arXiv research

Homology of partition algebras matches symmetric group homology under certain conditions.

Homologies of Jones and partition algebras match cyclic and symmetric groups.

A new algebra for Frobenius manifolds solves PDEs and constraints.

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.

New method connects neural networks to diagrammatic algebra.

New basis for permutation equivariant layers reduces computation costs.

Abstract studies 3-manifolds and vertex algebras, expanding known connections.

Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence between such manifold submetries and maximal Laplacian alge…

New bound on partition function proves Kähler-Einstein stability.

We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1)…

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and optimization applications. The problem is of an exponential complexity with respect to t…

Proves super-version of index theorem from algebraic cobordism invariants.

New algebra defined for Legendrian submanifolds, preserving key invariants.

We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from a closed one, but it is then assumed that the removed set is endowed with a part…

Let $M$ be a complete Riemannian manifold and assume that $M$ is partitioned by a hypersurface $N$. In this paper we introduce a novel class of functions $C_{\mathrm{w}}(M)$ on noncompact manifolds, which is slightly larger than the algebra of Higson functions. Out of $φ$ that belongs to $C_{\mathrm{w}}(M)$ we construc…

Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a Hermitean bundle with connection on M. We define a coarse multi-partitioned index of the…

We propose a new description of 3d $\mathcal{N}=2$ theories which do not admit conventional Lagrangians. Given a quiver $Q$ and a mutation sequence $m$ on it, we define a 3d $\mathcal{N}=2$ theory $\mathcal{T}[(Q,m)]$ in such a way that the $S^3_b$ partition function of the theory coincides with the cluster partition f…

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogu…

Generalizes Roe's theorem to noncompact hypersurfaces.

Abstract M5 branes on ADE singularities yields BPS spectrum and partition functions.

Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.

A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically, algebraic conditions for their symplecticity are derived.

We construct a topological Chern-Simons sigma model on a Riemannian three-manifold M with gauge group G whose hyperkahler target space X is equipped with a G-action. Via a perturbative computation of its partition function, we obtain new topological invariants of M that define new weight systems which are characterized…

The paper calculates a formula for knot complements using holomorphic curves.

Eguchi-Hori-Xiong and S. Katz proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. In this paper, we give a proof to this conjecture for the genus 0 part.

A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complet…

Novel threefold partitioning of $L^2$-spinor space on cone links.

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack la…

We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to braids with 4 strands and more. The proof uses a partition of the Cayley graph …

Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue of the normalized laplacian matrix of G. There is a basic fact in algebraic graph theory that lambda_k > 0 if and only if G has at most k-1 connected components. We prove a robust version of this fact. If lambda_k>0, then for some 1\leq \ell\l…

We outline a proof of a remarkable conjecture of Labastida-Mari{ñ}o-Ooguri-Vafa about certain new algebraic structures of quantum link invariants and the integrality of infinite family of new topological invariants. Our method is based on the cut-and-join analysis and a special rational ring characterizing the structur…

Mathematical construction of Chern-Simons partition function using reflection positivity.

For a topological space $X$, we introduce a criterion for the $\rm FI$ module $H^i({\rm Conf}_n(X))$ to be finitely generated and give several applications. For instance, if $C$ is a finite connected $CW$ complex, then $X = C \times \mathbb{R}^2$ satisfies the criterion. Our main tool is a spectral sequence that we der…

Develops a Gaussian model to compute the Alexander polynomial of knots.

It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quate…

Enhances POU-Nets with probabilistic noise model for efficient spatial data clustering.

Clarifies the structure of quantum states using algebraic methods.

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed f…

Virtual links were introduced by Kauffman in 1999. We characterize the virtual link invariants that are partition functions of vertex models (as considered by de la Harpe and Jones), both in the real and in the complex case. We show that for any fixed number of states, these invariants form an affine variety. Basic tec…

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many…

Network detection is an important capability in many areas of applied research in which data can be represented as a graph of entities and relationships. Oftentimes the object of interest is a relatively small subgraph in an enormous, potentially uninteresting background. This aspect characterizes network detection as …

A new filter reduces density fitting to a linear solve, improving performance on nonlinear systems.

We study singularities of algebraic curves associated with 3d N=2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T_K labeled by knots, whose partition functions package Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the defining equation, call…

This paper provides a construction of a quantum statistical mechanical system associated to knots in the 3-sphere and cyclic branched coverings of the 3-sphere, which is an analog, in the sense of arithmetic topology, of the Bost-Connes system, with knots replacing primes, and cyclic branched coverings of the 3-sphere …

Given a point (the "spider") on a rectangular box, we would like to find the minimal distance along the surface to its opposite point (the "fly" - the reflection of the spider across the center of the box). Without loss of generality, we can assume that the box has dimensions $1\times a\times b$ with the spider on one …

Given an open cover of a paracompact topological space X, there are two natural ways to construct a map from the cohomology of the nerve of the cover to the cohomology of X. One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the Mayer-Vietoris double complex, …

The study limits how many parts regular simplicial partitions can overlap.

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received …