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Let μ be a Borel measure on a compactum X. The main objects in this paper are s-ideals Ipdimq, J0pμq, Jf pμq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the s-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with nonhomogeneous s-ideals J0pμq or Jf pμq, the completions of the quotient Boolean algebras BorelpXq{J0pμq or BorelpXq{Jf pμq may be homogeneous. We discuss the topic in a more general setting, involving calibrated s-ideals. [ABSTRACT FROM AUTHOR]

Let F be a quadratic form in N ≥ 2 variables defined on a vector space V ⊆ KN over a global field K, and Ƶ ⊆ KN be a finite union of varieties defined by families of homogeneous polynomials over K. We show that if V \ Ƶ contains a nontrivial zero of F, then there exists a linearly independent collection of small-height zeros of F in V \ Ƶ, where the height bound does not depend on the height of Ƶ, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace W of the quadratic space (V, F) such that W is not contained in Ƶ. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit. [ABSTRACT FROM AUTHOR]

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. [ABSTRACT FROM AUTHOR]

In an earlier paper, we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras. Here, we consider full quivers (as well as pseudo-quivers) as a combinatoric tool in order to describe PI-varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. [ABSTRACT FROM AUTHOR]

Let $G$ be a reductive $ \mathfrak{p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak{p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem. [ABSTRACT FROM AUTHOR]

In this paper, we first give a slight improvement of Yamanoi's truncated second main theorem for holomorphic maps into abelian varieties. We then use the result to study the uniqueness problem for such maps. The results obtained generalize and improve E. M. Schmid's uniqueness theorem for holomorphic maps into elliptic curves. In the last section, we consider algebraic dependence for a finite collection of holomorphic curves into an abelian variety. [ABSTRACT FROM AUTHOR]

This paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We present a very general definition of quadratic and Koszul duality functors backed up by explicit examples. This generalizes the work of Beilinson, Ginzburg, and Soergel, 1996, in two substantial ways: We work in the setup of graded categories, i.e. we allow infinitely many idempotents and also define a ``Koszul'' duality {it functor} for not necessarily Koszul categories. As an illustration of the techniques we reprove the Koszul duality (Ryom-Hansen, 2004) of translation and Zuckerman functors for the classical category $mathcal {O}$ in a quite elementary and explicit way. From this we deduce a conjecture of Bernstein, Frenkel, and Khovanov, 1999. As applications we propose a definition of a ``Koszul'' dual category for integral blocks of Harish-Chandra bimodules and for blocks outside the critical hyperplanes for the Kac-Moody category $mathcal {O}$. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $12$. Counterexamples are given if $\varrho =2$. [ABSTRACT FROM AUTHOR]

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth. [ABSTRACT FROM AUTHOR]

Let $\mu$ be a Radon measure on $\mathbb{R}^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq Cr^n$, for some fixed $0

Let Sg (g ≥ 2) be a closed surface of genus g. Let K be any real number field, and let A be any quaternion algebra over K such that A ⊗K ℝ = M2(ℝ.). We show that there exists a hyperbolic structure on Sg such that K and A arise as its invariant trace field and invariant quaternion algebra. [ABSTRACT FROM AUTHOR]

Assume that D is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of relative cluster tilting objects, and T[1]-cluster tilting objects in D, which are a generalization of cluster-tilting objects. When D is 2-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let Λ = EndDop(T) be the opposite algebra of the endomorphism algebra of T. We show that there exists a bijection between T[1]-cluster tilting objects in D and support τ-tilting Λ- modules, which generalizes a result of Adachi--Iyama--Reiten [τ-tilting theory, Compos. Math. 150 (2014), no. 3, 415-452]. We develop a basic theory on T[1]-cluster tilting objects. In particular, we introduce a partial order on the set of T[1]-cluster tilting objects and mutation of T[1]-cluster tilting objects, which can be regarded as a generalization of "cluster-tilting mutation". As an application, we give a partial answer to a question posed in Adachi--Iyama--Reiten, loc. cit. [ABSTRACT FROM AUTHOR]

It is shown that there exists a compact set X in CN (N ≥ 2) such that X̂\ X is nonempty and the uniform algebra P(X) has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese X such that R(X) has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in CN is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein. [ABSTRACT FROM AUTHOR]

We explore orthogonal polynomials on the unit circle whose Schur parameters are {cαn}∞n=1, where 0 < ∣α∣, ∣c∣ < 1. Specifically, we derive two different generating functions. The first can be represented explicitly in terms of sums of a q-hypergeometric type and used to derive explicitly the underlying orthogonal polynomials, while the second obeys a functional differential equation and can be used to determine the asymptotic behaviour of these polynomials. Extending these constructs to orthogonal polynomials of the second kind, we are able to construct the Carathéodory function and examine the underlying orthogonality measure. [ABSTRACT FROM AUTHOR]

We give an Erdmann-Nakano type theorem for the finite quiver Hecke algebras RΛ0 (β) of affine type Dℓ+1(2), which tells their representation type. If RΛ0 (β) is not of wild representation type, we may compute its stable Auslander-Reiten quiver. [ABSTRACT FROM AUTHOR]

We prove an analogue of the Sato-Tate conjecture for Drinfeld modules. Using ideas of Drinfeld, J.-K. Yu showed that Drinfeld modules satisfy some Sato-Tate law, but did not describe the actual law. More precisely, for a Drinfeld module φ defined over a field L, he constructs a continuous representation ρ∞: WL → Dx of the Weil group of L into a certain division algebra, which encodes the Sato-Tate law. When φ has generic characteristic and L is finitely generated, we shall describe the image of ρ∞ up to commensurability. As an application, we give improved upper bounds for the Drinfeld module analogue of the Lang-Trotter conjecture. [ABSTRACT FROM AUTHOR]

We prove that if V is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, V has a difference term, and V has a finite residual bound, then V is finitely axiomatizable. This provides a common generalization of R. McKenzie's finite basis theorem for congruence modular varieties with a finite residual bound, and R. Willard's finite basis theorem for congruence meet-semidistributive varieties with a finite residual bound. [ABSTRACT FROM AUTHOR]

We give a diagrammatic description of Popa's symmetric enveloping algebras associated to planar algebra subfactors. As an application we construct a natural family of derivations on these factors and compute a certain free entropy dimension type quantity. [ABSTRACT FROM AUTHOR]

We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms. [ABSTRACT FROM AUTHOR]

We prove that there exist only finitely many commuting squares of finite dimensional *-algebras of fixed dimension, satisfying a "large second relative commutant" condition. We show this by studying the local minima of w → dim(A ∩ wBw*), where A,B are fixed subalgebras of some *-algebra C and w ε C is a unitary. When applied to lattices arising from subfactors satisfying a certain extremality- like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors. [ABSTRACT FROM AUTHOR]

One studies the Zariski tangent cone TX →π X to an affine variety X and the closure &Tmacron;X of ?-1(Reg(X)) in TX. One focuses on the comparison between TX and &Tmacron;X, giving sufficient conditions on X in order that TX = &Tmacron;X. One considers, in particular, the question of when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Another problem considered here is to understand the impact of the Cohen-Macaulayness or normality of &Tmacron;X on the local structure of X. [ABSTRACT FROM AUTHOR]