On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

On exterior powers of endomorphisms

Let V be an n-dimensional vector space and T?Hom(V,V). The first result shows that if C_m(T), the mth compound of T, possesses a basis of eigenvectors, then it possesses a basis consisting of decomposable eigenvectors in the mth Grassman space over V. The paper also contains a simplified proof of a recent result of S. Belcerzyk on traces of compounds as well as conditions for the equality of fixed coefficients in the polynomials det(λA+μX) and det(λB+μX).     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

On the vanishing of (co)homology over local rings

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.     

On the rigidity of the cotangent complex at the prime 2

In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65-87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455-487] a conjecture was posed to the effect that if R→A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of A-modules, of the associated cotangent complex L_A/R satisfies: . The aim of this paper is to initiate an approach for solving this conjecture when R has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ₂ and the André operation ? as it acts on Tor_R(A,?), ? any residue field of A. Second, we prove the conjecture is valid in two cases: when and when R is a Cohen-Macaulay ring.     

The logarithmic cotangent complex

We define the cotangent complex of a morphism of fine log schemes, prove that it is functorial, and construct under certain restrictions a transitivity triangle. We also discuss its relationship with deformation theory.     

On the vanishing of homology for modules of finite complete intersection dimension

We prove rigidity type results on the vanishing of stable Ext and Tor for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of pre-rigidity, which generalizes phenomena for modules of finite complete intersection dimension and complexity one. Using this concept, we prove results on length and vanishing of homology modules.     

The numerical radius of exterior powers

Let λ₁ and α₁ be respectively the eigenvalue of largest modulus and largest singular value of a linear operator A. Then A is called radial if |λ₁| = α₁. This paper is concerned with an examination of radial compound matrices. It turns out that the radial property for compound matrices is equivalent to an investigation of the case of equality in the classical inequalities of H. Weyl relating products of eigenvalues and singular values.     

Minors of symmetric and exterior powers

We describe some of the determinantal ideals attached to symmetric, exterior and tensor powers of a matrix. The methods employed use elements of Zariski's theory of complete ideals and of representation theory.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

On the vanishing of homology in random Čech complexes

We compute the homology of random ?ech complexes over a homogeneous Poisson process on the d‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erd?s ‐Rényi phase transition, where the ?ech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017     

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo \mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and \mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo \mathbbZ[i]{\mathbb{Z}[i]} the fractional part of z and write {z} for this, in general, complex number lying in the unit square S:={z ? \mathbbC:0 ￡ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over \mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ α ⁿ  +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤  1 and x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence z_n ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ α ⁿ −z _n , n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at (1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large.