arXiv research

The paper connects Riemann surface length spectra to Brownian loop measures.

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central…

For non-smooth surfaces, the measure of Brownian loops is derived using the Polyakov-Alvarez formula.

Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.

Study the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.

Study on the mass of Brownian loops on Riemann surfaces as genus grows.

Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …

Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring comodules, enabling curvature to be defined under general algebraic circumstanc…

Framework learns robust control policies from expert demonstrations.

New algorithms reduce dynamic regret in online MDPs with changing losses.

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge κ$, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of $κ$, the dimension, di…

We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …

New framework uses simplicial and categorical methods to detect market inconsistencies.

In this article, we discuss a probabilistic interpretation of McShane's identity as describing a finite measure on the space of embedded paths though a point.

Let $G$ be a simply connected compact Lie group. Let $L_e(G)$ be the based loop group with the base point $e$ which is the identity element. Let $ν_e$ be the pinned Brownian motion measure on $L_e(G)$ and let $α\in L^2(\wedge^1T^{\ast}L_e(G),ν_e)\cap {\mathbb D}^{\infty,p}(\wedge^1T^{\ast}L_e(G),ν_e)$ $(1<p<2)$ be a cl…

Theoretical model for iterative user discovery in recommender systems.

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…

Given a compact orientable surface $Σ$, let $\Cal S(Σ)$ be the set of isotopy classes of essential simple loops on $Σ$. We determine a complete set of relations for a function from $\Cal S(Σ)$ to $\bold Z$ to be a geometric intersection number function. As a consequence, we obtain explicit equations in $\bold R^{\Cal S…

New algorithm LOOP learns infinite-horizon AMDPs efficiently with function approximation.

Develops a framework for analyzing multi-agent and many-body systems with feedback loops.

The Turaev cobracket, a loop operation introduced by V. Turaev, which measures self-intersection of a loop on a surface, is a modification of a path operation introduced earlier by Turaev himself, as well as a counterpart of the Goldman bracket. In this survey based on the author's joint works with A. Alekseev, Y. Kuno…

PDA method optimizes neural networks with global convergence rate analysis.

This paper is an exposition of the relationship between Witten's Chern-Simons functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory…

Study smooth loops and loop bundles, relating to $G_{2}$-structures.

Novel algorithm reduces computational burden in IRL with finite-time guarantees.

This work discusses a closed-loop control strategy for complex systems utilizing scarce and streaming data. A discrete embedding space is first built using hash functions applied to the sensor measurements from which a Markov process model is derived, approximating the complex system's dynamics. A control strategy is t…

We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent ind…

The paper proves T-duality and Hori formulae for winding loop spaces.

Paper establishes loop space T-duality formulae and refines earlier work.

What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…

The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.

Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational…

Kähler manifold loop space inherits Kähler structure and is complete.

We propose a method for learning cyclic causal models from a combination of observational and interventional equilibrium data. Novel aspects of the proposed method are its ability to work with continuous data (without assuming linearity) and to deal with feedback loops. Within the context of biochemical reactions, we a…

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…

Introduces string structures linking to loop spaces.

A method for fair representation learning through bi-level optimization and implicit differentiation.

Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

Training-free looped transformers improve model performance without additional training.

Study path spaces and their homology, extending loop products and coproducts.

This paper reformulates the $p$-adic Littlewood Conjecture using infinite loops.

Uniform systole bounds for arithmetic orbifolds and number fields.

New basis and Schur-Weyl duality for loop Hecke algebra defined.

A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry ov…