A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
Computes circular area and spherical volume invariants via integrals.
problem Computing circular area and spherical volume invariants for curves and surfaces.
method Using the Divergence Theorem, express area and volume integrals as line and surface integrals against kernels, then compute analytically on triangulated meshes.
result Simple algorithm for computing spherical volume invariant for triangulated surfaces without discretizing ambient space.
We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…
We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl_2. This implies an integrality result on the…
Study on integrability of geodesic flows on Heisenberg group.
problem Integrability of geodesic flows on Heisenberg group.
method Investigation of two classes of normal geodesic flows associated with left-invariant sub-Riemannian metric.
result Left-left configuration is completely integrable in non-commutative sense, while left-right configuration exhibits non-commutative integrability in dimensions > 5.
We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite ty…
In a previous paper, we generalized the definition of the framed Kontsevich integral initially presented by Le and Murakami. We also defined an isotopy invariant Zf that is well-behaved under band sum moves. Using this invariant we study the construction of the LMO invariant, the Wheeling Theorem, and th…
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the pert…
A `total Chern class' invariant of knots is defined. This is a universal Vassiliev invariant which is integral `on the level of Lie algebras' but it is not expressible as an integer sum of diagrams. The construction is motivated by similarities between the Kontsevich integral and the topological Chern character.
This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich.
We deduce from the work of Chen, that the restriction morphism from closed free iterated integrals to closed iterated integrals on loops is onto. We use this to show that the module of higher order invariants of smooth functions is generated by free closed iterated integrals.
The integrality of the Kontsevich integral and perturbative invariants is discussed. We show that the denominator of the degree n part of the Kontsevich integral of any knot or link is a divisor of (2!3!...n!)4(n+1)!. We also show that the denominator of of the degree n part of the universal perturbative invari…
We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …
We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to order six. We present an application to online (stroke-trajectory based) charac…
We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal …
A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric poss…