arXiv subjects

Dennis Sullivan

Publications and source records attributed to Dennis Sullivan.

At least 19 recordsLinked to original sources

arXiv2025

Coding with the transverse intersection algebra

The concept of a fluid algebra was introduced by Sullivan over a decade ago as an algebraic construct which contains everything necessary in order to write down a form of the Euler equation, as an ODE whose solutions have invariant quantities which can be identified as energy and enthalpy. The natural (infinite-dimensional) fluid algebra on co-exact 1-forms on a three-dimensional closed oriented Riemannian manifold leads to an Euler equation which is equivalent to the classical Euler equation which describes non-viscous fluid flow. In this paper, the recently introduced transverse intersection algebra associated to a cubic lattice of An-Lawrence-Sullivan is used to construct a finite-dimensional fluid algebra on a cubic lattice (with odd periods). The corresponding Euler equation is an ODE which it is proposed is a `good' discretisation of the continuum Euler equation. This paper contains all the explicit details necessary to implement numerically the corresponding Euler equation. Such an implementation has been carried out by our team and results are pending.

math.AP↗

arXiv2025

Infinite-order combinatorial Transverse Intersection Algebra TIA via the probabilistic wiggling model

This paper constructs a graded-commutative, associative, differential Transverse Intersection Algebra TIA {on the torus (in any dimension) with its cubical decomposition by using a probabilistic wiggling interpretation. This structure agrees with the combinatorial graded intersection algebra (graded by codimension) defined by transversality on pairs of `cuboidal chains' which are in general position. In order to define an intersection of cuboids which are not necessarily in general position, the boundaries of the cuboids are considered to be `wiggled' by a distance small compared with the lattice parameter, according to a suitable probability distribution and then almost always the wiggled cuboids will be in general position, producing a transverse intersection with new probability distributions on the bounding sides. In order to make a closed theory, each geometric cuboid appears in an infinite number of forms with different probability distributions on the wiggled boundaries. The resulting structure is commutative, associative and satisfies the product rule with respect to the natural boundary operator deduced from the geometric boundary of the wiggled cuboids. This TIA can be viewed as a combinatorial analogue of differential forms in which the continuity of space has been replaced by a lattice with corrections to infinite order. See the comparison to Whitney forms at the end of the paper. For application to fluid algebra we also consider the same construction starting with the $2h$ cubical complex instead of the $h$ cubical complex. The adjoined higher order elements will be identical to those required in the $h$ cubical complex. The $d$-dimensional theory is a tensor product of $d$ copies of the one-dimensional theory.

math.AT↗

arXiv2025

The combinatorial transverse intersection algebra

This paper constructs (with challenging obstacles) on the three torus with its cubical decomposition: Firstly, a combinatorial graded intersection algebra (graded by the codimension) which is commutative and associative defined by transversality on the usual chains which are in general position. This, (with extra elements added) on the entire $h$-cubulated three torus whose differential satisfies the product rule and which agrees with the set theoretic intersection product appropriately weighted. The construction is characterized given these properties (see Comprehensive Theorem below). The challenge is to minimally adjoin infinitesimal elements when the geometric elements have glancing but transversal intersections weighted in such a way that the associativity (and commutativity) is not destroyed and the Leibniz product rule for the boundary operator is restored. Secondly, there is a $2h$ subcomplex introduced in Sullivan arXiv:1811.00086 and discussed further in Lawrence-Sullivan-Ranade arXiv:2011.07505 which shares the above good properties when ideal elements are added AND which also has a star bijection between degree zero and degree three and between one and degree two. This is introduced for the purposes of computations of 3D fluid motion, incompressible, with or without viscosity. For the latter purposes one only needs the good properties in dimensions zero, one and two, where the situation is a bit better. It is a new feature that the three good properties are respected by the crumbling chain mappings from coarse to finer subdivisions. The star operator does not cooperate with crumbling and is the sole reason in this discrete approximation for the Kolmogorov cascade to finer scales.

math.GT↗

arXiv2019

Structure on the Top Homology and Related Algorithms

We explore the special structure of the top-dimensional homology of any compact triangulable space $X$ of dimension $d$. Since there are no $(d+1)$-dimensional cells, the top homology equals the top cycles and is thus a free abelian group. There is no obvious basis, but we show that there is a canonical embedding of the top homology into a canonical free abelian group which has a natural basis up to signs. This embedding structure is an invariant of $X$ up to homeomorphism. This circumstance gives the top homology the structure of an (orientable) matroid, where cycles in the sense of matroids correspond to the cycles in the sense of homology. This adds a novel topological invariant to the topological literature. We apply this matroid structure on the top homology to give a polynomial-time algorithm for the construction of a basis of the top homology (over $\mathbb{Z}$ coefficients).

math.AT↗

arXiv2019

Compl\'ement to the Thurston 3D-Geometrization

Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries" (see Introduction and Figure 1). These geometric components have finite volume for the hyperbolic geometries (the H labeled vertices). They also have finite volume for each of the two geometries appearing as Seifert fibrations (the S labeled vertices). The remaining pieces (the I labeled vertices) have Euclidean geometries of linear volume growth. Then these vertex geometries are combined topologically to recover the original manifold. This, by cutting off the toroidal ends and then gluing the torus boundaries by affine mappings (indicated by the labeled edges in Figure 1). The point of this work is to make the affine gluing respect an interpretation of the metric geometry in terms of a new notion of `` regional Lie generated geometry". The vertex regions use four geometries in Lie form combined in the overlap edge regions via affine geometry. The Theorem solves, using Geometrization, a 45 year old question/approach to the Poincar\'{e} Conjecture. This was described in a '76 Princeton Math dept. preprint and finally documented in the 1983 reference by Thurston and the second author.

math.GT↗

arXiv2018

Lattice Hydrodynamics

Using the combinatorics of two interpenetrating face centered cubic lattices together with the part of calculus naturally encoded in combinatorial topology, we construct from first principles a lattice model of 3D incompressible hydrodynamics on triply periodic three space. Actually the construction applies to every dimension, but has special duality features in dimension three.

math.AP↗

arXiv2018

Characters for Complex Bundles and their Connections

The paper combines several fortunate mini miracles to achieve its two objectives. These were woven together in a several year's effort to answer a question raised by Iz Singer a decade ago. Our answer is accessible to the topologist, to the differential geometer and to the analyst who appreciates the statement of the Index theorem of Atiyah,Patodi,Singer for manifolds with boundary. The mini miracles are these: a] The Conner Floyd miracle that complex bordism tensored over the Todd genus and the Bott miracle that stable complex vector bundles respectively satisfy the axioms of a generalized homology theory and of a generalized cohomology theory. b] That these theories, with the covariant and contravariant geometric representations indicated, stably almost complex (SAC) manifolds modulo product relations and stable complex bundles, are not only related by Alexander duality but they are also related by Pontryagin duality. c] The abstract corollary of b] that stable complex bundles have a complete system of numerical invariants and that these can be computed by integrals of chern weil characteristic forms over manifolds with boundary reduced modulo integers, thanks to the APS Index Theorem. d] The adiabatic limit argument of the appendix to the last section showing a direct sum connection on the total space of a riemannian family of Riemannian manifolds with connection is Chern Simons equivalent in the limit to the Levi Civita connection of the direct sum metric. This allows the invariants to be described by the eta invariants of odd SAC manifolds reduced mod integers.

math.KT↗

arXiv2017

Generalized Euler classes, differential forms and commutative DGAs

In the context of commutative differential graded algebras over $\mathbb Q$, we show that an iteration of "odd spherical fibration" creates a "total space" commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its "fibrations" with "fiber" of finite cohomological dimension, the induced map on cohomology is injective.

math.AT↗

arXiv2017

A formula for topology/deformations and its significance

The formula is $\partial{e}=({\rm ad}_e)b+\sum_{i=0}^\infty{\frac{B_i}{i!}}({\rm ad}_e)^i(b-a)\>,$ with $\partial{a}+{1\over2}[a,a] =0$ and $\partial{b}+{1\over2}[b,b] =0$, where $a$, $b$ and $e$ in degrees $-1$, $-1$ and 0 are the free generators of a completed free graded Lie algebra $L[a,b,e]$. The coefficients are defined by ${x\over{e^x-1}}=\sum_{n=0}^\infty{B_n\over{}n!}x^n$. The theorem is that (I) this formula for $\partial$ on generators extends to a derivation of square zero on $L[a,b,e]$, (II) the formula for $\partial{e}$ is unique satisfying the first property, once given the formulae for $\partial{a}$ and $\partial{b}$, along with the condition that the "flow" generated by $e$ moves $a$ to $b$ in unit time. The immediate significance of this formula is that it computes the infinity cocommutative coalgebra structure on the chains of the closed interval. It may be derived and proved using the geometrical idea of flat connections and one parameter groups or flows of gauge transformations. The deeper significance of such general DGLAs which want to combine deformation theory and rational homotopy theory is proposed as a research problem.

math.AT↗

arXiv2016

Hybrid quantum systems for enhanced nonlinear optical susceptibilities

Significant effort has been expended in the search for materials with ultra-fast nonlinear-optical susceptibilities, but most fall far below the fundamental limits. This work applies a theoretical materials development program that has identified a promising new hybrid made of a nanorod and a molecule. This system uses the electrostatic dipole moment of the molecule to break the symmetry of the metallic nanostructure that shifts the energy spectrum to make it optimal for a nonlinear-optical response near the fundamental limit. The structural parameters are varied to determine the ideal configuration, providing guidelines for making the best structures.

physics.optics↗

arXiv2014

The Cumulant Bijection and Differential Forms

According to Jae Suk Park, physicists use "canonical coordinate systems" to compute correlations in perturbative quantum field theories. One may interpret these canonical coordinate systems as equivalences of generalized differential Lie algebras. In this note we discuss these flattenings in one particular setting and refer to them as "cumulant bijections". The main point we make is that these cumulant bijections are functorial for deformation retracts. The discussion is completely self contained and based on well known universal properties.

math.AT↗

arXiv2013

Transverse string topology and the cord algebra

We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.

math.GT↗

arXiv2010

The Mayer-Vietoris Property in Differential Cohomology

In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differential cohomology functor J^ associated to any Z-graded cohomology functor J(, Z) which, in each degree, assigns to a point a finitely generated group. The approach is to show that the result follows from Diagram 1, the commutative diagram we take as a definition of differential cohomology, and Diagram 2, which combines the three Mayer-Vietoris sequences for J*(, Z), J*(, R) and J*(, R/Z).

math.AT↗

arXiv2010

A finite time blowup result for quadratic ODE's

This short paper is dedicated to Mauricio Peixoto who abstracted the practical theory of ODE's and to David Rand who applied the subsequent powerful abstract theory to practical problems.

math.AT↗

arXiv2010

The Mayer-Vietoris Property in Differential Cohomology

In [1] it was shown that K^, a certain differential cohomology functor associated to complex K-theory, satisfies the Mayer-Vietoris property when the underlying manifold is compact. It turns out that this result is quite general. The work that follows shows the M-V property to hold on compact manifolds for any differential cohomology functor J^ associated to any Z-graded cohomology functor J(,Z) which, in each degree, assigns to a point a finitely generated group. The approach is to show that the result follows from Diagram 1, the commutative diagram we take as a definition of differential cohomology, and Diagram 2, which combines the three Mayer-Vietoris sequences for J*(,Z), J*(,R) and J*(,R/Z).

math.DG↗

arXiv2010

Algebra, Topology and Algebraic Topology of 3D Ideal Fluids

There is a remarkable and canonical problem in 3D geometry and topology: To understand existing models of 3D fluid motion or to create new ones that may be useful. We discuss from an algebraic viewpoint the PDE called Euler's equation for incompressible frictionless fluid motion. In part I we define a "finite dimensional 3D fluid algebra", write its Euler equation and derive properties related to energy, helicity, transport of vorticity and linking that characterize this equation. This is directly motivated by the infinite dimensional fluid algebra associated to a closed riemannian three manifold whose Euler equation as defined above is the Euler PDE of fluid motion. The classical infinite dimensional fluid algebra satisfies an additional identity related to the Jacobi identity for the lie bracket of vector fields. In part II we discuss informally how this Jacobi identity can be reestablished in finite dimensional approximations as a Lie infinity algebra. The main point of a developed version of this theory would be a coherence between various levels of approximation. It is hoped that a better understanding of the meaning of the Euler equation in terms of such infinity structures would yield algorithms of computation that work well for conceptual reasons.

math.AT↗

arXiv2008

Structured vector bundles define differential K-theory

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.

math.AT↗

arXiv2008

The homotopy invariance of the string topology loop product and string bracket

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h_* that supports an orientation of the M_i 's.

math.GT↗