Mathematical Sciences Faculty ScholarshipCopyright (c) 2020 Binghamton University All rights reserved.
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Recent documents in Mathematical Sciences Faculty Scholarshipen-usWed, 18 Nov 2020 16:17:40 PST3600Connecting Play to STEM Concepts, Practices and Processes: Review of Research on Play within STEM Learning Environments (publication in fourth edition of the International Encyclopedia of Education)
https://orb.binghamton.edu/mathematics_fac/10
https://orb.binghamton.edu/mathematics_fac/10Tue, 03 Nov 2020 07:51:57 PST
As play is diminishing and the need to prepare students to enter the STEM workforce is rising, we conducted a literature review to examine how play in STEM learning environments may address issues in STEM, and conversely, how STEM learning environments can be framed as a context for human development through play. Findings highlight the value of all types of play for development within STEM across a range of learning environments. Yet, the scholarship seems to point to adults as vital to the play STEM learning environment as they provide connections between play and STEM concepts, practices and processes.
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Amber Simpson et al.A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group
https://orb.binghamton.edu/mathematics_fac/9
https://orb.binghamton.edu/mathematics_fac/9Thu, 04 May 2017 10:00:10 PDT
The twin building of a Kac–Moody group G encodes the parabolic subgroup structure of G and admits a natural G–action. When G is a complex Kac–Moody group of hyperbolic type, we construct an embedding of the twin building of G into the lightcone of the compact real form of the corresponding Kac–Moody algebra. When G has rank 2, we construct an embedding of the spherical building at infinity into the set of rays on the boundary of the lightcone.
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Lisa Carbone et al.Weyl Groups of some Hyperbolic Kac-Moody Algebras
https://orb.binghamton.edu/mathematics_fac/8
https://orb.binghamton.edu/mathematics_fac/8Thu, 04 May 2017 10:00:06 PDT
We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac-Moody algebras T ++ n , obtained by a process of double extension from a Cartan matrix of finite type Tn, whose corresponding generalized Cartan matrices are symmetric.
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Alex J. Feingold et al.The 3-State Potts Model and Rogers-Ramanujan Series
https://orb.binghamton.edu/mathematics_fac/7
https://orb.binghamton.edu/mathematics_fac/7Thu, 04 May 2017 10:00:02 PDT
We explain the appearance of Rogers-Ramanujan series inside the tensor product of two basic A (2) 2 -modules, previously discovered by the first author in [F]. The key new ingredients are (5, 6) Virasoro minimal models and twisted modules for the Zamolodchikov W3-algebra.
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Alex J. Feingold et al.Hyperbolic Weyl Groups and the Four Normed Division Algebras
https://orb.binghamton.edu/mathematics_fac/6
https://orb.binghamton.edu/mathematics_fac/6Thu, 04 May 2017 09:59:58 PDT
We study the Weyl groups of hyperbolic Kac–Moody algebras of ‘over-extended’ type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K = R, C, H, O, respectively. A crucial role is played by integral lattices of the division algebras and associated discrete matrix groups. Our findings can be summarized by saying that the even subgroups, W+, of the Kac– Moody Weyl groups, W, are isomorphic to generalized modular groups over K for the simply laced algebras, and to certain finite extensions thereof for the non-simply laced algebras. This hints at an extended theory of modular forms and functions.
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Alex J. Feingold et al.A Hyperbolic Kac-Moody Algebra and the Theory of Siegel Modular Forms of Genus 2
https://orb.binghamton.edu/mathematics_fac/5
https://orb.binghamton.edu/mathematics_fac/5Thu, 04 May 2017 09:32:37 PDTAlex J. Feingold et al.A New Perspective on the Frenkel-Zhu Fusion Rule Theorem
https://orb.binghamton.edu/mathematics_fac/4
https://orb.binghamton.edu/mathematics_fac/4Thu, 04 May 2017 09:16:01 PDT
In this paper we prove a formula for fusion coefficients of affine Kac–Moody algebras first conjectured by Walton [M.A. Walton, Tensor products and fusion rules, Canad. J. Phys. 72 (1994) 527–536], and rediscovered by Feingold [A. Feingold, Fusion rules for affine Kac–Moody algebras, in: N. Sthanumoorthy, Kailash Misra (Eds.), Kac–Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac–Moody Algebras and Applications, Jan. 28–31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, in: Contemp. Math., vol. 343, American Mathematical Society, Providence, RI, 2004, pp. 53–96]. It is a reformulation of the Frenkel–Zhu affine fusion rule theorem [I.B. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992) 123–168], written so that it can be seen as a beautiful generalization of the classical Parthasarathy–Ranga Rao–Varadarajan tensor product theorem [K.R. Parthasarathy, R. Ranga Rao, V.S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Math. (2) 85 (1967) 383–429].
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Alex J. Feingold et al.Subalgebras of Hyperbolic Kac-Moody Algebras
https://orb.binghamton.edu/mathematics_fac/3
https://orb.binghamton.edu/mathematics_fac/3Thu, 04 May 2017 09:15:58 PDT
The hyperbolic (and more generally, Lorentzian) Kac-Moody (KM) Lie algebras $\cA$ of rank r+2>2 are shown to have a rich structure of indefinite KM subalgebras which can be described by specifying a subset of positive real roots of $\cA$ such that the difference of any two is not a root of $\cA$. Taking these as the simple roots of the subalgebra gives a Cartan matrix, generators and relations for the subalgebra. Applying this to the canonical example of a rank 3 hyperbolic KM algebra, $\cF$, we find that $\cF$ contains all of the simply laced rank 2 hyperbolics, as well as an infinite series of indefinite KM subalgebras of rank 3. It is shown that $\cA$ also contains Borcherds algebras, obtained by taking all of the root spaces of $\cA$ whose roots are in a hyperplane (or any proper subspace). This applies as well to the case of rank 2 hyperbolics, where the Borcherds algebras have all their roots on a line, giving the simplest possible examples.
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Alex J. Feingold et al.Fusion Rules For Affine Kac-Moody Algebras
https://orb.binghamton.edu/mathematics_fac/2
https://orb.binghamton.edu/mathematics_fac/2Thu, 04 May 2017 08:54:10 PDT
This is an expository introduction to fusion rules for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Many explicit examples are included with figures illustrating the rank 2 cases. New results relating fusion coefficients to tensor product coefficients are proved, and a conjecture is given which shows that the Frenkel–Zhu affine fusion rule theorem can be seen as a beautiful generalization of the Parasarathy Ranga Rao Varadaragan tensor product theorem. Previous work of the author and collaborators on a different approach to fusion rules from elementary group theory is also exlained.
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Alex J. FeingoldSingularities of the eta function of first-order differential operators
https://orb.binghamton.edu/mathematics_fac/1
https://orb.binghamton.edu/mathematics_fac/1Fri, 19 Aug 2016 13:16:21 PDT
We report on a particular case of the paper [7], joint with Raphael Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n - 1, n - 3, n - 5, ....
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Paul Loya et al.