arXiv research

New proof of index theorem for topological manifold bundles.

Study parametrized cobordism categories for smooth bundles, proving index theorem.

Bökstedt and Madsen defined an infinite loop map from the embedded $d$-dimensional cobordism category of Galatius, Madsen, Tillmann and Weiss to the algebraic $K$-theory of $BO(d)$ in the sense of Waldhausen. The purpose of this paper is to establish two results in relation to this map. The first result is that it exte…

Develops a new geometric cobordism theory using smooth Thom stacks.

The paper develops a theory for identifying the best arm in non-parametric multi-armed bandits with a fixed budget.

New map constructed from equivariant spectra for manifold study.

Study on GANs learning distributions, deriving rates and regularization.

Study shows reinforcement learning is possible with once-per-episode feedback.

We propose a theory "a la Conley" for cone fields using a notion of relaxed orbits based on cone enlargements, in the spirit of space time geometry. We work in the setting of closed (or equivalently semi-continuous) cone fields with singularities. This setting contains (for questions which are parametrization independe…

Using the descriptive method of log-periodic power laws (LPPL) based on a theory of behavioral herding, we use a battery of parametric and non-parametric tests to demonstrate the existence of an antibubble in the yields with maturities larger than 1 year since October 2000. The concept of ``antibubble'' describes the e…

Paper proposes a method to estimate Transfer Entropy using Copula Entropy.

Theory explains power-law distributions without complex models.

Proves Farrell-Jones Conjecture for specific groups and spectra.

Developed a theory of local convexity for second order differential equations on Lie algebroids.

We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of super-replication and utility maximization from terminal wealth in a bond market. Finally, w…

Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a ma…

This paper defines a theory of cobordism for virtual knots and studies this theory for standard and rotational virtual knots and links. Non-trivial examples of virtual slice knots are given. Determinations of the four-ball genus of positive virtual knots are given using the results of a companion paper by the author an…

The paper develops a theory of $C^\infty$-superrings and their superschemes.

Unified theory of optimal transport for random measures.

Levy processes, which have stationary independent increments, are ideal for modelling the various types of noise that can arise in communication channels. If a Levy process admits exponential moments, then there exists a parametric family of measure changes called Esscher transformations. If the parameter is replaced w…

Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semi-Riemannian manifolds.

A theory of feature geometry using spectral analysis of weight matrices.

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

The paper develops a theory for clustering graphs sampled from a graphon model.

We develop a theory of `non-uniformly local' tent spaces on metric measure spaces. As our main result, we give a remarkably simple proof of the atomic decomposition.

We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology.

SSL theory improves representation learning from raw data.

We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…

A theory of deep learning is emerging, focusing on training dynamics and statistics.

We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in $\mathcal{E}_n$-algebra.

We develop a theory for solving continuous time optimal stopping problems for non-linear expectations. Our motivation is to consider problems in which the stopper uses risk measures to evaluate future rewards.

Develops a theory to make learning solutions fair and safe.

This paper connects gerbes and uncertainty in particle physics.

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allow…

A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes a relatively accessible example of such a thing, suggested by the wall-crossing formulas of Donaldson theory.

The study provides a theory for causal machine learning with generalization bounds.

The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that predicts optimally. An analogous result is shown for universal induction.

Theory broadens GFlowNets to handle continuous spaces.

Developed a theory of ultradifferentiable sheafs with applications.

Financial derivatives have often been criticized as casino-style betting instruments. It turns out that many naive ways of making them are indeed equivalent to gambling. Fortunately, this inadvertent effect can be understood and prevented. We present a theory of product design which achieves that.

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

The paper develops a theory for speculative decoding acceptance criteria.

We consider the Frenet-Serret geometry of null curves in a three and a four-dimensional Minkowski background. We develop a theory of deformations adapted to the Frenet-Serret frame. We exploit it to provide a Lagrangian description of the dynamics of geometric models for null curves.

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.

In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.

We develop a theory of Nobeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina and is its counterpart in the realm of complete spaces. In particular, the Nobeling manifold characterization conjecture is proven.

This paper develops a theory of graded manifolds in differential geometry.

The paper develops a theory of surplus invariance in vector lattices.