Related papers: Experimentation in the Schubert Calculus
Categorical Quantum Mechanics, and graphical calculi in particular, has proven to be an intuitive and powerful way to reason about quantum computing. This work continues the exploration of graphical calculi, inside and outside of the…
Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation…
We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the…
In modern science, computer models are often used to understand complex phenomena, and a thriving statistical community has grown around analyzing them. This review aims to bring a spotlight to the growing prevalence of stochastic computer…
There has been no lack of coverage in the past few years in scientific journals of the topic of quantum computation. Rightly so, as this is a novel idea with--so far--at least one very important practical application (prime factorisation)…
We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and…
Computational studies of basic models of strongly-correlated electron systems can provide guidance in the search for new materials as well as insight into the physical mechanisms responsible for their properties. Here, we illustrate this by…
A multiplication on persistence diagrams is introduced by means of Schubert calculus. The key observation behind this multiplication comes from the fact that the representation space of persistence modules has the structure of the Schubert…
Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of…
Excellent computer simulations are done for a purpose. The most valid purposes are to explore uncharted territory, to resolve a well-posed scientific or technical question, or to make a design choice. Stand-alone modeling can serve the…
The unique and beautiful character of certain mathematical results and proofs is often considered one of the most gratifying aspects of engaging with mathematics. We study whether this perception of mathematical arguments having an…
The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of…
Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate…
This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a…
We show that dissipative classical dynamics converging to a strange attractor can be simulated on a quantum computer. Such quantum computations allow to investigate efficiently the small scale structure of strange attractors, yielding new…
We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a…
We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result. The unusual feature of the use of computers here was that a computer generated but human…
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear…
We study Smarandache sequences of numbers, and related problems, via a Computer Algebra System. Solutions are discovered, and some conjectures presented.
Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces it has to be redesigned when applied to other…