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En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los…
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…
This is a survey of old and new problems and results in additive number theory.
We present a question bank consisting of over 250 multiple-choice and true-false questions covering a broad range of material typically taught in an introductory undergraduate course in numerical analysis or scientific computing. The…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…
I discuss a seemingly unlikely confluence of topics in algebra, numerical computation, and computer vision. The motivating problem is that of solving multiples instances of a parametric family of systems of algebraic (polynomial or rational…
The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns…
We present a conceptually clear introduction to quantum theory, deriving the theory from scratch from the point of view of quantum information. Different subsets of these lectures were taught to a wide variety of audiences, including…
Section headings: 1 Qubits, gates and networks 2 Quantum arithmetic and function evaluations 3 Algorithms and their complexity 4 From interferometers to computers 5 The first quantum algorithms 6 Quantum search 7 Optimal phase estimation 8…
Graphs are a natural representation for systems based on relations between connected entities. Combinatorial optimization problems, which arise when considering an objective function related to a process of interest on discrete structures,…
A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour…
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states…
Computational thinking has been a recent focus of education research within the sciences. However, there is a dearth of scholarly literature on how best to teach and to assess this topic, especially in disciplinary science courses. Physics…
This tutorial introduces quantum computing with a focus on the applicability of formal methods in this relatively new domain. We describe quantum circuits and convey an understanding of their inherent combinatorial nature and the…
This is an REU paper written for the University of Chicago REU, summer 2017. The main purpose of this note is to collect some of the many combinatorial models for MV cycles that exist in the literature. In particular, we will investigate MV…
Drawing appropriate diagrams is a useful problem solving heuristic that can transform a given problem into a representation that is easier to exploit for solving it. A major focus while helping introductory physics students learn problem…
This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each…
Problem decomposition--the ability to break down a large task into smaller, well-defined components--is a critical skill for effectively designing and creating large programs, but it is often not included in introductory computer science…
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
Seven years ago (2016), we began integrating Robotics into our Computer Science curriculum. This paper explores the mission, initial goals and objectives, specific choices we made along the way, and why and outcomes. Of course, we were not…