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1. Central Extensions and Almost Representations

   https://doi.org/10.1093/imrn/rnaf261  

   Dadarlat, Marius; Glebe, Forrest (September 2025, International Mathematics Research Notices)

   Abstract For a sequence of unital tracial $$C^{*}$$-algebras $$(A_{n},\tau _{n}),$$ we construct a canonical central extension of the unitary group $$U(\ell ^\infty (\mathbb{N},A_{n})/c_{0}(\mathbb{N},A_{n}))$$ by $$Q(\mathbb{R})=c_{0}(\mathbb{N},\mathbb{R})/\mathbb{R}^\infty ,$$ using de la Harpe–Skandalis pre-determinant. For an asymptotic group homomorphism $$\rho _{n}: \Gamma \to U(A_{n}),$$ the corresponding pullback of the canonical central extension gives a 2-cohomology class in $$H^{2}(\Gamma ,Q(\mathbb{R})),$$ which obstructs the perturbation of $$(\rho _{n})$$ to a sequence of true homomorphisms of groups $$\pi _{n}:\Gamma \to GL(A_{n})$$. The pairing of the obstruction class with elements of $$H_{2}(\Gamma ,\mathbb{Z})$$ yields numerical invariants in $$\tau _{n\,*} (K_{0}(A_{n}))$$ that subsume the winding number invariants of Kazhdan, Exel, and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group $$\Gamma $$ into the general linear group of any sequence of tracial unital Banach algebras. In that case, the obstruction class belongs to $$H^{2}(\Gamma ,Q(\mathbb{C})),$$ where $$Q(\mathbb{C})=c_{0}(\mathbb{N},\mathbb{C})/\mathbb{C}^\infty .$$ As an application, we show that 2-cohomology obstructs various stability properties under weaker assumptions than those found in existing literature. In particular, we show that the full group $$C^{*}$$-algebra $$C^{*}(\Gamma )$$ of a discrete group $$\Gamma $$ is not $$C^{*}$$-stable if $$H^{2}(\Gamma ,\mathbb{R})\neq 0$$ and in fact, $$\Gamma $$ is not stable in operator norm with respect to tracial von Neumann algebras. 

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2. Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

   https://doi.org/10.1007/s00209-022-03193-3  

   Larsen, Michael; Taylor, Jay; Tiep, Pham Huu (February 2023, Mathematische Zeitschrift)

   Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups. 

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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. Non-vanishing of class group L -functions for number fields with a small regulator

   https://doi.org/10.1112/S0010437X20007472  

   Khayutin, Ilya (November 2020, Compositio Mathematica)

   null (Ed.)

   Let $$E/\mathbb {Q}$$ be a number field of degree $$n$$ . We show that if $$\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$$ then the fraction of class group characters for which the Hecke $$L$$ -function does not vanish at the central point is $$\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$$ . The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $$\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$$ associated to the maximal order of $$E$$ , and the escape of mass of the torus orbit associated to the trivial ideal class. 

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5. Cohomology of finite tensor categories: Duality and Drinfeld centers

   https://doi.org/10.1090/tran/8548  

   Negron, Cris; Plavnik, Julia (March 2022, Transactions of the American Mathematical Society)

   We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

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