Math Gems

Spring 2021:

Interested in giving a talk, but don't know how to start? Check out this guide on what the journey to giving a MathGems talk looks like.

Schedule:

MathGems meets online this semester, Tuesdays at 5pm. The Zoom link was sent out to all UNC math undergrads. If you'd like to attend, and have not received the link, please fill out this google form to get added to the mailing list. The schedule is listed below:

February 2 - Nick Tapp-Hughes, "Two proofs of a needle-dropping problem"
February 9 - No talk.
February 16 - No talk, UNC wellness day.
February 23 - Luke Conners, "2.5 Brief Proofs of Arrow's Impossibility Theorem: How Philosophy of Mathematics Informs Mathematical Philosophy"
March 2 - No talk.
March 9 - Cooper Schoone, "Measure-Preserving Transformations and Glasser's Master Theorem"
March 16 - Will Davis, "Sliding in Plane Sight: Solvability of the Fifteen Puzzle"
March 23 - No talk.
March 30 - No talk.
April 6 - Wesley Hamilton, "An Algebraic Proof of Morley's Theorem"
April 13 - Scott Hallyburton, "Two Applications of Abel's Identity"
April 20 - (7pm EST) - Alvis Zhao, "Geometric Optics of Hyperbolic Boundary Value Problems"
April 27 - David Yavenditti, "Differentiating Under the Integral Sign"
- Abstract: While still a student at M.I.T. and Princeton, Richard Feynman, the eventual Nobel laureate in physics, developed a reputation among his peers for being especially skilled at computing definite and improper integrals. How? Feynman had learned one particular technique unfamiliar to his colleagues, and his friends would come to him for help only after exhausting their other, more familiar techniques. In this talk, we shall explore Feynman's technique, known as differentiating under the integral sign, the Leibniz Rule, or sometimes even Feynman integration.
  
  Using differentiation under the integral sign, we shall compute explicit examples of integrals via this technique which would be intractable using elementary methods. We explore some useful strategies for applying this technique effectively, as well as its technicalities and limitations. Finally, we consider a few variations on the original method.
  
  Prerequisites: This talk shall assume a calculus background through multivariable calculus (Math 233 at UNC-Chapel Hill), with a few results from an introductory class in ordinary differential equations (Math 383). Some of the more technical results will use concepts from multivariable real analysis (at the level of Math 522), but these will be used sparingly.
  
  Logistical Note: A link to the lecture slides will be made at the beginning of the talk so the audience can more easily follow the talk despite the constraints of Zoom.

Abstracts are available here.

Material from previous MathGems iterations:

Beamer template

Here are the slides and the Beamer template used in the talk "Mining, Refining, and Selling your Gems".

Resources for finding topics

Some websites:

Some books (almost all of which can be found at the UNC library):

“100 Great Problems of Elementary Mathematics,” by Heinrich Dorrie
“Proofs from THE BOOK,” by Aigner and Ziegler; some specific historical suggestions are
1. Chapter 8: Some irrational numbers (especially connections to the Hermite 1873 paper)
2. Chapter 11: Lines in the plane and decompositions of graphs
3. Chapter 14: Cauchy’s rigidity theorem (especially where Cauchy’s original proof was wrong)
4. Chapter 19: Sets, functions, and the continuum hypothesis (Theorem 4 onwards)
5. Chapter 25: Cotangent and the Herglotz trick (especially Euler’s approach)
6. Chapter 26: Buffon’s needle problem
“Mathematical Gems, vol. 1, 2, 3,” by Ross Honsberger
“Proofs that Really Count,” by Arthur Benjamin and Jennifer J. Quinn
“Mathematical Delights,” by Ross Honsberger
“Charming Proofs,” by Claudi Alsina and Roger B. Nelsen
“Mathematical Omnibus,” by Dmitry Fuchs and Serge Tabachnikov
“God Created the Integers,” by Stephen Hawking; some specific historical suggestions are
1. Euclid, Book I, VII, IX.1, or IX.2
2. Archimedes, “Measurement of a Circle” or “The Sand Reckoner”
3. Diophantus, Book II
4. Descartes, Book I
5. Newton
6. Euler (any)
7. Gauss, Section III
8. Cauchy, Third Lecture
9. Galois, “On groups and equations and Abelian integrals”
10. Riemann, “On the number of prime numbers less than a given number”
"A source book in Mathematics," by David Eugene Smith

Presentation tips:

Here are some resources on giving presentations:

Previous seminar schedule:

Abstracts are available here.

Fall 2020:

September 1 - Wesley Hamilton, "Mining, Refining, and Selling your Gems"
September 8 - Cooper Schoone, "Space-Filling Curves & the Hahn-Mazurkiewicz Theorem"
September 15 - No talk.
September 22 - David Yavenditti, "The Calkin-Wilf tree"
September 29 - Cameron Kass, "Conditions for the Solvability of Equations by Radicals"
October 6 - Will Davis, "Let Me Count the Ways: a Look into the Catalan Numbers"
October 13 - Scott Hallyburton, "The Equivalence of the Theorems of Hex and Brouwer"
October 20 - Luke Conners, "Planes, Braids, and (Inner) Automorphisms: The Search for Closure in a Tangled World"
October 27 - No talk.
November 3 - No talk; go vote!
November 10 - No talk.

Spring 2020:

January 21st – David Yavenditti, “The Geometry of Numbers and Lagrange’s Four-Square Theorem”
January 28th – Alvis Zhao, “An introduction to distributions”
February 4th – Cooper Faile, “An Identity for Cotangent and the Herglotz Trick”
February 11th – No meeting.
February 18th (6.30-7.30pm) – Nick Tapp-Hughes,“The Brachystochrone Problem and the Birth of Optimal Control Theory”
February 25th – Wesley Hamilton, “The Knight’s tour and related chess puzzles”
March 3rd – No meeting.
March 10th – Spring break, no meeting.
March 17th – Cameron Kass, TBA (cancelled due to coronavirus).
March 24th – Dylan O’Connor, TBA (cancelled due to coronavirus).
March 31st – Kexuan Yang, TBA (cancelled due to coronavirus).
April 7th – Cooper Schoone, TBA (cancelled due to coronavirus).
April 14th – (cancelled due to coronavirus).
April 21st – (cancelled due to coronavirus).
April 28th – David Yavenditti, “The Calkin-Wilf tree” (cancelled due to coronavirus).

Spring 2019:

January 23rd – Informational meeting, invitation for presentations
January 30th – CMC event, no meeting
February 6th – Wesley Hamilton, “A finite number of proofs of the infinitude of primes”
February 13th – Yicheng Wang, “The POWER of a point”
February 20th – Cooper Faile, “Arithmetic Progressions of Three Squares”
February 27th – No meeting
March 6th – No meeting
March 13th – Spring break, no meeting!
March 20th – Pranav Arrepu, rescheduled
March 27th – no meeting; CMC meeting
April 3rd – Rescheduled due to building issues
April 10th – Pranav Arrepu, “ Power Series Before Newton and Leibniz”
April 17th (5-6pm) – Simon Bertron, “Mathematical Theorems You Never Knew Existed Because They Can’t Be Proved”
April 24th (room PH 385) – David Yavenditti, “`Think Deeply of Simple Things’: The Philosophy of The Ross Mathematics Program”

Fall 2018:

September 4th – Wesley Hamilton, “Fun with Fourier Series”
September 11th – Hurricane watch, no meeting
September 18th – David Yavenditti, “On Transcendental Numbers and Approximations by Rational Numbers”
September 25th – Daniel Cantwell, “On relativistic group laws”
October 2nd – No meeting
October 9th – Cooper Faile, “Proving Brouwer’s fixed point theorem with a board game”
October 16th – Arunabha Debnath, “The Optimal Stopping Problem”
October 23rd – James Haberberger, “All finite division rings are fields”
October 30th – No meeting; CMC event
November 6th – No meeting; CMC event
November 13th – Simon Bertron, “Selections from Counterexamples in Analysis“
November 20th – No meeting; Thanksgiving break
November 27th – No meeting
December 4th – Dylan O’Connor, “ An Ellipse is a Ring: Constructing Addition and Multiplication on Conic Sections in Projective Planes”

Spring 2018:

January 17th – Snow day, no meeting
January 24th – Informational Meeting
January 31st – Wesley Hamilton, “Some interesting Taylor series”
February 7th – No meeting
February 14th – Daniel Cantwell, “On the cardinality of stuff”
February 21st – Cooper Faile, “On Monsky’s Theorem”
February 28th – Arunabha Debnath, “On Godel’s Incompleteness Theorem”
March 1st – Yicheng Wang, “Information theory: an informal, informative introduction”
March 14th – Spring break, no meeting
March 21st – Daniel Pezzi, “A Derivation and Exploration of the Heisenberg Uncertainty Principle”
March 28th – No meeting
April 4th – Daniel Cantwell, “Carrying is a cocycle”
April 11th – Simon Bertron, “The remaining 193 proofs are left as an exercise to the reader”
April 18th – Kaylee Stanton, “On the Basel problem”
April 25th – David Yavenditti, “Elementary Integrability”

Fall 2017:

September 7th – Informational Meeting
September 14th – Wesley Hamilton, “Advanced `trig identities’ ”
September 21st – Andrew Prudhom, “The Thue-Morse sequence and infinite chess”
September 28th – Simon Bertron, “The Banach-Tarski paradox”
October 5th – Cooper Faile, “Linear recurrence relations and the look-and-say sequence”
October 12th – No meeting
October 19th – Fall break, no meeting
October 26th – Kaylee Stanton, “Intersections of hypercubes and hyperspheres”
November 2nd – David Yavenditti, “The Geometry of Numbers and Lagrange’s Four-Square Theorem”
November 9th – No meeting
November 16th – Jieying Li, “The Simulation of Non-competitive Absorption Models”
November 23rd – Thanksgiving break, no meeting
November 30th – No meeting

Spring 2017:

January 25th – Informational Meeting
February 1st – Wesley Hamilton, “An Algebraic Proof of Morley’s Theorem”
February 8th – Molly Merritt, “A Discussion of Computer-Aided Proofs”
February 15th – Logan Tatham, “The Jones Polynomial and the Tait Conjecture”
February 22nd – No meeting
March 1st – Simon Bertron, “On the Dinitz Problem”
March 8th – No meeting
March 15th – Spring Break, no meeting
March 22nd – Stanley Sun, “Ramanujan’s Sums”
March 29th – Nuch Aminian, “Dynamical Systems and the Poincare-Bendixson Theorem”
April 5th – No meeting
April 12th – Pranav Arrepu, “On Hadamard’s Determinant Problem”
April 19th – Cooper Faile, Surreal Numbers
April 26th – Yicheng Wang, Zeckendorf’s Theorem

Feel free to email me at wham at live dot unc dot edu with any questions!

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