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Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an appropriate reduction mechanism to rule out the possibility of completeness by simply invoking Tarski's Undefinability of Truth theorem. We also use the proof strategy of Theorem A to obtain other incompleteness results. [ABSTRACT FROM AUTHOR]

Let M be the collection of the multipliers of the Drury-Arveson space H2n, n ≥ 2. In a recent paper [Adv. Math. 335 (2018), pp. 372-404], Aleman et al. showed that for ƒ ∈ H2n, the condition sup|z|<1 Re(ƒ,K_zƒ) < ∞ is sufficient for the membership ƒ ∈ M. We show that this condition is not necessary for ƒ ∈ M. Moreover, we show that the condition sup|z|<1 Re(ƒ,Kzƒ) < ∞ only captures a nowhere dense subset of M. [ABSTRACT FROM AUTHOR]

We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations. [ABSTRACT FROM AUTHOR]

By Orlov's formula, the derived category of blow up X = BlpX' → X' contains Db(X') as a semiorthogonal component. This raises an interesting question: does there exist a variety X' such that Db(X') does not admit an exceptional collection of maximal length, but Db(X) admits an exceptional collection of maximal length? We give such an example when X' is a minimal Enriques surface. [ABSTRACT FROM AUTHOR]

Given a collection of boundary divisors in the moduli space \overline {\mathcal {M}}_{0,n} of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in \overline {\mathcal {M}}_{g,n}. [ABSTRACT FROM AUTHOR]

For any d∈N and any function ƒ:(0,∞) → [0,1] with ƒ(R)\to 0 as R\to ∞, we construct a set A ⊆ Rd and a sequence Rn → ∞ such that | x-y |t ¬ Rn for all x,y∈A and μ (A∩BRn)\ ≥ ƒ(Rn)μ(BRn) for all n∈N, where BR is the ball of radius R centered at the origin and μ is a Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on Rd. [ABSTRACT FROM AUTHOR]

Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a finite collection of strongly irreducible Heegaard surfaces Si and incompressible surfaces Kj such that any strongly irreducible Heegaard surface is a Haken sum Si + ∑j nj Kj, up to one-sided associates of the Heegaard surfaces. [ABSTRACT FROM AUTHOR]