Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and si...

The nonabelian tensor square G⊗G of the group G is the group generated by the symbols g ⊗ h, where g, h ∈ G, subject to the relations gg′ ⊗ h = (gg′ ⊗ h)(g ⊗ h) and g ⊗ hh′ = (g ⊗ h)(g ⊗ hh′) for all g, g, h, h′ ∈ G, where gg′ = gg′g−1 is conjugation on the left. Following the work of C. Miller [18], R. K. Dennis in [10] introduced the nonabelian tensor square which is a specialization of the m...

Underlying most digital signal processing (DSP) algorithms is the group Z/N of integers modulo N , which is taken as the data indexing set. Recently, An and Tolimieri [1] considered a different class of index set mappings, which arise when the underlying group is nonabelian, and successfully apply them to 2D image data. Advantages of indexing signals with nonabelian groups are not limited to im...

In this paper we introduce the conjugate graph ΓG associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if ΓG ∼= ΓS , where S is a finite nonabelian simple group which satisfy Thompson’s conjecture, then G ∼= S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then ...

Lifting supersymmetric quantum mechanics to loop space yields the superstring. A particle charged under a fiber bundle thereby turns into a string charged under a 2-bundle, or gerbe. This stringification is nothing but categorification. We look at supersymmetric quantum mechanics on loop space and demonstrate how deformations here give rise to superstring background fields and boundary states, ...

We study a class of N = 1 supersymmetric U(N) gauge theories and find that there exist vacua in which the low-energy magnetic effective gauge group contains multiple nonabelian factors, ∏ i SU(ri), supported by light monopoles carrying the assiciated nonabelian charges. These nontrivially generalize the physics of the so-called r-vacua found in softly broken N = 2 supersymmetric SU(N) QCD, with...

In 2006, Vasil’ev posed the problem: Does there exist a positive integer k such that there are no k pairwise nonisomorphic nonabelian finite simple groups with the same graphs of primes? Conjecture: k = 5. In 2013, Zvezdina, confirmed the conjecture for the case when one of the groups is alternating. We continue this work and determine all nonabelian simple groups having the same prime graphs a...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. let $g$ be a finite nonabelian $p$-group. it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic, or $g/z(g)$ is powerful, then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattini subgro...

We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of...

We use the multimonopole moduli space as a tool for studying the properties of BPS monopoles carrying nonabelian magnetic charges. For configurations whose total magnetic charge is purely abelian, the moduli space for nonabelian breaking can be obtained as a smooth limit of that for a purely abelian breaking. As the asymptotic Higgs field is varied toward one of the special values for which the...