Classifies metric symplectic Lie algebras with quadratic extensions.
problem Classifying metric symplectic Lie algebras.
method Standard model, quadratic cohomology sets, isomorphism classes.
result Complete list of metric symplectic Lie algebras in special cases.
The paper classifies Poisson structures on toric manifolds and computes their cohomology.
problem Classifying real Poisson structures on complex toric manifolds of type (1,1) and computing their cohomology. method Investigates algebraic and differential Poisson structures, focusing on smooth and generically non-degenerate cases.
result Computes the first two cohomology groups for algebraic Poisson structures.
Non-Euclidean number rings have non-integral Steinberg modules.
problem Characterizing when Steinberg modules are generated by integral elements.
method Analyzing special linear groups over non-Euclidean imaginary number rings.
result Steinberg modules are not generated by integral elements in non-Euclidean rings.
Classifies symplectic Lie algebras with degenerate center.
problem Understanding symplectic Lie algebras with degenerate center.
method Standard model, quadratic cohomology sets, classification scheme.
result Complete list of 6-dimensional nilpotent symplectic Lie algebras.
New techniques prove bounded acyclicity results for semi-simplicial sets.
problem Homological stability of semi-simplicial sets and groups.
method Combinatorial techniques in simplicial bounded cohomology.
result Slope-1/2 stability results for certain groups. The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a fu…
Derives integral formula for Hodge and Teichmüller norms.
problem Relationship between Hodge and Teichmüller norms.
method Integrals and cohomology classes.
result Comparison between Hodge and Teichmüller norms.
The paper explores quadratic differentials in spherical CR geometry and their properties.
problem Understanding quadratic differentials in spherical CR geometry.
method Analyzing the Rumin complex, defining differential operators, and studying quasiconformal maps.
result Definition and properties of quadratic differentials in spherical CR geometry.
The paper defines and studies the first Pontryagin class for quadratic Lie 2-algebroids.
problem Defining and studying the first Pontryagin class for quadratic Lie 2-algebroids.
method Detailed study of transitive Lie 2-algebroids, introduction of quadratic Lie 2-algebroids, definition of first Pontryagin class, construction of quadratic Lie 2-algebroids.
result The first Pontryagin class is the obstruction class for the existence of a CLWX-extension and trivial for certain quadratic Lie 2-algebroids.
The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…
We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary r…
We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for …
The study examines the asymptotic behavior of cohomology groups of algebraic group subgroups.
problem Understanding the asymptotic behavior of cohomology groups of algebraic group subgroups.
method Analyzing the dimensions of cohomology groups and their approximation to ℓ2-Betti numbers. result The dimensions of cohomology groups approximate ℓ2-Betti numbers with a controlled error term. Graph cohomology solves symplectic problems in surface mapping groups.
problem Compute the symplectic decomposition of Torelli group and its cohomology.
method Graph cohomology and ideas from graph cohomology.
result Effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group.
Geometrically generates Fukaya categories of Weinstein manifolds.
problem Generating Fukaya categories of Weinstein manifolds.
method Geometric approach using surgery formula and Floer cohomology.
result Wrapped Fukaya category generated by index n critical points.
This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a …
Study Euler characteristics and fundamental groups of compact manifolds.
problem Analyzing Euler characteristics and fundamental groups of compact manifolds.
method Using universal coverings and cohomology, the paper explores properties of 1-forms and 2-forms on compact manifolds.
result The Euler characteristic of compact manifolds with certain fundamental group properties is non-negative or positive.
We give a solution to the word problem for the singular braid monoid SB_n. The complexity of the algorithm is quadratic in the product of the word length and the number of the singular generators in the word. Furthermore we algebraically reprove a result of Fenn, Keyman and Rourke that the monoid embeds into a group an…
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
After defining cohomologically higher order BRST and anti-BRST operators for a compact simple algebra {\cal G}, the associated higher order Laplacians are introduced and the corresponding supersymmetry algebra Σ is analysed. These operators act on the states generated by a set of fermionic ghost fields transforming u…
Study on cohomology of special linear groups over Euclidean number rings.
problem Understanding cohomology of principal congruence subgroups of special linear groups.
method Analyzing cohomology groups and using properties of Tits buildings.
result Natural map between cohomology and homology is surjective and an isomorphism under certain conditions.
The study classifies 1-connected 7-manifolds with torsion-free second homology.
problem Classifying 1-connected 7-manifolds with specific homological properties.
method Generalizes a previous result using Kreck-Stolz invariants and quadratic refinements.
result New classification method for 2-connected and simply connected 7-manifolds.
Study differential forms on period domains using superconnections.
problem Define and analyze differential forms on period domains.
method Use superconnections to define forms, study their properties, and compute their cohomology.
result Explicitly compute the cohomology class of the forms and recover known forms in specific cases.
We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the…
Study reveals vanishing Massey products on compact complex surfaces, impacting their fundamental group structure.
problem Understanding the real homotopy type of compact complex surfaces.
method Analyzes Massey products and fundamental group presentations, providing explicit presentations in non-Kähler cases.
result Explicit presentations of fundamental groups based on first Betti numbers, vanishing Massey products beyond certain lengths.
Projective resolves symplectic Steinberg module for number rings.
problem Constructing a projective resolution for symplectic Steinberg module.
method Similar to special linear group, but more complex construction.
result Computed top degree cohomology of congruence subgroups.
For any regular Courant algebroid, we construct a characteristic class a la Chern-Weil. This intrinsic invariant of the Courant algebroid is a degree-3 class in its naive cohomology. When the Courant algebroid is exact, it reduces to the Severa class (in H^3_{DR}(M)). On the other hand, when the Courant algebroid is a …
The paper connects two descriptions of Teichmüller space tangent spaces using harmonic vector fields.
problem Describing tangent spaces to Teichmüller space in two different ways.
method Using harmonic vector fields inspired by harmonic maps to connect the two descriptions.
result A harmonic vector field on the upper half plane describes a connection on the universal Teichmüller curve.
New spherical Milnor spaces for diffeological groups with geometric and topological properties.
problem Understanding higher topological structures in diffeological spaces.
method Spherical Milnor construction based on quadratic normalization.
result Provides a natural setting for studying principal bundles with Z2-twists and higher cohomology. Let us consider a compact oriented riemannian manifold M without boundary and of dimension n=4k. The signature of M is defined as the signature of a given quadratic form Q. Two different products could be used to define Q and they render equivalent definitions: the exterior product of 2k-forms and the cup product of co…
New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form τ on the determinant line of the cohomology. Both τ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsi…
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
Study Morse-Bott functions on orthogonal groups, computing Betti numbers.
problem Computing Betti numbers of orthogonal groups using Morse-Bott functions.
method Detailed analysis of quadratic and linear Morse-Bott trace functions on O(n), using critical loci and indices. result New Morse-theoretic computation of mod 2 Betti numbers of SO(n). Study the geometry of matrix multiplication in deep neural networks.
problem Understanding the structure of matrix multiplication in deep neural networks.
method Using quiver representations and equivariant cohomology, determine codimension and irreducible components.
result Codimension and number of top-dimensional irreducible components of matrix multiplication are invariant under permutations and have specific log-canonical thresholds.
The paper revisits Bianchi identities for Riemann and Weyl tensors using advanced algebraic analysis.
problem Proving Bianchi identities for Riemann and Weyl tensors in arbitrary dimensions.
method Using Spencer cohomology and differential systems, the paper proves Bianchi identities for Riemann and Weyl tensors in arbitrary dimensions.
result The paper discovers that the number of generating Bianchi identities for the Weyl tensor varies depending on the dimension, with 9 second-order identities in n=4 and 5 third-order identities in n=3.
Let X(Σ) be a smooth projective toric variety for a complex torus T_\C. In this paper, a real T_\C-invariant Poisson structure Π_Σis constructed on the complex manifold X(Σ), the symplectic leaves of which are the T_\C-orbits in X(Σ). It is shown that each leaf admits a Hamiltonian action by a sub-torus of the compact …
The main new result here is the cancellation of global anomalies in the Type I superstring, with and without D-branes. Our argument here depends on a precise interpretation of the 2-form abelian gauge field using KO-theory; then the anomaly cancellation follows from a geometric form of the full Atiyah-Singer index theo…
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…
Graph complex acts on Poisson bi-vectors, producing universal cocycles.
problem Understanding the action of graph complex on Poisson bi-vectors.
method Using Lie derivatives and graph cocycles, the graph complex acts on Poisson bi-vectors.
result A uniform construction of universal cocycles for homogeneous Poisson bi-vectors.
New cohomology defined for Lie algebroids by odd cocycles.
problem Defining a new cohomology for Lie algebroids.
method Defining twisted cohomology by an odd cocycle in Lie algebroid cohomology.
result Twisted cohomology only depends on the Lie algebroid cohomology class of the odd cocycle.
Researchers twist Deligne cohomology for the first time.
problem No new problem introduced; existing Deligne cohomology is already versatile.
method Explicitly twisted Deligne cohomology by taking degree one twists of integral and de Rham cohomology.
result New properties of twisted Deligne cohomology are presented and illustrated.
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p, t^2 + kappa t + 1), where p is a prime number and t^2 + kappa t + 1 is irreduci…
Extends cohomology theory for infinite volume transformation groups.
problem Generalizing cohomology for infinite volume transformation groups.
method Introduces norm-controlled cohomology as a generalization of bounded cohomology.
result Establishes norm-controlled cohomology for infinite volume transformation groups.
New cohomology theory for Lie 2-algebras extends classical theory.
problem Classical cohomology theory limitations for Lie 2-algebras.
method Introduced a new cohomology theory for Lie 2-algebras.
result Second cohomology group classifies extensions of Lie 2-algebras.
New metric space cohomology relates to Riemannian manifold cohomology.
problem Understanding cohomology structures on Riemannian and contact manifolds.
method Relating Lq,p-cohomology to ℓq,p-cohomology and proving quasi-isometry invariance. result Quasi-isometry invariance and multiplicative structure of Lq,p-cohomology. Generics extended to new cohomologies.
problem Extending classical genera to new cohomologies.
method Constructing generalised characteristic classes for bordism cohomologies.
result Natural extension of classical genera to new cohomologies.