The geometry of Whitney’s critical sets

In this paper, the complete geometric characterizations, including decomposition and compression theorems, are obtained for a connected and compact set to be a critical set in Whitney’s sense, i.e., a set such that there exists a differentiable function critical but not constant on it. The problem to characterize these sets geometrically was posed by H. Whitney [21] in 1935. We also provide a complete geometrical characterization for monotone Whitney arc, i.e., there exists a differentiable function critical but also increasing along the arc. All examples appearing in the literature are monotone Whitney arcs, for example, the examples by Whitney [21] and Besicovitch [2], Norton’s t-quasi-arcs with Hausdorff dimension > t [14], and self-similar arcs [19]. Furthermore, after introducing the notion of homogeneous Moran arc, we can completely characterize all the monotone Whitney arcs of criticality > 1, which include t-quasi arcs and self-conformal arcs. Some applications to arcs which are attractors of Iterated Function Systems are discussed, including self-conformal arcs, self-similar arcs and self-affine arcs. Finally, we give an example of critical arc such that any of its subarcs fails to be a t-quasi-arc for any t, providing an affirmative answer to an open question by Norton.     

The geometry of the Wilks’s Λ random field

The statistical problem addressed in this paper is to approximate the P value of the maximum of a smooth random field of Wilks’s Λ statistics. So far results are only available for the usual univariate statistics (Z, t, χ², F) and a few multivariate statistics (Hotelling’s T ², maximum canonical correlation, Roy’s maximum root). We derive results for any differentiable scalar function of two independent Wishart random fields, such as Wilks’s Λ random field. We apply our results to a problem in brain shape analysis.     

The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence

One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_{ {\mathcal {X}}}One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let T _XT_{ {\mathcal {X}}} be the 3×3 matrix having only two nonzero entries (T _X)₁₂=(T _X)₂₁=1(T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1 and let T_\varLambda {\mathcal {T}}_{\varLambda } be the set of real, symmetric tridiagonal matrices with the same spectrum as T _XT_{ {\mathcal {X}}} . There exists a neighborhood U ì T_\varLambda \boldsymbol {{\mathcal {U}}}\subset {\mathcal {T}}_{\varLambda } of T _XT_{ {\mathcal {X}}} which is invariant under Wilkinson’s shift strategy with the following properties. For T₀ ? UT_{0}\in \boldsymbol {{\mathcal {U}}} , the sequence of iterates (T _k ) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry (T _k )₂₃. In fact, quadratic convergence occurs exactly when limT_k=T _X\lim T_{k}=T_{ {\mathcal {X}}} . Let X\boldsymbol {{\mathcal {X}}} be the union of such quadratically convergent sequences (T _k ): The set X\boldsymbol {{\mathcal {X}}} has Hausdorff dimension 1 and is a union of disjoint arcs X^s\boldsymbol {{\mathcal {X}}}^{\sigma} meeting at T _XT_{ {\mathcal {X}}} , where σ ranges over a Cantor set.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

The automorphisms of Petit’s algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut_F(K) and n≥m?1, where n is the order of σ and m the degree of f, and obtain partial results for n<m?1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.     

The converse of Schur’s theorem

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of G being nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).     

Hans Duistermaat’s contributions to Poisson geometry

Hans Duistermaat was scheduled to lecture in the 2010 School on Poisson Geometry at IMPA, but passed away suddenly. This is a record of a talk I gave at the 2010 Conference on Poisson Geometry (the week after the School) to share some of my memories of him and to give a brief assessment of his impact on the subject.     

‘There are great alterations in the geometry of late’. The rise of Isaac Newton’s early Scottish circle†

This paper traces the rise of three Scottish mathematicians – Colin Campbell, John Craig, and David Gregory – to become key figures in the dissemination and promotion of Newton’s mathematical ideas and natural philosophy in the 1680s. Two medical men – Archibald Pitcairne and his former student George Cheyne – both likewise captivated by the Principia, played minor roles in the story of Newton’s mathematics, while at the same time promoting the concept of mathematical medicine derived from his philosophical thought. Drawing on contemporary correspondence and previously unpublished papers, it considers how these men contributed to the scholarly perception of Newton and how, conversely, Newton used his increasing influence in order to encourage their work, most notably obtaining for Gregory the vacant chair in astronomy at Oxford in 1691.     

The lengths of proofs: Kreisel’s conjecture and Gödel’s speed-up theorem

We collect and compare several results that have been obtained so far in the attempts to prove a statement conjectured by Kreisel about the lengths of proofs. We also survey several results regarding a speed-up theorem announced by G?del in an abstract published in 1936, Finally, we connect this to Kreisel’s conjecture. Bibliography: 63 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 153–188.     

The image of Colmez’s Montreal functor

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL₂(Q _{?? p} ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL₂(Q _{?? p} ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.     

The mathematical background of Lomonosov’s contribution

This is a short overview of the influence of the mathematicians of the Enlightenment on the creative contribution of Mikhaĭlo Lomonosov.     

The matrix version of Hamburger’s theorem

Mathematical Notes - Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely...     

The equational complexity of Lyndon’s algebra

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.