Narrow Results

In this paper, we give a detailed introduction to the theory of (curved) $L_{\infty }$-algebras and $L_{\infty }$-morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer–Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of $L_{\infty }$-algebras and $L_{\infty }$-modules. In particular, one can interpret $L_{\infty }$-morphisms and morphisms of $L_{\infty }$-modules as Maurer–Cartan elements in certain $L_{\infty }$-algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of $L_{\infty }$-algebras.

We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L_∞-morphism and, by standard facts about L_∞-algebras, we obtain new conditions on obstructions to deform Hitchin pairs.

This paper discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper [Noncommutative deformation and flops, Duke Math. J.165(8) (2016) 1397–1414], the author and Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterizes their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants [Noncommutative width and Gopakumar–Vafa invariants, Manuscripta Math.148(3–4) (2015) 521–533]. Before reviewing these results, and others, the author gives a brief introduction to the rich geometry of flopping curves on 3-folds, starting from the resolutions of Kleinian surface singularities. This is based on a talk given at VBAC 2014 in Berlin.

We prove that a weak $\mathbb{Q}$-Fano $3$-fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak $\mathbb{Q}$-Fano $3$-fold. We also treat a stacky proof of the unobstructedness of deformations of a $\mathbb{Q}$-Fano $3$-fold.

We prove that the automorphism group of a general complete intersection $X$ in ${ℂ\mathbb{P}}^n$ is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the cohomology of $X$ faithfully with a few well-understood exceptions.

We present an approximate analytic solution of the Klein–Gordon equation in the presence of equal scalar and vector generalized deformed hyperbolic potential functions by means of parametric generalization of the Nikiforov–Uvarov method. We obtain the approximate bound-state rotational–vibrational (ro–vibrational) energy levels and the corresponding normalized wave functions expressed in terms of the Jacobi polynomial , where μ > -1, ν > -1, and x ∈ [-1, +1] for a spin-zero particle in a closed form. Special cases are studied including the nonrelativistic solutions obtained by appropriate choice of parameters and also the s-wave solutions.

We show how to get a noncommutative product for functions on spacetime starting from the deformation of the coproduct of the Poincaré group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on spacetime (ℝ⁴) can be identified as the set of functions on the Poincaré group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincaré group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features.

As is known, spacetime algebra fixes the coproduct on the diffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.

Consistent couplings among a set of scalar fields, two types of one-forms and a system of two-forms are investigated in the light of the Hamiltonian BRST cohomology, giving a four-dimensional nonlinear gauge theory. The emerging interactions deform the first-class constraints, the Hamiltonian gauge algebra, as well as the reducibility relations.

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semisimple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. The result can be used to construct invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.

We briefly report on some recent progresses in the computation of B-brane superpotentials for Type II strings compactified on Calabi–Yau manifolds, obtained by using a paramatrization of tubular neighborhoods of complex submanifolds, also known as local spaces. In particular, we propose a closed expression for the superpotential of a brane on a genus-g curve in a Calabi–Yau threefold for the cases in which there exists a holomorphic projection from the local space around the curve to the curve itself.

We study the general form of Möbius covariant local commutation relations in conformal chiral quantum field theories and show that they are intrinsically determined up to structure constants, which are subject to an infinite system of constraints. The deformation theory of these commutators is controlled by a cohomology complex, whose cochain spaces consist of linear maps that are subject to a complicated symmetry property, a generalization of the anti-symmetry of the Lie algebra case.

The systematics of energetic terms as they are taught in continuum mechanics deviate seriously from the standard doctrine in physics, resulting in a profound misconception. It is demonstrated that the First Law of Thermodynamics has been routinely reinterpreted in a sense that would make it subordinate to Bernoulli's energy conservation law. Proof is given to the effect that the Cauchy stress tensor does not exist. Furthermore, it is shown that the attempt by Gibbs to find a thermodynamic understanding for elastic deformation does not sufficiently account for all the energetic properties of such a process.

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty }$-algebras governing these deformation problems.

In this paper, we consider the action of Vect(S¹) by Lie derivative on the spaces of pseudodifferential operators $\mathrm{\Psi }{𝒟}{𝒪}$. We study the $𝔥$-trivial deformations of the standard embedding of the Lie algebra Vect(S¹) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle $T^{\ast }{S}^1$. We classify the deformations of this action that become trivial once restricted to $𝔥$, where $𝔥=𝔞𝔣𝔣(1)$ or $𝔰𝔩(2)$. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

In the context of normal forms, we study a class of slow–fast Hamiltonian systems on general Poisson fiber bundles with symmetry. Our geometric approach is motivated by a link between the deformation theory for Poisson structures on fibered manifolds and the adiabatic perturbation theory. We present some normalization results which are based on the averaging theorem for horizontal 2-cocycles on Poisson fiber bundles.

The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra; this leads to a new proof using multiplicative free resolutions. Specifically, our main result states that every such resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, where Bergman’s condition of “resolvable ambiguities” is precisely the first nontrivial component of the Maurer–Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing resolutions of algebras with monomial relations.

This study proposes an innovative computational strategy to predict the initiation of elastoplastic buckling in shell structures. This strategy is developed in connection with ABAQUS/Standard Finite Element (FE) code. Toward this objective, two constitutive frameworks are implemented as User MATerial subroutines (UMATs) into this FE code; namely, the incremental flow theory of plasticity and the total deformation theory. These frameworks are formulated under the plane-stress condition, which is particularly suitable for modeling sheet structures and which enhances computational efficiency. Elastoplastic buckling is detected by the Hill loss of uniqueness criterion, which establishes that buckling occurs when the global stiffness matrix, derived from the finite element computations, becomes singular. To determine this matrix and investigate its singularity, a Python script is developed and combined to the ABAQUS computations. The reliability and accuracy of this computational strategy are assessed through various representative numerical examples. The effect of some geometric and material parameters on the onset of elastoplastic buckling in both thin and thick plates, as well as cruciform columns, is investigated and compared to reference results from the literature. The findings of the present contribution can serve as useful reference guidelines for ABAQUS/Standard users, offering valuable insights for predicting the occurrence of elastoplastic buckling, even in metallic structures characterized by complex mechanical behavior and geometric configurations.

For a cuspidal newform $f=\sum a_n{q}^n$ of weight $k\ge 3$ and a prime $\lambda$ of $Q(a_n)$, the deformation problem for its associated mod $\lambda$ Galois representation is unobstructed for all primes outside some finite set. Previous results gave an explicit bound on this finite set for $f$ of squarefree level; we modify this bound and remove the squarefree hypothesis. We also show that if the $\lambda$-adic deformation problem for $f$ is unobstructed, then $f$ is not congruent mod $\lambda$ to a newform of lower level.

We consider certain $p$-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when $p=3$. The proof uses methods from deformation theory and mostly works for any odd prime $p$, but ultimately relies on the existence of a weight $1$ form in an auxiliary family which is available only for $p=3$. We end by giving several non-trivial examples of $3$-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

We consider the inverse Galois problem over function fields of positive characteristic $p$, for example, over the projective line. We describe a method to construct certain Galois covers of the projective line and other curves, which are ordinary in the sense that their Jacobian has maximal $p$-torsion. We do this by constructing Galois covers of ordinary semi-stable curves, and then deforming them into smooth Galois covers.