arXiv research

Schmutz Schaller and Thurston's approaches are dual.

Sum of Lagrange numbers equals a specific formula.

The study constructs hyperbolic manifolds with many systoles.

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.

Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C…

New hyperbolic 3-manifolds have many neighbors.

For any $\varepsilon>0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most $\varepsilon g$ for the systole fun…

Study finds bounds for systole length on arithmetic punctured spheres.

The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like $g^{\s…

Small sets of systoles fill hyperbolic surfaces of large genus.

The trace set of a Fuchsian group $Γ$ ist the set of length of closed geodesics in the surface $Γ\backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithm…

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that…

The diameter function is a topological Morse function.

Post-pandemic, work patterns shifted with fewer days in offices and a new midweek mountain.

Women belonging to the socially disadvantaged caste-groups in India have historically been engaged in labour-intensive, blue-collar work. We study whether there has been any change in the ability to predict a woman's work-status and work-type based on her caste by interpreting machine learning models using feature attr…

In this work we propose a heteroscedastic generalization to RVM, a fast Bayesian framework for regression, based on some recent similar works. We use variational approximation and expectation propagation to tackle the problem. The work is still under progress and we are examining the results and comparing with the prev…

This paper describes the work of Jesse Douglas on the Plateau problem, work for which he was awarded a Fields Medal in 1936, and considers the contributions Tibor Rado made in the 1930s.

Explains Uhlenbeck's work on maps and fields.

We discuss possible extensions of the classical Chern-Weil formalism to an infinite dimensional setup. This is based on joint work with Steven Rosenberg, joint work with Simon Scott and joint work with Jouko Mickelsson.

We present Michel Kervaire work on knots in higher dimensions.

Neumann's work connects 3-manifold invariants to quantum topology.

New proof shows spacetime energy is always positive in higher dimensions.

This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.

This paper applies quantum theory to cost accounting, focusing on WIP valuation.

Since the first work of Thomas Friedrich showing that isometric immersions of Riemann surfaces are related to spinors and the Dirac equation, various works appeared generalizing this approach to more general Spin-manifolds, in particular the case of submanifolds of Spin-manifolds of constant curvature. In the present w…

This is the first in a planned series of papers giving an alternate approach to Zlil Sela's work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a finitely generated group and F is a non-abelian free group.

Work maximization guides machine learning models in adaptive systems.

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

Consequential decision-making incentivizes individuals to strategically adapt their behavior to the specifics of the decision rule. While a long line of work has viewed strategic adaptation as gaming and attempted to mitigate its effects, recent work has instead sought to design classifiers that incentivize individuals…

The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presente…

We review recent works on analyzing the dynamics of gradient-based algorithms in a prototypical statistical inference problem. Using methods and insights from the physics of glassy systems, these works showed how to understand quantitatively and qualitatively the performance of gradient-based algorithms. Here we review…

Explores new perspectives in transverse index theory for Lie group actions.

First working model of a monostable tetrahedron described.

This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman--Turaev Lie bia…

RSAC improves lightweight continuous learning efficiency.

We survey Mirzakhani's work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of non-experts.

Study confirms asymptotic behavior of logarithmic balanced metric near infinity.

By reducing optimization to a sequence of smaller subproblems, working set algorithms achieve fast convergence times for many machine learning problems. Despite such performance, working set implementations often resort to heuristics to determine subproblem size, makeup, and stopping criteria. We propose BlitzWS, a wor…

In this work, we are going to find sufficient conditions on the initial triaxial Bianchi IX metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and…

In this work we offer a framework for reasoning about a wide class of existing objectives in machine learning. We develop a formal correspondence between this work and thermodynamics and discuss its implications.

We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi.

The average and median income dependence on work experience and time is analyzed and modeled for the USA. The original data set providing the mean and median income estimates in 10 year long intervals spans a long time period of almost 35 years ? from 1967 to 2003. A microeconomic model linking personal income, populat…

This paper is mostly a survey of recent work on sequences of locally symmetric spaces whose Riemannian volume goes to infinity. We also work out some applications to random surfaces.

This work simplifies IRL by using potential-based reward shaping.

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admis…

For each graph, we construct a bigraded chain complex whose graded Euler characteristic is a version of the Tutte polynomial. This work is motivated by earlier work of Khovanov, Helme-Guizon and Rong, and others.

Typical neural networks with external memory do not effectively separate capacity for episodic and working memory as is required for reasoning in humans. Applying knowledge gained from psychological studies, we designed a new model called Differentiable Working Memory (DWM) in order to specifically emulate human workin…

Clarifies structures in link homology theories using Frobenius extensions.