Papers

“The difference between a research paper and toilet paper is that toilet paper has crap only on one side.” — Anonymous academic joke

Research publications and preprints

• Mixed Tensor Products, Capelli Berezinians, and Newton’s Formula for gl(m|n)
  S. Erat, A. Kannan, S. Kanungo. Transformation Groups. Published (2025)
  [ arXiv ] [ journal ] [ PDF ]

• Representation Theory of the Twisted Yangians in Complex Rank.
  A. Kannan, S. Kanungo. Journal of Algebra. In review; submitted May 14, 2025
  [ arXiv ]

• Saturation Problems for Sequences.
  J. Geneson, S. Kanungo.
  In preparation

• On Product Formulas of Guillera and Sondow
  S. Kanungo, J. Schettler. American Mathematical Monthly. Submitted Oct 27, 2024.
  [ arXiv ]

• A weaker notion of the finite factorization property
  H. Jiang, S. Kanungo, H. Kim. Commun. Korean Math. Soc. 39 (2024), No. 2, pp. 313–329
  [ arXiv ] [ journal ] [ PDF ]
  

Expository

click on the title to see a short description

• David Gale's Subset Takeaway Game. Combinatorial Game Theory [ PDF ]

  We describe the natural interpretation of the game in terms of simplicial complexes, and the binary star reduction technique that shows that Subset Takeaway is a second player win for n < 7. We also look at Subset Takeway played on a graph, and compute the Grundy values for complete n-partite graphs and all bipartite graphs.
  
  
  Space-Filling Curves. Real Analysis [ arXiv ]

  We examine space-filling curves, which are surjective continuous maps from [0, 1] to some higher-dimensional space, usually the unit square [0, 1]². In particular, we define Peano’s curve and Lebesgue’s curve, and state some of their properties. We also discuss the Hahn-Mazurkiewicz theorem, which characterizes those subsets of Rⁿ that are the image of a space-filling curve. Finally, we discuss real-world applications of Hilbert curves, in particular Google’s S2 Cells.
  
  
  The Hahn-Banach Theorem and Applications. Functional Analysis [ PDF ]

  We explore the Hahn–Banach Theorem’s foundational role in functional analysis and its critical applications in mathematical finance. Through both conceptual exposition and a worked example, we demonstrate how the theorem enables the extension of pricing functionals and underpins the existence of risk-neutral measures in arbitrage-free markets.
  
  
  Sobolev Spaces and Applications to PDEs. Theory of PDEs [ PDF ]

  We provide an introduction to Sobolev spaces, a foundational concept in modern analysis and the theory of partial differential equations (PDEs). These spaces are useful for studying, among other things, the well-posedness of partial differential equations and their approximation using finite elements. We begin with a historical overview, tracing the development of weak derivatives and the shift from classical to variational formulations of PDEs. After establishing the basic definitions and presenting key examples, we survey central theorems such as the Sobolev Embedding Theorem and Rellich's Theorem, emphasizing their significance in ensuring existence, uniqueness, and regularity of solutions. Finally, we discuss a classic application to PDEs, the Elliptic Regularity Theorem. We aim to provide a self-contained and accessible introduction for students with a background in real analysis and the theory of PDEs.
  
  
  The Stochastic Gradient Descent Method. Stochastic Methods [ PDF ]

  Stochastic Gradient Descent (SGD) is a cornerstone algorithm in modern optimization, especially prevalent in large-scale machine learning. This paper introduces the theoretical foundation of SGD, contrasts it with deterministic gradient descent, and explores its convergence properties, practical implementation considerations, and typical applications in applied mathematics and data science. We also give some basic numerical simulations which showcase the strengths of different variants of SGD.
  
  
  On the S-matrix Conjecture for even n. Nonnegative Matrices [ PDF ]

  Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0,1]. This is known as the S-matrix conjecture. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. In this report we discuss Frankel and Urschel's proof of the S-matrix conjecture for all even n larger than a small constant.
  
  
  Chromatic Polynomials. Graph Theory [ PDF ]

  A general introduction to the theory of chromatic polynomials. We derive their salient properties, and describe some practicaI methods for computing them. We briefly discuss the connection between the theory of chromatic polynomials and map coloring problems.
  
  
  Lagrangian and Hamiltonian Mechanics. Dynamical Systems [ PDF ]

  we discuss the basics of Lagrangian and Hamiltonian dynamics. We derive the Euler-Lagrange equations fro D’Alembert’s principle, show that they are equivalent to Hamilton’s principle of least actions, and finally use them to derive Hamilton’s equations. We also provide some examples to illustrate the use of Lagrangian and Hamiltonian dynamics.
  
  
  Brun's Theorem. Number Theory [ PDF ]

  The mathematician Viggo Brun was born in Sweden in 1885. He is known for his outstanding contributions to the field of number theory. In his early career, Brun focused on analytic number theory and prime number theory. One of his most famous achievements was Brun’s theorem in 1915, which gave an upper bound on the distance between prime numbers.
  
  
  Large Gaps Between Primes. Analytic Number Theory [ PDF ]

  We give a summary of the upper bounds that have been obtained for the maximal prime gap, G(x), over the last century, particularly Rankin’s lower bound, and the improvement to it discovered independently by Ford-Green-Konyagin-Tao and Maynard in 2014. We go over a sketch of Rankin’s, Ford-Green-Konyagin-Tao’s, and Maynard’s proofs of their bounds, omitting technical details but still presenting the main ideas.
  
  
  Dynamical Billiards. Ergodic Theory [ PDF ]

  The field of dynamical billiards studies the motion of a ball bouncing within a billiard table, which is bounded by a smooth, closed curve. The ball's movement adheres to two key properties: it always travels in a straight line, and the angle of incidence equals the angle of reflection at the boundary. The latter property is an empirical observation from physics. In this paper, we analyze the dynamics of various billiard tables in R², employing Euclidean geometric methods to investigate and classify their ergodic behavior. Specifically, we examine the ergodicity of billiards within circular and annular (circular ring) boundaries and present some results on elliptic billiards. Additionally, we explore examples of chaotic billiards, where chaos is characterized by the lack of correlation between the starting point and subsequent positions after many bounces. In such cases, even a slight variation in the initial conditions can lead to significantly divergent trajectories. Finally, we conclude with a discussion of a physical application of billiards.
  
  
  Arithmetic Dynamics. Number Theory [ PDF ]

  We explore the field of arithmetic dynamics, which lies at the intersection of discrete dynamical systems and number theory. Discrete dynamical systems focus on the iterative behavior of functions, while number theory examines the properties of integers. Combining these two areas gives rise to arithmetic dynamics, where we investigate the number-theoretic properties of orbits of integers and rational numbers under the iteration of polynomials and rational functions. The core idea of arithmetic dynamics is to consider a function mapping a set to itself and analyze its behavior under repeated iteration. In this paper, we begin by defining the set of p-adic numbers and presenting key results related to them. We then examine an application of arithmetic dynamics, establishing a connection to dynamical systems in the p-adic numbers.
  
  
  Minimal surfaces. Differential Geometry [ PDF ]

  We introduce the theory of minimal surfaces. In the chapter on Geodesics, we considered the problem of finding the shortest distance between two points. We investigate the higher dimensional analogue of this, where we find ways to construct a surface of "minimal" area with a given boundary. Such surfaces can be represented by soap films, where the surface tension of the film ensures that it attains a shape with the minimal surface area. Minimal surfaces can be found in anything from the event horizons of black holes, to biomolecules for drug delivery, to the designs of roofs.
  
  
  Mixing-time estimates for riffle shuffle. Markov Chains [ PDF ]

  We talk about one of the most well-known shuffling methods, called the riffle shuffle or dovetail shuffle. We are interested in the number of shuffles that will make the deck of n cards well-mixed, or close to uniformly random.
  
  
  Classifying groups of certain orders. Abstract Algebra [ PDF ]

  We first discuss the question of which integers n have exactly one group of order n, namely the cyclic group Z/nZ. We will see that these are the integers that are relatively prime to the Euler totient function φ(n). Then we discuss how many groups there are of order p ³ for each prime p. We end with a couple of interesting results and conjectures pertaining to groups of squarefree order.