ESP Biography


Major: Physics, Mathematics

College/Employer: Columbia University

Year of Graduation: 2020

Picture of Benjamin Church

Brief Biographical Sketch:

Not Available.

Past Classes

  (Clicking a class title will bring you to the course's section of the corresponding course catalog)

M860: Introduction to Handwaving in Splash Fall 2019 (Oct. 27, 2019)
Have you ever thought that mathematics is done with too many damn symbols? If yes, then this is the class for you. We will dive into some of the most important and abstract results in higher mathematics using only silly informal language. This class will give you insight into some of the deepest results in mathematics but with their statements and proofs handwaved off to infinity.

M877: A Grand Tour of Rotations, Quaternions, The Hopf Fibration and Spin with a side of Lie Groups in Splash Fall 2019 (Oct. 27, 2019)
We will begin a fantastical journey into some of the most beautiful and useful geometric objects in modern mathematics, Lie groups, by asking the simple question: how do we represent rotations in 3D space. This question will lead us to define a strange algebraic object, the quaternions, investigate the mysterious topology of spheres living in four (and more) dimensions, marvel at a beautiful image of a sphere in dimension four decomposed by the Hopf fibration, and finally discuss how these fantastical higher-dimensional geometric objects are, in fact, realized in the strange world of quantum mechanics as spin.

S738: Einstein's Annus Mirabilis in Splash Fall 2018 (Oct. 28, 2018)
In this class we will investigate the four groundbreaking papers published all within one year (the miraculous year) which took Albert Einstein from a nobody patent clerk to a household name. In these papers, Einstein proved the existence of atoms, laid the groundwork for quantum mechanics, and unveiled his special theory of relativity.

M739: Godel's Incompleteness Theorems in Splash Fall 2018 (Oct. 28, 2018)
Godel's Incompleteness Theorems are the most famous results in mathematical logic. These theorems prove the impossibility of David Hilbert's project of finding a set of formal axioms from which all of mathematics (specifically Godel consideres number theory) can be derived. The further requirement is to prove that these axioms are consistent i.e. cannot derive any contradictions. In a landmark paper, Kurt Godel proved both these projects are impossible. He showed that any reasonable (we will discuss what is required to make an axiom system reasonable) axiom system capable of expressing integer arithmetic must necessarily either be inconsistent i.e. derive a contradiction or incomplete i.e. cannot prove all true statements. We will dive into the murky depths of mathematica logic to discuss the proof of this remarkable theorem.

M748: Measure Theory and Vitali Sets in Splash Fall 2018 (Oct. 28, 2018)
Given an arbitrary set of points in the real numbers, how do you measure its length? When the set is made from intervals this is quite clear but what about crazier sets like the rational numbers inside the reals. Can the notion of length be generalized to any number of dimensions? We will explore all these questions and develop the field of measure theory, the rigorous foundation for the study of size and probability. Along the way we will find that inevitably (up to choices for the axioms of set theory) any measure function, that is a choice for the length of each set, must fail to give a reasonable answer for some sufficiently nast set. We will give a description of a Vitali set which is an example of such an immeasurable set. If time permits, we will discuss the implications of immeasurable sets to the famed Banach-Tarski paradox which allows one sphere to be turned into two through cuts and rotations alone.

S624: Way Too Much General Relativity to Fit in 3 Hours in Splash Spring 18 (Mar. 31, 2018)
A mathematically rigorous introduction to general relativity, Einstein's legendary theory of gravity. This class will explore the motivation behind geometric curvature and its connection to gravity. Topics will include: the basic theory of manifolds and differential geometry, geodesics, the Einstein Field Equations, Black Holes, Gravitational Waves, and (time permitting) the cosmological constant and models of the universe. Get ready to learn about this beautifully geometric theory and the bizarre consequences of Einstein's elegant field equations connecting energy to space-time curvature, $$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} $$

M631: Constructible Regular Polygons and Galois Theory in Splash Spring 18 (Mar. 31, 2018)
The ancient Greeks asked a simple question, what shapes could be geometrically constructed using nothing more than a compass and a straightedge? Quickly it was discovered that equilateral triangles, squares, regular pentagons, regular hexagons could all easily be constructed. But then progress came to a screeching halt. Constructing the regular heptagon eluded mathematicians for a thousand years until Gauss proved its impossibility in 1796. Better still, Gauss proved the incredible theorem: a regular n-gon can be constructed if and only if $$ n = 2^k \cdot p_1 \cdot p_2 \cdots p_r$$ where the numbers $$p_i$$ are primes of the form $$p_i = 2^{2^m} + 1$$ (known as Fermat primes). The proof of this fact will lead us to the development of Galois theory, one of the greatest developments in modern mathematics. This two millennium old problem will bring us face to face with an unsolved problem in modern mathematics: are there infinitely many Fermat primes? Equivalently, as Gauss showed, are there infinitely many constructable odd-sided regular polygons?

S685: Introduction to Statistical Mechanics in Splash Spring 18 (Mar. 31, 2018)
In introductory physics classes, one spends a lot of time thinking about the motions of a small number of particles. But in real life, things aren't made of one or two particles, they are made of at least a trillion trillion atoms. How can a physicist ever hope to understand the properties of real materials and objects given this complexity? The answer is statistical mechanics, asking not what a material will do but rather what is the most probable state to find it in. In this class we will introduce the methods of statistical physics, understand fundamental quantities such as entropy and pressure from the viewpoint of statistical ensembles and, time permitting, apply these ideas to the problem of spontaneous magnetization in magnetic materials.

M564: Constructing a Transcendental Number in Splash Fall 2017 (Nov. 04, 2017)
A complex number is called algebraic if it is the solution so some polynomial with rational coefficients. The question remains, are all complex numbers algebraic? The answer, given by Cantor, is a resounding NO. Almost all complex numbers (in a technical sense) are not algebraic, rather, they are transcendental because they "transcend" algebraic definition. However, Cantor's proof does not construct an explicit example. In this class we will undertake the task of constructing explicitly a family of transcendental numbers. On the way, we will prove Cantor's theorem and define a measure of how irrational a number is.

S565: Relativistic Quantum Theory in Splash Fall 2017 (Nov. 04, 2017)
An intense introduction to the mathematics of relativistic quantum mechanics. This class will introduce Lorentz transformations, relativistic mass-energy, the Klien-Gordon equation (and why it fails), and finally the magnificent Dirac equation. We will show that the odd phenomenon of spin is a necessary consequence of including relativity in quantum mechanics.

M566: p-adic number theory and geometry in Splash Fall 2017 (Nov. 04, 2017)
The p-adics are an alternate metric completion of the rational numbers. Metric completions will be discussed as will the geometry of p-adic numbers.

M567: The Most Integral Parts of Number Theory in Splash Fall 2017 (Nov. 04, 2017)
The basics of number theory focusing on the divisor sum function, the greatest common divisor, and Bezout's identity. The class will culminate with the proof of the Euclid-Euler theorem which gives a correspondence between primes of the form $$2^p - 1$$ and perfect numbers.

M575: Some Prime Results in Number Theory in Splash Fall 2017 (Nov. 04, 2017)
Further topics in number theory focusing on Euler's phi function and multiplicative order. The class will culminate with the proof of the primitive root theorem which describes the multiplicative structure of modular arithmetic. Applications of the primitive root theorem such as quadratic equations in modular arithmetic and Carmichael numbers will be discussed if time permits.