# Abstracts

Spring 2021:

• February 2 - Nick Tapp-Hughes, "Two proofs of a needle-dropping problem"

• Abstract: Originally posed in 1777, Buffon's needle problem is a delightful introduction to continuous probability. We will discover two elegant proofs of the problem: one with calculus and one without. As we deal with lines and line segments, we will unexpectedly find π. Time permitting, we will use the result of the problem to approximate π by experimentation.

• 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"

• Abstract: Social choice theory, the study of how to fairly aggregate individual preferences into a collective decision, has vexed and intrigued mathematicians, philosophers, and social scientists since its inception in the 18th century. The field was dominated by the close examination of pre-determined voting systems for desirable properties for centuries, until Kenneth Arrow took the reverse approach: Given a set of reasonable 'fairness' criteria, can we design a voting system that satisfies all our desires? His 1951 paper "Social Choice and Individual Values" resolved this question negatively. In this talk, we'll explore 2.5 proofs of Arrow's Theorem. Along the way, we'll benefit from the power of placing a problem in the proper framework to bolster our intuition and simplify our reasoning. This talk is intended to be self-contained; prerequisites are minimal for an undergraduate audience. The source material is John Geanakoplos's 1996 paper "Three Brief Proofs of Arrow's Impossibility Theorem".

• March 2 - No talk.

• March 9 - Cooper Schoone, "Measure-Preserving Transformations and Glasser's Master Theorem"

• Abstract: The field of measure theory studies how to measure the size of sets, and how to apply these measurement techniques to develop powerful tools for integration. A natural question which arises in measure theory is: “For a given measure, which functions map sets of a given size to sets of the same size?”. In this talk, after giving a high-level overview of the basics of measure theory, we’ll discuss these types of functions, which are known as “measure-preserving transformations”. We’ll then prove Glasser’s master theorem, a little-known integration theorem which characterizes all rational functions that preserve the standard measure on the real line.

• March 16 - Will Davis, "Sliding in Plane Sight: Solvability of the Fifteen Puzzle"

• Abstract: The Fifteen puzzle is a classic slide puzzle in which the numbers 1-15 are placed in a 4-by-4 grid and must be rearranged into the correct order. However, not every Fifteen puzzle is even able to be solved. We will use the symmetric group and the taxicab metric to develop an invariant on a particular slide puzzle that we can use to determine whether a particular configuration of this puzzle is actually solvable or not.

• March 23 - No talk.

• March 30 - No talk.

• April 6 - Wesley Hamilton, "An Algebraic Proof of Morley's Theorem"

• Abstract: Morley's theorem states that the intersections of pairwise adjacent trisectors of a triangle form an equilateral triangle. Namely, given a triangle ABC, trisect the angles at vertices A, B, and C, and look at where the trisectors for A and C (closest to the line segment AC) intersect, where the trisectors for A and B intersect, and where the trisectors for B and C intersect; these three intersections together form an equilateral triangle. While this statement sounds like a proposition in Euclid's Elements, the first proof was discovered in 1899 by F. Morley (and not by N. Bonaparte). Many more proofs were found in the past 100 years, including one by Conway, though these all generally rely on compass and ruler constructions, and/or trigonometric relations, and/or other concepts from elementary geometry. In this talk I'll present a more recent proof using certain symmetries of the plane and group theoretic ideas, and show how to produce 18 such equilateral triangles using these trisectors.

• April 13 - Scott Hallyburton, "Two Applications of Abel's Identity"

• Abstract: Abel’s Identity transforms the partial sum of a product of two sequences into an often-illuminating form. This talk will focus on two main applications of this identity to theorems in analysis and number theory. The first half of the talk will focus on the application of Abel’s identity to alternating-type summations and the convergence of a power series on its boundary. The second half will provide an extended version of Abel’s identity that considers sequences derived from continuously differentiable functions. From this we will examine how this identity leads to an asymptotic formula for the partial sums of (p_n^-1) where (p_n) is the sequence of prime numbers.

• April 20 - (7pm EST) - Alvis Zhao, "Geometric Optics of Hyperbolic Boundary Value Problems"

• Abstract: Geometric optics is synonym to short wavelength asymptotic analysis of solutions to systems of partial differential equations. The method of geometric optics (or WKB method) is a typical method for finding approximate solutions. However, the approximate solutions are not always justified (ie. in a sense “close to” the exact solutions). In this talk, we consider an example relating to systems that arise from the study of Mach stem and Vortex sheets. I’ll present in the first part of the talk, how we construct geometric optics solutions, and in the second half, how we use a uniform estimate to justify the approximate solutions.

• 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.

Fall 2020:

• September 1 - Wesley Hamilton, "Mining, Refining, and Selling your Gems"

• Abstract: Interested in presenting a mathematical gem, but don't know how to start? What do gems look like? How do you go from a "raw gem" to a nice, polished presentation? Find out in this overview of what MathGems is all about! We'll look at the process from start to finish, with specific examples of Gems that have been presented in the past.

• September 8 - Cooper Schoone, "Space-Filling Curves & the Hahn-Mazurkiewicz Theorem"

• Abstract: Space-filling curves are pathological functions defined on a subset of the real line that call into question our intuitive notions about dimension and measure. In this talk, I will discuss the Peano curve, a prototypical example of such a function, building up to the proof of the Hahn-Mazurkiewicz theorem, which characterizes exactly which subsets of Euclidean space are the images of space-filling curves.

• September 15 - No talk.

• September 22 - David Yavenditti, "The Calkin-Wilf tree"

• Abstract: The Calkin–Wilf Tree is a remarkable mathematical object. Like some fractals, it is very simple to describe, but that simplicity belies its deep, intricate underlying structure and patterns. In this talk, we shall introduce the Calkin–Wilf Tree, the closely related Calkin–Wilf Sequence and Insert Sum Here'' Sequence, and present an overview of some of the properties of these objects.
Prerequisites for this talk are minimal for an undergraduate audience, and the presentation is designed to be as self-contained as possible. That said, the more familiarity one has with concepts like countability, mathematical induction, Euler's totient/phi function, Fibonacci numbers, binary expansions of positive integers, and continued fractions, the more deeply one will be able to appreciate this talk's results.
Note: A link to the lecture slides will be made available before the talk begins so that the audience has greater opportunity to understand this new material.

• September 29 - Cameron Kass, "Conditions for the Solvability of Equations by Radicals"

• Abstract: The fundamental problem of algebra might be considered finding expressions for the quantities at which equations evaluate to zero. We refer to these special quantities as an equation’s roots. For many years, mathematicians grappled with the problem of expressing the roots of polynomial equations by radicals. Niels Henrik Abel would prove that no such formula exists in general for polynomial equations of degree 5 in 1824, and in 1831, Évariste Galois would provide a method for investigating the solvability of polynomial equations by radicals in general. I will discuss motivating examples and provide a summary of Galois’s findings. The purpose of this talk will be to develop the background and machinery necessary to understand the ideas Galois presented.

• October 6 - Will Davis, "Let Me Count the Ways: a Look into the Catalan Numbers"

• Abstract: From trees to polygons to mountain ranges to permutations, there are many seemingly-unrelated problems in combinatorics which can be all solved using an integer sequence formed by the Catalan Numbers. In this talk, we will develop a few different constructions of this sequence, investigate some of the counting problems described by this sequence, and identify properties of these problems which we can intuit from the properties of the sequence itself. Prerequisites for this talk are minimal for an undergraduate audience, and the talk is meant to be self-contained. All relevant terms will be defined, and lots of pictures will be provided.

• October 13 - Scott Hallyburton, "The Equivalence of the Theorems of Hex and Brouwer"

• Abstract: A version of Brouwer’s Fixed Point theorem states that a continuous map from the closed unit square in Euclidean space into itself must have a fixed point. Many proofs of this theorem exist, some using tools including combinatorial identities and Stoke’s theorem. In this talk I will discuss how the game of Hex, a simple variant of tic-tac-toe, can be used to prove Brouwer’s theorem in two-dimensions. The essential feature of Hex, known as the Hex Theorem, is that no game can end in a draw. It turns out, in fact, that the Hex and Brouwer theorems are equivalent. I will also discuss how a generalization of Hex to n players leads to Brouwer’s theorem in the general n-dimensional case.

• October 20 - Luke Conners, "Planes, Braids and Inner Automorphisms: Why Links and Braids are Knot so Different"

• Abstract: Knots are inherently topological objects, and as anyone who's tried to untangle their headphones knows, it can be difficult to decide whether a strand is even knotted! On the other hand, braids admit elegant algebraic descriptions in terms of generators and relations. In this talk, we'll explore the Reidemeister moves for distinguishing equivalent knots and take an algebraic and a topological look at the braid group, with the eventual goal of understanding the celebrated theorems of Alexander and Markov relating the two settings. This talk is intended to be self-contained; prerequisites are minimal for an undergraduate audience. All relevant terms will be defined, and many diagrams will be provided to help guide our intuition.

• October 27 - No talk.

• November 3 - No talk.

• 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”

• Abstract: The theory of distributions is a generalization of the classical analysis, with makes it possible to deal in a systematic way with difficulties that are not justified rigorously. Moreover, it provides a new and wider framework, and a more perspicuous language in which one can reformulate and study classical problems. Its influence has been particularly pervasive and fruitful in the theory of linear partial differential equations. In the first half of this talk, I will present the definition and operations of distributions with examples and see how it “generalizes” the classical functions; the second half I will have a glimpse of tempered distributions the extension of Fourier transforms and gave an example of how it helps solving PDEs.

• February 4th – Cooper Faile, “An Identity for Cotangent and the Herglotz Trick”

• Abstract: In 1748 Euler found a partial fraction expansion of the cotangent function. At this talk I will present Gustav Herglotz’s proof that cotangent is indeed equal to this expansion. Time permitting, I may also show how this identity gives us a way to easily calculate the Riemann zeta function for positive even integers.

• February 11th – No meeting.

• February 18th (6.30-7.30pm) – Nick Tapp-Hughes,“The Brachystochrone Problem and the Birth of Optimal Control Theory”

• Abstract: After the invention of calculus in the late 17th century, the race was on to find what problems could be solved using these new methods. The brachystochrone problem, the problem of finding the curve of a ramp which leads a marble to roll from one point to another in minimal time, was now in reach. I will
present Johann Bernoulli’s 1697 solution of the brachystochrone problem, as well as some of the ideas of Newton, Leibniz, Legendre, Euler, and Lagrange which led to the development of modern optimal control theory.

• February 25th – Wesley Hamilton, “The Knight’s tour and related chess puzzles”

• Abstract: In chess, each piece can either move horizontally, vertically, diagonally, or some combination of these directions… except the knight. The knight can only move in “L” shapes, so naturally one can ask whether or not a knight can reach every square in the standard chessboard; this is called the problem of a knight’s tour. In this talk, we’ll discuss the knight’s tour, some simple heuristics for finding knight’s tours, and then look at variants of this tour for the other pieces. This talk will include a combination of puzzles and gems.

• 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 29th – CMC event, no meeting

• February 6th – Wesley Hamilton, “A finite number of proofs of the infinitude of primes”

• Abstract: In this talk, we’ll explore a few proofs of the infinitude of primes. The different proofs each convey a different flavor of math: the original proof by Euclid, a number theoretic proof, a group theoretic proof, and an analytic proof. Time permitting, we’ll also see a proof using notions from topology.

• February 13th – Yicheng Wang, “The POWER of a point”

• Abstract: In this talk we will explore the power of a point theorem. We’ll talk about what it is, and use it to derive certain interest results related to geometric inversion, which is the mapping that maps the inside of a circle to the outside of said circle. We’ll conclude by talking about some cool results of geometric inversion, specifically the Steiner Porism.

• February 20th – Cooper Faile, “Arithmetic Progressions of Three Squares”

• Abstract: In this talk we will explore arithmetic progressions of three squares. This will begin by finding all such integer progressions by looking at rational points on a circle. Then I will give a quick overview of elliptic curves and use an elliptic curve to describe arithmetic sequences of the squares of rationals with a common difference n.

• 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”

• Abstract:
A student of calculus is likely familiar with the power series for the sine and cosine functions along with their derivations based on Taylor’s theorem. Yet, it is not often taught that the power series for these functions were known in various parts of the world centuries prior to the formal development of Calculus by Newton (1643 – 1727) and Leibniz (1646 – 1716). The primary aim of this talk is to provide an exposition on the derivations of these series relying on Arabic and Sanskrit sources. The proofs require only elementary geometric observations and should be accessible to those who have not studied calculus. The talk will also discuss the results of Scottish mathematician James Gregory (1638 – 1675) and Japanese mathematician Takebe Kenko (1664 – 1739) as an illustration of how power series were studied as part of independent efforts throughout the world.

• April 17th – Simon Bertron, “Mathematical Theorems You Never Knew Existed Because They Can’t Be Proved”

• Abstract: This talk will be a meta-mathematical analysis of the limits of first-order Peano arithmetic. Simplifying drastically, Gödel’s Incompleteness Theorem states that any axiomatization of arithmetic is inherently incomplete and the ideas in this talk can be thought of as an extension to that principle. We will consider theorems which may be stated in the language of finite arithmetic but not proved using finite arithmetic. We will discuss how one may prove such a theorem (using methods other than finite arithmetic) and how one might prove that such a theorem cannot be proved by finite arithmetical means. We will also discuss several ideas that appear in the study of these topics, including ordinal numbers and “fast-growing functions”.

• April 24th (room PH 385) – David Yavenditti, “`Think Deeply of Simple Things’: The Philosophy of The Ross Mathematics Program”

• Abstract: The Ross Mathematics Program is an intensive summer program in mathematics for high school students. The program has alumni who have gone onto distinction in many fields, especially in STEM. Some alumni have even produced Ross-inspired programs elsewhere, including PROMYS at Boston University.
In this talk, we shall focus on the approach and philosophy of the Ross program, demonstrating what makes its approach unique and valuable.

Fall 2018:

• September 4th – Wesley Hamilton, “Fun with Fourier Series”

• Abstract: Fourier series are a fundamental tool in modern analysis; they also provide a plethora of surprising, and seemingly magic, identities involving \pi. In this talk, we will explore a convergent series whose sum is unchanged after its terms are all squared, cubed, and even raised to the fourth power. Mathematica will be featured prominently, both as a tool to experiment with series, and a tool to evaluate exact representations of certain quantities. No experience with Fourier series is required. A knowledge of integration by parts is all that is needed.

• September 11th – Hurricane watch, no meeting

• September 18th – David Yavenditti, “On Transcendental Numbers and Approximations by Rational Numbers”

• Abstract: Since ancient Greece, mathematicians have known that the set real numbers consists of the set of rational numbers and the set of irrationals. Among the reals, all rationals and many irrationals are algebraic. This means they are roots to nontrivial polynomials with integers coefficients. For example, the rational number $\frac{5}{7}$ is a root to the polynomial $7x-5$, and the irrational number $\sqrt{2}$ is a root to the polynomial $x^2-2$. In~1806, Legendre conjectured that there exist real numbers which are \emph{not} algebraic—today called transcendental numbers—but he was unable to prove this. This talk shall explicitly construct an example of a transcendental number via a method of Liouville. Namely, we explore properties of real numbers that (in a sense to be made precise) can be \emph{very} well approximated by rational numbers. Most of this talk assumes prerequisites no more advanced than high-school level algebra. In particular, only one result from calculus, the Mean Value Theorem, shall be used in the proof of this talk’s main result.

• 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”

• Abstract: Hex was discovered independently in the 1940s by Piet Hein and John Nash. In this talk, I will show that the impossibility of a draw in hex (the hex theorem) is equivalent to the Brouwer fixed point theorem.

• October 16th – Arunabha Debnath, “The Optimal Stopping Problem”

• Abstract: We are often encountered by situations where we are given a set of choices one by one, and we have to pick the best one when we see it without backtracking, but we have no information or hueristic by which we really figure out which option is the best just when we see it. How can we go about ensuring the greatest chance that we will end up with the optimal choice? This problem is called the Optimal Stopping Problem, and in Tuesday’s talk, I will be discussing the general
form of the problem, and its solution.

• October 23rd – James Haberberger, “All finite division rings are fields”

• Abstract: Rings are important structures in abstract algebra, with applications in topological spaces and theoretical physics. This talk will cover the basic properties of a Ring, and will define them in relation to more typical Field structures. After providing background, the talk will follow with Witt’s proof that 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

• Abstract: Much of mathematics is concerned with proving theorems. That is, establishing the truth of a true statement. Math GEMS especially seems to place an emphasis on true statements and their most truthy truthliness. There is, however, another way! One can expend just as much time and energy demonstrating the falsity of a false statement. And the principal way one does this is through a counterexample. Lucky for you, I have found a book dedicated entirely to the elegant and pathological construction of counterexamples in everyone’s favorite undergraduate-friendly discipline: analysis. This talk will explore a few of my favorites and how they can be just as illuminating, informative, and surprising as any theorem.

• 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”

• Abstract: Projective geometry’s main appeal lies in the beautiful symmetry of its axioms: two lines meet in one point and two points are joined by one line. On a projective plane, parallelism is non-existent. In this talk we will explore how that change in axioms from Euclidean geometry affects the classical subject of geometric construction. Using tools gleaned from Jürgen Richter-Gebert’s Perspectives on Projective Geometry including harmonic points, projective transformations, and quadrilateral sets, we will learn how to construct addition and multiplication in the projective plane, first on a line, and then on ellipses, parabolas, and hyperbolas.