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1. Computing cohomology groups that classify bundles of strongly self-absorbing C∗-algebras

   https://doi.org/10.1142/S1793525324500110  

   Dadarlat, Marius; McClure, James E; Pennig, Ulrich (March 2024, Journal of Topology and Analysis)

   Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory. 

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2. Residually finite dimensional algebras and polynomial almost identities

   https://doi.org/10.1142/S0219498822500384  

   Larsen, Michael; Shalev, Aner (February 2022, Journal of Algebra and Its Applications)

   Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon, 

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3. Some remarks on the Erdős Distinct subset sums problem

   https://doi.org/10.1142/S1793042123500860  

   Steinerberger, Stefan (September 2023, International Journal of Number Theory)

   Let [Formula: see text] be a set of positive integers, [Formula: see text] denoting the largest element, so that for any two of the [Formula: see text] subsets the sum of all elements is distinct. Erdős asked whether this implies [Formula: see text] for some universal [Formula: see text]. We prove, slightly extending a result of Elkies, that for any [Formula: see text], [Formula: see text] with equality if and only if all subset sums are [Formula: see text]-separated. This leads to a new proof of the currently best lower bound [Formula: see text]. The main new insight is that having distinct subset sums and [Formula: see text] small requires the random variable [Formula: see text] to be close to Gaussian in a precise sense. 

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4. Random anti-commuting Hermitian matrices

   https://doi.org/10.1142/S2010326324500199  

   McCarthy, John E; McCarthy, Hazel T (October 2024, Random Matrices: Theory and Applications)

   We consider pairs of anti-commuting [Formula: see text]-by-[Formula: see text] Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of [Formula: see text]-by-[Formula: see text] blocks on which the first matrix has eigenvalues [Formula: see text] and the second has eigenvalues [Formula: see text]. We call [Formula: see text] the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling. 

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5. Tri-plane diagrams for simple surfaces in S4

   https://doi.org/10.1142/S0218216523500414  

   Allred, Wolfgang; Aragón, Manuel; Dooley, Zack; Goldman, Alexander; Lei, Yucong; Martinez, Isaiah; Meyer, Nicholas; Peters, Devon; Warrander, Scott; Wright, Ana; et al (June 2023, Journal of Knot Theory and Its Ramifications)

   Meier and Zupan proved that an orientable surface [Formula: see text] in [Formula: see text] admits a tri-plane diagram with zero crossings if and only if [Formula: see text] is unknotted, so that the crossing number of [Formula: see text] is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in [Formula: see text], proving that [Formula: see text], where [Formula: see text] denotes the connected sum of [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text] and [Formula: see text] unknotted projective planes with normal Euler number [Formula: see text]. In addition, we convert Yoshikawa’s table of knotted surface ch-diagrams to tri-plane diagrams, finding the minimal bridge number for each surface in the table and providing upper bounds for the crossing numbers. 

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