Fall 2017

Moonshine For All

Speaker: Neekon Vafa and Xavier Gonzalez, Harvard Undergraduates

Date: 11/28/2017

Abstract: "Monstrous Moonshine" is one of the most fantastical—and whimsically named—stories of modern mathematics. Those who attended Brian Warner's "scary MONSTER" Halloween Math Table have already enjoyed the first part of this saga. We will recount the rest of the story with relish before presenting some of our results from Ken Ono's summer REU (Research Experience for Undergrads) at Emory, where we worked together along with undergrads S. DeHority of UNC and R. Van Peski of Princeton. Acquaintance with characters of representations of finite groups would be helpful for understanding our discussion of the MONSTER group. We will then review the theory of modular forms to provide the necessary background for an appreciation of the original Moonshine conjecture and an explanation of our results. Our talk can serve as a continuation of Brian's talk, but by no means was attending it necessary for comprehending ours.

Knots and the Spirit World

Speaker: Sander Kupers, Harvard University

Date: 11/14/2017

Abstract: In 1878 professor Johann Zöllner claimed to have proof of the existence of a spirit world, by doing experiments with medium Henry Slide and using the mathematics of knots in three and four dimensions. These mathematical ideas were cutting-edge research at the time, and are still of interest. I'll explain them, and whether his claims stood up to scrutiny (spoiler: they did not).

Introduction to Morse Theory

Speaker: Rohil Prasad, Harvard Undergraduate

Date: 11/07/2018

Abstract: A common question that one encounters in a multivariable calculus class is, given a smooth function F from R2 to R, what are the local maxima and minima? As in the single variable case, the maxima and minima are at the critical points of the function, which is where the gradient vanishes. However, not all of the critical points are maxima or minima. There are also weird critical points called saddle points, which look like maxima in one direction and minima in another. In Morse theory, we replace R2 with a smooth manifold M of dimension n. The function F is now a smooth function from M to R, and the situation gets even crazier. The critical points can be maxima, minima, or anything in between. However, it turns out that we can use these critical points to our advantage to obtain a lot of information about the topology of the manifold M. In particular, the critical points serve as a sort of "roadmap" for the construction of a decomposition of M into nice topological shapes called handles. The handle decomposition and its associated "calculus" are a key tool in some of the most powerful results in the classification of manifolds. The talk will be self-contained, and references will be given for anyone who would like to work through the more technical details. We will work to (roughly) define manifolds, define Morse functions and look at a few examples, and then talk about the handle decomposition.

The Monster Group and Scary Character Tables

Speaker: Brian Warner, Harvard Undergraduate

Date: 10/31/2017

Abstract: The fundamental theorem of arithmetic allows us to write any natural number as a unique prime factorization. One can imagine classifying---so to speak---every natural number by the primes that it is made up of. Such classifications are nice and comfortable, they let us imagine building up complicated numbers from simple fundamental components. It is natural (no pun intended) to wonder if we can do this for other mathematical objects, like groups. This is in some sense the goal of the classification theorem for finite simple groups. However, what would otherwise be a very nice classification, like that for integral numbers, is muddled by a collection of 27 groups, known as the ``sporadic groups'', who refuse to play nice. This talk will focus on the largest of these groups: the Monster Group. As we describe this frighteningly large beast, we will along the way find ourselves learning about classification theorems, supersingular primes, Galois groups, and how in the world anyone could begin to effectively describe a group of order 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The only background needed is some familiarity with groups, as several basic facts and definitions will be assumed and not shown/proven.

Cayley Graphs and the Grothendieck Construction

Speaker: David Spivak, MIT

Date: 10/24/2017 

Abstract: Category theory has become the central gateway through which to learn pure mathematics. It organizes concepts and highlights the commonalities that appear throughout the field into a coherent mathematical framework. The Grothendieck construction is one example. It generalizes two important ideas from group theory: the Cayley graph of a group and the semi-direct product of groups. Because of time constraints, I only have time to talk about one of these during this talk, namely Cayley graphs. I'll assume the audience knows what groups and group homomorphisms are. But I will define categories, functors, the Grothendieck construction, and the Cayleygraph of a group during the talk.

6561101970383!

Speaker: Noam Elkies, Harvard University

Date: 10/17/2017

Abstract: Let n = 6561101970383. How many digits does n! have? Once that’s solved: how did I find this n (which is the smallest of its kind past n = 1), and is its size surprising? As usual for problems that depend on arithmetic base 10, this property of n! is of only recreational significance, but the analysis leads to some unexpected uses of familiar ideas ranging from differential and integral calculus to the Euclidean algorithm to computational complexity. 

How to Compute a Cobordism Group

Speaker: Carlos Albors-Riera, Harvard Undergraduate

Date: 10/10/2017

Abstract: The classification of manifolds up to cobordism was a major accomplishment in topology. In this talk I'll define what it means for two manifolds to be ``cobordant'' and motivate the study of ``cobordism groups.'' Afterwards, I'll sketch the surprising connection between cobordism and homotopy theory that paved the way for the computation of these groups. Prerequisites: A basic familiarity with abstract algebra and topology.

Poincare-Birkhoff: an island of stability within chaos

Speaker: Davis Lazowski, Harvard Undergraduate

Date: 10/03/2017

Abstract: Often, we are interested in the time evolution of dynamical systems described by differential or difference equations. A natural question is: if the equations of the system change `a little bit', what do we know about the resulting system? In this talk, I will discuss a version of the Poincaré-Birkhoff theorem which helps us start to answer this question, by telling us what happens around `nice' solutions of the system. I will draw lots of pictures to give intuition for the subject, and mention applications too, and historical motivation from physics. The talk will assume no prerequisites other than calculus, though at limited points some proofs might use basic topology or slightly more advanced analysis.

Tuesday, Sep 26 (6:00pm, SC 232)

Title: Math Experiences Math Table (Sponsored by Crews)

Several students shared their experiences in mathematics. If you are a Harvard student and have any questions about summer programs, feel free to look up their emails on the Harvard system and send them your questions!

Serina Hu: PRISE summer research

Vaughan McDonald: PROMYS math camp counselor, REU summer research

Davis Lazowski, Michele Tienni: Trip to China with Prof. Jaffe

Natalia Pacheco-Tallaj: REU summer research

Hanna Mularczyk: Budapest summer/semester abroad in math

Shyam Narayanan: REU summer research, finance internship, PRISE summer research

Luke Melas-Kyriazi: Harvard-MIT math tournament, finance internship

Vector Bundles and K-theory

Speaker: Cameron Krulewski, Harvard Undergraduate

Date: 09/19/2017

Abstract: In algebraic topology, we are interested in using algebraic structures to study topological spaces. That is, we like to associate an algebraic invariant to a topological space such as a sphere or a torus, and use that invariant to discover properties of the space and distinguish it from other spaces. One such invariant is K-theory, which associates a ring to a topological space. While K-theory has many sophisticated properties, it actually has a very geometric and visual definition in terms of structures called vector bundles. In this talk, we will explore how vector bundles can be constructed over topological spaces, and how operations between them can allow us to define a ring. Some familiarity with the definition of algebraic groups and rings and with the concept of continuity will be helpful, but this talk will feature many clarifying examples

Adapting to a Chaotic World

Speaker: Rosalie Belanger-Rioux, Harvard University

Date: 09/12/2017

Abstract: Finding the roots of functions is quite challenging. For degree 2 polynomials we have the quadratic equation. However, beyond polynomials of degree 4, or for generic functions, no such formulas exist. We then need to resort to approximating the roots of our function. A classic approach to this problem is called Newton's Method, which often creates fractals. Fractals are beautiful, but tend to cause misleading or wrong solutions. This talk will present many beautiful pictures of the chaos caused by Newton's method, but will also attempt to do away with the chaos by using an adaptive Newton's method. Adaptive methods are an incredible tool used in many branches of computational mathematics, often based on some basic principles I will describe. This talk dovetails on last week's talk, but you do not need to have attended last week to come this week. My presentation will be elementary, while alluding to some deep mathematics along the way.