authors documented that I. Schur actively and successfully worked in many branches of mathematics, such as for example, finite groups, matrices, alge- braic equations (where he gave examples of equations with alterna- ting Galois groups), number theory, divergent series, integral equa- tions, function theory. We cannot add anything important to this, saturated with factual and emotional details, article of O’Connor 122 L.A.
  • Man and Machine Thinking About the Smooth 4-Dimensional Poincaré

    Man and Machine Thinking About the Smooth 4-Dimensional Poincaré

    Quantum Topology 1 (2010), xxx{xxx Quantum Topology c European Mathematical Society Man and machine thinking about the smooth 4-dimensional Poincar´econjecture. Michael Freedman, Robert Gompf, Scott Morrison and Kevin Walker Mathematics Subject Classification (2000).57R60 ; 57N13; 57M25 Keywords.Property R, Cappell-Shaneson spheres, Khovanov homology, s-invariant, Poincar´e conjecture Abstract. While topologists have had possession of possible counterexamples to the smooth 4-dimensional Poincar´econjecture (SPC4) for over 30 years, until recently no invariant has existed which could potentially distinguish these examples from the standard 4-sphere. Rasmussen's s- invariant, a slice obstruction within the general framework of Khovanov homology, changes this state of affairs. We studied a class of knots K for which nonzero s(K) would yield a counterexample to SPC4. Computations are extremely costly and we had only completed two tests for those K, with the computations showing that s was 0, when a landmark posting of Akbulut [3] altered the terrain. His posting, appearing only six days after our initial posting, proved that the family of \Cappell{Shaneson" homotopy spheres that we had geared up to study were in fact all standard. The method we describe remains viable but will have to be applied to other examples. Akbulut's work makes SPC4 seem more plausible, and in another section of this paper we explain that SPC4 is equivalent to an appropriate generalization of Property R (\in S3, only an unknot can yield S1 × S2 under surgery"). We hope that this observation, and the rich relations between Property R and ideas such as taut foliations, contact geometry, and Heegaard Floer homology, will encourage 3-manifold topologists to look at SPC4.