Related papers: Mental Geometry
We consider the problem of learning a manifold from a teacher's demonstration. Extending existing approaches of learning from randomly sampled data points, we consider contexts where data may be chosen by a teacher. We analyze learning from…
In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into…
Geometric diagrams are critical in conveying mathematical and scientific concepts, yet traditional diagram generation methods are often manual and resource-intensive. While text-to-image generation has made strides in photorealistic…
Clifford geometric algebras of multivectors are treated in detail. These algebras are build over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear, that…
Our starting point is a particular `canvas' aimed to `draw' theories of physics, which has symmetric monoidal categories as its mathematical backbone. In this paper we consider the conceptual foundations for this canvas, and how these can…
In the limited workspace model, we consider algorithms whose input resides in read-only memory and that use only a constant or sublinear amount of writable memory to accomplish their task. We survey recent results in computational geometry…
This paper wants to show how practical geometry, created to give a concrete help to people involved in trade, in land-surveying and even in astronomy, underwent a transformation that underlined its didactical value and turned it first into…
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…
Neural algorithmic reasoning aims to capture computations with neural networks by training models to imitate the execution of classical algorithms. While common architectures are expressive enough to contain the correct model in the weight…
Large Language Models (LLMs) demonstrate ever-increasing abilities in mathematical and algorithmic tasks, yet their geometric reasoning skills are underexplored. We investigate LLMs' abilities in constructive geometric problem-solving one…
"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing,…
Generative AI has transformed the economics of information production, making explanations, proofs, examples, and analyses available at very low cost. Yet the value of information still depends on whether downstream users can absorb and act…
In this paper, we focus on the two tasks of 3D shape abstraction and semantic analysis. This is in contrast to current methods, which focus solely on either 3D shape abstraction or semantic analysis. In addition, previous methods have had…
How do we imagine visual objects and combine them to create new forms? To answer this question, we need to explore the cognitive, computational and neural mechanisms underlying imagery and creativity. The body of research on deep learning…
Mathematics is often perceived as difficult or inaccessible, yet meaningful engagement can arise in unexpected places. In this article we describe a multi-year exploration of mathematical outreach through games, puzzles, exhibitions, and…
We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can…