arXiv research

Resolves anomaly cancellation for M5-brane in M-theory.

We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).

We extend Matveev's theory of complexity for 3-manifolds, based on simple spines, to (closed, orientable, locally orientable) 3-orbifolds. We prove naturality and finiteness for irreducible 3-orbifolds, and, with certain restrictions and subtleties, additivity under orbifold connected sum. We also develop the theory of…

In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the su…

Enhances content moderation with culturally-aware models.

This paper surveys fairness notions in ML and recommends the most suitable one for real-world scenarios.

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing rep…

An NQ-manifold is a non-negatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual…

Detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency Deutsche Aktienindex data. The tail index ($α$), the Renyi exponents based on the box counting algorithm for the graph ($d_q$) and the generalized Hurst ex…

We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoret…

Survey on metric SYZ conjecture and non-archimedean geometry.

Ambitwistor string matches superstring chiral integrands at zero tension.

Options financial instruments designed to protect investors from the stock market randomness. In 1973, Fisher Black, Myron Scholes and Robert Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation. Herein, we derive the Black-Scholes equation for the o…

These are the notes of the lectures delivered by the author at CIME in June 2018. The main purpose of the notes is to provide an overview of the techniques used in the construction of the triply graded link homology. The homology is space of global sections of a particular sheaf on the Hilbert scheme of points on the p…

Generative models with both discrete and continuous latent variables are highly motivated by the structure of many real-world data sets. They present, however, subtleties in training often manifesting in the discrete latent being under leveraged. In this paper, we show that such models are more amenable to training whe…

Recent work has demonstrated that embeddings of tree-like graphs in hyperbolic space surpass their Euclidean counterparts in performance by a large margin. Inspired by these results and scale-free structure in the word co-occurrence graph, we present an algorithm for learning word embeddings in hyperbolic space from fr…

Fractal learning rate schedules accelerate vanilla gradient descent.

Probabilistic classifiers output a probability distribution on target classes rather than just a class prediction. Besides providing a clear separation of prediction and decision making, the main advantage of probabilistic models is their ability to represent uncertainty about predictions. In safety-critical applicatio…

We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-…

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal $G$-bundle with connection and a class in $H^4(BG, \ZZ)$ for a compact semi-simple Lie group $G$. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply …

The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwi…

We analyse the moduli spaces of superconformal field theories (SCFTs). For N=2 we find an enhanced moduli space which in geometrical terms corresponds to tori with two independent complex structures. To explain the precise relation with the moduli space of SCFTs on K3 surfaces as described by Aspinwall and Morrison, we…

This article reviews mathematical insights into neural networks and machine learning.

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra v and its automorphism 2-group Aut(v). Gauge transformat…

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion …

Study non-orientable link cobordisms using Floer homologies to prove inequalities.

In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a …

New tool lassos connects virtual and surface link diagrams, changing primeness rules.

Recent work on fairness in machine learning has focused on various statistical discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute, features, and outcome. While convenient to work with, observational crite…

Based on the Multifractal Detrended Fluctuation Analysis (MFDFA) and on the Wavelet Transform Modulus Maxima (WTMM) methods we investigate the origin of multifractality in the time series. Series fluctuating according to a qGaussian distribution, both uncorrelated and correlated in time, are used. For the uncorrelated …

There has recently been much work on the "wide limit" of neural networks, where Bayesian neural networks (BNNs) are shown to converge to a Gaussian process (GP) as all hidden layers are sent to infinite width. However, these results do not apply to architectures that require one or more of the hidden layers to remain n…

In [4]: `The Riley slice of Schottky space', (Proc. London Math. Soc. 69 (1994), 72-90), Keen and Series analysed the theory of pleating coordinates in the context of the Riley slice of Schottky space R, the deformation space of a genus two handlebody generated by two parabolics. This theory aims to give a complete des…

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is relate…

We solve structure learning for cyclic linear causal models using observational data.

New approach confirms Kruskal-Szekeres extension for Schwarzschild spacetime.

Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Fin…

For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…

New methods avoid spectral pollution in transfer operators for accurate analysis.

Unified framework for risk evaluation under uncertainty.

New insights on bias in multi-armed bandits under conditional sampling.

A graphical model is a structured representation of the data generating process. The traditional method to reason over random variables is to perform inference in this graphical model. However, in many cases the generating process is only a poor approximation of the much more complex true data generating process, leadi…

GPflux simplifies deep Gaussian processes for Python.

Machine learning (ML) and artificial intelligence (AI) algorithms are now being used to automate the discovery of physics principles and governing equations from measurement data alone. However, positing a universal physical law from data is challenging without simultaneously proposing an accompanying discrepancy model…

``An orbifold is a space which is locally modeled on the quotient of a vector space by a finite group.'' This sentence is so easily said or written that more than one person has missed some of the subtleties hidden by orbifolds. Orbifolds were first introduced by Satake under the name ``V-manifold'' and rediscovered by…

Algorithms for hyperparameter optimization abound, all of which work well under different and often unverifiable assumptions. Motivated by the general challenge of sequentially choosing which algorithm to use, we study the more specific task of choosing among distributions to use for random hyperparameter optimization.…

In this article, we study the convergence of Mirror Descent (MD) and Optimistic Mirror Descent (OMD) for saddle point problems satisfying the notion of coherence as proposed in Mertikopoulos et al. We prove convergence of OMD with exact gradients for coherent saddle point problems, and show that monotone convergence on…

Much attention has been devoted recently to the development of machine learning algorithms with the goal of improving treatment policies in healthcare. Reinforcement learning (RL) is a sub-field within machine learning that is concerned with learning how to make sequences of decisions so as to optimize long-term effect…

Quite a number of distinct versions of Bartnik's definition of quasi-local mass appear in the literature, and it is not a priori clear that any of them produce the same value in general. In this paper we make progress on reconciling these definitions. The source of discrepancies is two-fold: the choice of boundary cond…