What is Expositio?
A public repository for expository academic papers, with an integrated scholarly record.
Expositio is a place to read and cite expository work. Papers are published as stable,
citable records, while remaining connected to later versions, editorial decisions,
and related context.
Some papers are early drafts; others have been peer reviewed or formally published.
Each paper remains readable on its own, while preserving the history behind it.
The result is both a reading room and a record: a place to browse by field, read
papers in stable form, and follow their development over time.
v1
In this paper, we use multiple techniques to find the complete set of solutions to Pell and negative Pell equations. We prove the existence of a nontrivial solution to the former using Dirichlet's Approximation Theorem and then find all solutions from the minimal positive soluti…
Mathematics / Number theory
v1
Sparse linear algebra is the study of linear systems Ax = b in which the matrix
A contains overwhelmingly many zero entries. Such systems arise throughout applied mathematics and engineering from the discretization of partial differential equations on large grids, to network an…
Mathematics / Numerical analysis
v2
The representations of the Symmetric Groups S_n carry a very rich combinatorial structure. In particular, their representations have connections to seemingly unrelated problems, including to symmetric polynomials and the theory of Young Tableau. We demonstrate some classical res…
Mathematics / Combinatorics
v1
This paper builds up and proves the Poincaré-Hopf Theorem for compact smooth manifolds. Beginning with the introduction of key concepts in differential topology, including smooth manifolds, coordinate charts, tangent spaces, and mappings, we prove foundational theorems like the …
Mathematics / Manifolds and cell complexes
v1
The symmetric groups are fundamental objects in abstract algebra, and for
nearly every value of n, their automorphism groups are completely inner. The unique
exception is the symmetric group S_6, which possesses a nontrivial outer automorphism. This
paper develops the theory…
Mathematics / Group theory and generalizations
v1
This paper gives an expository introduction to arithmetic progressions in Ramsey theory and additive combinatorics. We begin with the language of colorings, using small examples to illustrate how simple coloring rules can force surprisingly rigid structure. The main result discu…
Mathematics / Combinatorics
