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1. Representation Homology of Topological Spaces

   https://doi.org/10.1093/imrn/rnaa023  

   Berest, Yuri; Ramadoss, Ajay C; Yeung, Wai-Kit (February 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, $${{\mathbb{S}}}^1$$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in $${\mathbb{R}}^3$$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $${\mathbb{R}}^3$$. 

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2. Harmonic analysis on certain spherical varieties

   https://doi.org/10.4171/JEMS/1381  

   Getz, Jayce R; Hsu, Chun-Hsien; Leslie, Spencer (October 2023, Journal of the European Mathematical Society)

   Braverman and Kazhdan proposed a conjecture, later refined by Ngô and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B. Liu and later the first two authors proved these conjectures for certain spherical varieties Y built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on Y: We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on the affine closures of Braverman–Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory. 

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3. Spanning Configurations and Representation Stability

   https://doi.org/10.37236/11136  

   Pawlowski, Brendan; Ramos, Eric; Rhoades, Brendon (January 2023, The Electronic Journal of Combinatorics)

   Let $$V_1, V_2, V_3, \dots $$ be a sequence of $$\mathbb {Q}$$-vector spaces where $$V_n$$ carries an action of $$\mathfrak{S}_n$$. Representation stability and multiplicity stability are two related notions of when the sequence $$V_n$$ has a limit. An important source of stability phenomena arises when $$V_n$$ is the $$d^{th}$$ homology group (for fixed $$d$$) of the configuration space of $$n$$ distinct points in some fixed topological space $$X$$. We replace these configuration spaces with moduli spaces of tuples $$(W_1, \dots, W_n)$$ of subspaces of a fixed complex vector space $$\mathbb {C}^N$$ such that $$W_1 + \cdots + W_n = \mathbb {C}^N$$. These include the varieties of spanning line configurations which are tied to the Delta Conjecture of symmetric function theory. 

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4. Free quotients of favorable Calabi-Yau manifolds

   https://doi.org/10.1007/JHEP07(2022)116  

   Gray, James; Wang, Juntao (July 2022, Journal of High Energy Physics)

   A bstract Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. 

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5. Random Interpolating Sequences in the Polydisc and the Unit Ball

   https://doi.org/10.1007/s40315-022-00448-2  

   Dayan, Alberto; Wick, Brett D.; Wu, Shengkun (March 2023, Computational Methods and Function Theory)

   Abstract We study almost surely separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0–1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov–Sobolev spaces $$B_{2}^{\sigma }\left( \mathbb {B}_{d}\right) $$ B 2 σ B d , in the range $$0 < \sigma \le 1 / 2$$ 0 < σ ≤ 1 / 2 . For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) and its multiplier algebra $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) : in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) -interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $$0-1$$ 0 - 1 law for random interpolating sequences for $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) . 

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