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.
Introduces minimal surfaces to undergraduates.
Invites readers to a mathematical journey with historical and contemporary mathematicians.
These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The…
Teaches advanced differential topology to master students with minimal background.
Derives Black-Scholes model using central limit theorem.
This book teaches differential geometry of curves and surfaces in 3D Euclidean space.
This article simplifies differential geometry concepts for physics students.
By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the …
Benchmark for math reasoning models from human proofs.
Explains Arnold's J+ invariant for curves, using basic math.
An introductory course on hyperbolic geometry for advanced students.
Brief introduction to deep learning for math students.
Introduces geometric control theory for students.
This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced undergraduate and beginning graduate students. Applications include brief introductions to …
Undergraduate proves Morse inequalities using Witten's method.
A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.
Baccalaureate institutions seek to integrate statistics into data science.
This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular for studying constant mean curvature (CMC) surfaces, that was developed by J. Dorfmeister, F. Pedit and H. Wu. There already exist a number of other introductions to this method, but all of them require a …
These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphas…
A self-contained introduction is presented of the notion of the (abstract) differentiable manifold and its tangent vector fields. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear maps to abstract spaces (manifolds) and diffeomorphisms is emphasized. Necessar…
Bayesian model identifies skill difficulties and student subgroups in engineering education.
In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by interpreting parametrized geodesics as the paths traced out in time by the motion of…
Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
Researchers tabulate mosaic numbers of prime knots up to eight crossings.
The paper presents instructive interdisciplinary applications of constrained mechanics calculus in economics on a level appropriate for the undergraduate physics education. The aim of the paper is: 1. to meet the demand for illustrative examples suitable for presenting the background of the highly expanding research fi…
Develops a method to improve price prediction algorithms using macro-financial indicators.
In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the de…
Explains how geometry and statistics intertwine, focusing on information geometry.
This is an expository paper designed to introduce undergraduates to the Atiyah-Singer index theorem 50 years after its announcement. It includes motivation, a statement of the theorem, an outline of the easy part of the heat equation proof. It includes counting lattice points and knot concordance as applications.
Predicts student success and major using first courses.
Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.
This expository article introduces the topic of roots in a compact Lie group. Compared to the many other treatments of this standard topic, I intended for mine to be relatively elementary, example-driven, and free of unnecessary abstractions. Some familiarity with matrix groups and with maximal tori is assumed. This ar…
When undergraduates ask me what geometric group theorists study, I describe a theorem due to Gromov which relates the groups with an intrinsic geometry like that of the hyperbolic plane to those in which certain computations can be efficiently carried out. In short, I describe the close but surprising connection betwee…
Accessible proof of Gauss-Bonnet theorem using vector calculus.
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and invariants, characteristic classes). Thus main ideas of algebraic topology are presented w…
We classify invariant Lagrangians of the form depending at most quadratically on the variables and , where is a Lorentz metric and is a tensor field of arbitrary rank on a smooth manifold. As a corollary, we prove a conjecture of Bray'…
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
New proof shows a link problem is hard without complex links.
This is a survey article on Morse theory based on lectures to graduate students and advanced undergraduates. After a brief review of standard material, mostly without proofs, the Morse theory of complex Grassmannian manifolds is worked out in detail. In contrast to standard treatments, gradient flow lines and their str…
This note is purely expository. A subset N of the plane is affine ambient homogeneous if for each x,y in N there exists an affine transformation taking x to y and N to itself. The result of D. Repovs, E. V. Scepin and the author on such subsets is presented, together with discussion, corollaries and generalizations. At…
Explains the Borromean rings, icosahedron, and Poincaré homology sphere.
StoryGen uses images to generate coherent text from multiple images.
Teaches matrix calculus for machine learning and optimization.
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We rela…
This note is purely expository. In the course of the Kolmogorov-Arnold solution of Hilbert's 13th problem on superpositions there appeared the notion of basic embedding. A subset K of R^2 is basic if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for each point …
Our goal here is to discuss the pricing problem of European and American options in discrete time using elementary calculus so as to be an easy reference for first year undergraduate students. Using the binomial model we compute the fair price of European and American options. We explain the notion of Arbitrage and the…
Paper introduces a continuous convexity measure for compact sets.
New model suggests universe emerges from single particle quantum mechanics.