Related papers: Some Comments on Multiple Discovery in Mathematics

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition…

While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many…

Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call "mathematical data science". In this paradigm, one studies mathematical objects collectively rather than…

As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to…

We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are introduced. In fact, infinitely many new…

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…

Based on various strategies and a new general doubling operator, we obtain several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem. A simple non-directed graph proof which is especially suitable for a calculus course…

Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…

$ $[This paper is a (self contained) chapter in a new book, Mathematics and Computation, whose draft is available on my homepage at https://www.math.ias.edu/avi/book ]. We survey some concrete interaction areas between computational…

Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much…

We discuss coincidences of pairs (f_1, f_2) of maps between manifolds. We recall briefly the definition of four types of Nielsen numbers which arise naturally from the geometry of generic coincidences. They are lower bounds for the minimum…

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp…

This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall…

Deductive and abductive reasoning are two critical paradigms for analyzing knowledge graphs, enabling applications from financial query answering to scientific discovery. Deductive reasoning on knowledge graphs usually involves retrieving…

Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We…

Philosophers now seem to agree that frequentism is an untenable strategy to explain the meaning of probabilities. Nevertheless, I want to revive frequentism, and I will do so by grounding probabilities on typicality in the same way as the…

Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some…

Some tools and ideas are interchanged between random matrix theory and multivariate statistics. In the context of the random matrix theory, classes of spherical and generalised Wishart random matrix ensemble, containing as particular cases…