Sample student papers
- Magical Numbers and Where to Find Them: pdf, source code: .tex, .bib
- Similarities Between Integers and Polynomials: pdf, souce code: .tex, .bib
- Reverse Hex: pdf, .tex, .bib, Hexgraph.png, Hexgraph2.png, Hexgraph3.png, Ygraph.png
Resources for paper topics and references
Week 12
Week 11
Week 10
Week 9
- Week 9 possible paper contents:
- Gaussian integers
- Comparison Gaussian integers with integers and how the Gaussian integers behave remarkably like the integers
- Week 9 (future!) talk suggested references:
- Hilbert’s 11th Problem: Quadratic forms by Jeremy Teitelbaum
- See the online “Number Theory” textbooks posted under “Week 8”
Week 8
- Week 8 possible paper contents:
- A short section of your paper could be on historical anecdotes in number theory (see the introduction sections of Apostol and Davenport)
- Overview of infinite series, series test for determining convergence/divergence (especially the limit comparison test), and applying the tests to infinite series useful for studying prime numbers
- Famous results about infinite series related to prime numbers (from Euler, 1737 to the present day)
- Background needed for stating famous theorems about prime numbers, rates of growth, their relationships to each other and to other integers, etc.
- Definitions and concepts needed to state the Riemann hypothesis
- Famous results and conjectures related to prime numbers, for example, the Goldbach’s conjecture.
- Explain Yitang Zhang’s 2013 result, Polymath8’s others’ results inspired by his (without proofs), the Twin Prime Conjecture, and a brief history of Zhang, Polymath8, and other mathematicians involved
- A short section of your paper could be your explanation of the proof that there are inifinitely many primes (as explained by the speaker), other proofs using different methods (like infinite series)
- A section of your paper could be a proof sum 1/p (where p are primes) diverges (see Apostol and Davenport below), and/or a proof that the harmonic series diverges (see your Calculus textbook)
- Some concepts from the theory of complex analysis that are related to prime numbers or infinite series
- Group theory of integers and its quotient groups
- Modular Arithmetic, with the goal of explaining the hypotheses of Dirichlet’s Theorem of Arithmetic Progressions and/or the non-admissible cases of the k-tuples conjecture (on prime constellations).
- The asymptotic and error notations (like the tilde symbol, big-oh, little-oh, etc.) that are used to state the Prime Number Theorem can also be understood via Taylor series (just where x—>0 instead of infinity).
- The Eratosthenes-Legendre sieve (also sometimes called just the Legendre sieve I) is a formalized version of the 10x10 sieve presented on the slides
- “Partial Summation”, “Abel Summation”, “Euler-Maclaurin formula” or what Apostol called the “Euler summatory function” (Thm 3.1 pg. 54 of Apostol’s book) are all basically the same thing, and are used a lot in this subject to relate sums to integrals in a quantitative way. They are all basically still Calculus, just used in an unusual way.
- Others (talk to me first)
- Week 8 suggested references:
- Hilbert’s 8th Problem: Problems about prime numbers (Riemann hypothesis, Goldbach’s conjeture, and the twin prime problem) by Brandon Alberts
- Slides used by the speaker
- The books used by the speaker to write the talk
- Textbook “Introduction to Analytic Number Theory” by Apostol, 1976 (the UConn library owns a paper copy)
- Textbook “Multiplicative Number Theory” by Davenport (currently the UConn library owns the 1st and 2nd edition in paper - I have put in a request for an electronic 3rd version)
- Journal article “The History of the Primality of One: A Selection of Sources” by Caldwell, Reddick, and Xiong (free online)
- Producing Prime Numbers via Sieve Methods by John Friedlander in Analytic Number Theory : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, July 11-18, 2002
- References for Yitang Zhang (as a person) and results related to his famous result
- Your Calculus (or Real Analysis) textbook for overview of infinite series and series test for convergence/divergence
- Free undergraduate Elementary Number Theory textbook (for definitions and classical proofs)
- A free undergraduate “intro to proofs” textbook which is relevant to primes and elementary number theory: A gentle introduction to the art of mathematics by Fields
- Another intro to proofs free textbook: Book of Proof by Hammack, 3rd edition
- Complex Analysis textbooks (for basic complex analysis concepts):
- A free undergraduate Complex Analysis textbook A first course in Complex Analysis by Beck, Marchesi, Pixton, Sabalka
- The textbook often used at UConn for the Complex Variables course is “Complex Variable and Applications” by Churchill and Brown (the latest edition is the 9th edition, but there are older editions available for free online, and our library owns a paper copy from the 2004 edition)
- Riemann Hypothesis references:
- Numberphile YouTube video: “The Key to the Riemann Hypothesis”
- Article by Conrad, Keith (2010): “Consequences of the Riemann hypothesis”
- Article by Stein, William and Mazur, Barry (2007): “What is Riemann Hypothesis?”
- In the Wikipedia page, check out the “External links” and see if any is easy to read.
- In the Wikipedia page, check out the “Popular Expositions”
- Website: Sieve of Eratosthenes, number square, view multiples and prime numbers
- If you are looking for a specific resource for a specitic topic, I will ask the speaker to help me find it
Week 7 (Number Theory)
- Week 7 possible paper contents:
- Continued fractions, continued fractions which converge to famous ratios (using limit method from Calculus 1), overview of solved and open problems (continued fraction is related my research)
- Proving that e is irrational using infinite series (from Calculus 2) or another method
- Algebraic vs transcendental and rational vs irrational, finding roots of a single-variable polynomial, overview of solved and open problems
- Overview of transcendental number results, from Hermite (1873) to present (although it’s no longer a hot topic)
- Algebraic independence, overview of solved and open problems
- Famous transcendental numbers such as pi, tau, e:
- the importance of e in the world of complex numbers,
- important formulas and applications involving e and pi,
- applications of e in financial mathematics
- Other famous irrational numbers:
- Lucas numbers and other Fibonacci-like sequences and the limit of their ratios,
- continued fractions which converge to phi, continued fractions for Fibonacci and Lucas numbers.
- A section of your paper could be the Lucas number analog of Jim Fowler’s video computation for Fibonacci numbers
- Week 7 talk suggested references:
- Hilbert’s 7th problem by Keith Conrad
- A few books used by the speaker to write the talk:
- Burger and Tubbs, “Making Transcendence Transparent,” 2004
- Robert Tubbs, “Hilbert’s 7th Problem: Solutions and Extensions”, Springer’s book 2016 - UConn library owns an electronic copy so you can download this for free!
- Section 1.7 for transcendence of pi, in: Hadlock, “Field Theory and Its Classical Problems”, 1978
- Chapter 7 for transcendence of e and pi in: Jones, Morris, Pearson “Abstract Algebra and Famous Impossibilities”, Springer-Verlag, 1991
- Lecture notes will be part of a survey on irrationality of pi, e, etc by K. Conrad (but not yet)
- “e”: The Story of a Number, a book by Eli Maor, 2015 (latest edition, originally published in 1994), owned by the UConn library so it is free to download
- The Math Fun Facts page “e is irrational” by F. Su
- The references at the bottom of a survey on irrationality by K. Conrad
- Your Calculus textbook for many facts about Pi, e, their relationship, and other famous numbers
- A complex analysis textbook for many facts about Pi, e, their relationship, and other famous numbers, see suggested complex analysis textbooks under “Week 8”
- Tons of references at the bottom of the Wikipedia entry on pi: en.wikipedia.org/wiki/Pi - More recent references, starting from the 90s, may be easier for you to read
- You should also be aware that 2 times pi day should be a thing!! tauday.com and Youtube video: Pi Is (still) Wrong
- See the references cited in the sample paper pi and e
- Chapter 17: Polynomials in the online textbook Abstract Algebra: Theory and Applications by Judson
- See also the number theory and “intro proofs” textbooks suggested under “Week 8”
- See also the number theory and “intro proofs” textbooks suggested under “Week 8”
- Article Irrational Numbers by Niven, an older publication not written in LaTeX
- Video of Fibonacci numbers, recurrences, limit of ratios, etc by Jim Fowler
- Bottom of Wikipedia pages (ask me to help you get free access to a reference)
Week 6
- Week 6 possible paper contents:
- A section of your paper can be on the proof that every positive single-variable polynomial is the sum of two squares (see slides 5,6 from the speaker)
- Overview of positive polynomials and formal power series rings, theorems, examples, counterexamples, positive rational functions – needs to be more than following the book/slides
- Positive Laurent polynomials in field of rational functions in cluster algebras (related to my research) and other rings
- Positive power series, generating functions for positive sequences
- Finding integer solutions to polynomial questions (Diophantine equations), overview of solved and open problems
- Sums of Squares for integers, see Textbook Number Theory: In Context and Interactive (with SageMath) by Crisman
- Positive integer sequences like somos sequences, somos-like sequences, Cube recurrence (look for a reference from this century)
- Positive integer arrays which form a Conway - Coxeter frieze pattern (related to my research)
- Others (talk to me first)
- Week 6 talk suggested references:
Week 5
- Week 5 possible paper contents (if you like Linear Algebra and theoretical math):
- Overview of symmetric polynomials, Schur polynomial, Molien’s theorem - needs to be more than simply following the lecture notes
- Overview of group acting on sets (like numbers, a field, a vector space, cosets) - needs to be more than simply following a textbook
- Background and definitions needed for stating one or two theorems in invariant theory of finite groups
- Symmetries (like reflective or rotational symmetry) of a polygon or a 3D object forming a finite group
- Others (talk to me first)
- Week 5 talk possible references:
Week 4
- Week 4 possible paper contents:
- Banach-Tarski paradox (related to set-theory, axiom of choice (Hilbert’s 1st problem) and abstract algebra (group theory)): decompose a ball into finitely many subsets which can be put back together into two identical copies of the original ball
- Equidecomposability of 2-dimensional shapes
- Platonic solids (tetrahedron, cube, octahedran, dodecahedron, icosahedron), other famous 3D and higher-dimension polytopes such as permutahedron, associahedron, generalized associahedron, hypercube
- Objects (Catalan objects, permutations, Baxter objects) that can be arranged as associahedron and other famous polytopes (associahedron is related to my research)
- Simplicial polytopes, graphs of polytope, polytopes as lattices (lattices are related to my research)
- Others (talk to me first)
- Week 4 talk suggested references:
- Hilbert’s 3rd Problem: Could two polyhedra have the same volume but not decomposable into finitely many congruent polyhedra? by Michael Biro
- A chapter about Hilbert’s Third Problem in the book “Proofs from the Book” by Aigner and Ziegler, 2018: Link to free download with UConn Net ID. Older versions are available to be physically checked out from the library.
- Slides of a similar talk given at University of Minnesota
- The speaker used an older book “Hilbert’s third problem” (sections 7, 10, and 13) by Vladimir Boltianskiĭ. It is available via Interlibrary loan.
- References for Catalan numbers:
- References for permutahderon, associahedron, and Catalan objects:
- Bottom of Wikipedia pages (ask me to help you get free access to a reference)
Week 3
- Week 3 Possible paper topics:
- Overview of countable sets vs uncountable sets (may include proofs)
- Overview of Cantor-style set theory
- Introduction to elementary concepts in first-order mathematical logic
- Fractals, algorithms for creating fractals-like objects like Sierpinski triangle, Sierpinski carpet, Koch curve, and L-systems
- Dragon curves
- Fractals in Computer Science
- Some famous proofs by Cantor (if you like writing proofs)
- Others (talk to me first)
- Week 3 talk suggested references:
Week 2
- Week 2 talk suggested references:
- Slides courtesy of K. Conrad
- Four more references on the last slide of Slides courtesy of K. Conrad
- Wikipedia table of Hilbert problems
- Book by Gray The Hilbert Challenge (Prof. Gunawan has a copy checked out from the library)
- Bottom of Wikipedia pages (ask me to help you get free access to a reference)
LaTeX
LaTeX Offline
If you would like to download a LaTeX distribution and front-end/compiler to your laptop, try:
A simple visual editor for writing diagrams and converting it to Tikz github.com/yishn/tikzcd-editor, try it out at https://tikzcd.yichuanshen.de
JSTOR
Writing convention
AMS Style Guide (PDF)
Wikipedia:Manual of Style/Mathematics
Writing proofs
Advice on Mathematical Writing by K. Conrad
Examples of proofs by induction by K. Conrad
Jobs requiring math skills
Math students, connect with prospective employers here: MAA Career Resource Center
Miscellaneous
Use CamScanner app or another scan app to scan your hand-written homework into PDF.
