A lightning talk is a short presentation meant to quickly introduce a
topic or concept to an audience. At SUMS, we host lightning talks
related to mathematics and related fields and industries. In other
words, if your talk is marginally related to math, we'd love to
host your talk!

Our lightning talks typically last 15 minutes to an hour. Anyone can
give a lightning talk, including, but not limited to, undergraduate
and graduate students, faculty, and staff. Like almost all of our
other events, attendance is open to the public.

If you'd like to give a talk, please sign up
here.

After filling out the form, all you'll have to do is present your
talk. We can take care of event logistics and publicity for you!

Homotopy and the Hopf Fibration

Zack Garza | Jun 4, 2018

Homotopy theory provides tools for studying topological spaces -
roughly speaking, the homotopy groups πn(X) of a space X can be
defined in terms of maps Sn → X from n dimensional spheres into it,
yielding a method of “probing” a space for n dimensional holes.

This talk will introduce some of the computational and theoretical
tools used in homotopy theory. It will also cover some of the
background needed to understand the Hopf Fibration, a map
constructed by Heinz Hopf in the 1930s which wraps S3 around S2 in a
nontrivial way. This has led to surprising consequences, including
the existence of higher homotopy groups of spheres.

(There will also be several visualizations that demonstrate the
beautiful structure that emerges!)

Systems, Convolution, and Fourier Analysis

Eric Gelphman | May 20, 2018

We will first show how the response of an nth order linear system can be mathematically described by the operation of convolution by solving the ODE that describes the response of an nth order linear system using a Green’s Function. We will then go over how convolution can be transformed into multiplication by the Fourier and Laplace transforms. Finally, we will discuss applications to both electrical and biological systems.

To infinity and beyond, but why stop there?

Jonathan Lim | Mar 1, 2018

”I can’t talk about our love story, so I will talk about math. I am not a mathematician, but I know this: There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities.” - John Green Although this argument may seem intuitive, when it comes to infinities, there are many times when our intuitions are wrong. In fact, the cardinality or the size of the interval [0,1], [0,2], and even [0,1,000,000] are all the same. Why is this true? Well come out to our next Lightning talk about cardinalities for a soft introduction to the puzzling idea of infinity and beyond!

Topological Fixed Point Theorems

Zack Garza | Mar 1, 2018

If you’re ever bored in Math class, try pulling out your calculator, putting in a random number, and then start hitting the cosine button. Before long, the digits will stop changing, which denotes a fixed point of the function. This idea generalizes to different kinds of maps and spaces, and using the tools of topology we can discover some counter-intuitive results about the world we live in - for example, can you ever truly mix a cup of coffee? And is it possible for two opposite points on a planet to have the exact same temperature and atmospheric pressure? These questions can be answered using the Brouwer Fixed-Point theorem and the Borsuk-Ulam theorem, which are the topics we’ll be discussing in this talk.

Homology and the Snake Lemma

Zack Garza | Nov 20, 2017

Homology is an algebraic tool that has proved to be useful in many
areas of Mathematics, and in recent years has even been used in
applied settings such as large scale data analysis.

This talk is motivated by the movie “It’s My Turn” from 1980, which
features an infamous scene involving a result known as “The Snake
Lemma.” Surprisingly, this scene is mathematically precise, and is
even referenced as a proof in a text by Weibel’s “Introduction to
Homological Algebra”!

The goal of this talk will be to provide a brief overview of some of
the mathematics used in this scene, and ultimately to formulate what
the Snake Lemma is and an example of how it can be used. In the
process, we will also cover some of the basics of homology and its
generalization in homological algebra.

Algebraic Geometry: A Historical Primer

Zack Garza | Oct 25, 2017

Algebraic Geometry is often touted as an abstruse, difficult to penetrate subject - however, these two branches of Mathematics have been intertwined since the time of the Greeks! In this talks, we will discuss Algebraic Geometry from a historical perspective, leading up to a brief, high-level modern overview of “the Nullstellensatz”, a theorem which provides a dictionary between the worlds of algebra and geometry.

Boolean Fourier Analysis

Peter Amidon | May 26, 2017

Boolean Fourier Analysis has proven to be a remarkably versatile tool in various fields in theoretical computer science. In this talk I will introduce the basics of the technique and show how the simple change-of-basis that is Fourier analysis can solve many problems. In addition I will show a Fourier-analysis based proof of the famous Arrow’s Theorem which shows an approximate version of it: if a ranked-choice voting system has a high probably of choosing a Condorcet winner, then in some way it is “close” to being a dictatorship.

Vibrations on a Drumhead

Simon Hu | Apr 28, 2017

In this talk, we will discuss the problem of vibrations on a
drum-head. Specifically, we find an analytical solution to the wave
equation in two dimensions by using Bessel’s functions.

We first derive the Bessel’s function, which is a solution to a
Second-Order, Linear, Ordinary Differential Equation. Then we
discuss, shortly, the wave equation and the fundamental solution.
Finally, we will discuss the problem of vibrations on a drum-head.
If time permits, we will discuss applications of this problem.

Brownian Motion

Kim Udomprapasup | Mar 17, 2017

This talk will discuss the mathematical theory of Brownian motion.
We will begin with the description of Brownian motion, then move on
to some of its interesting properties. Its connections to other
areas in mathematics will also be mentioned, for example, harmonic
analysis and PDE.

If time permits, we will get a glimpse of stochastic calculus as
well.

Introduction to Category Theory, Part 2

Zack Garza | Feb 24, 2017

Initial and terminal objects, products and coproducts, use of universal property in quotient and polynomial ring constructions, pullbacks/pushforwards, adjunction, monoids, monads.

The Galilean Group and Mechanics, both Classical and Quantum

David Moore | Feb 24, 2017

This talk will discuss the mathematical formulation of
nonrelativistic classical and quantum mechanics. Both boil down to
ordinary and partial differential equations, and both possess the
galilean group as their symmetry group.

We will start at the group and its corresponding Lie algebra,
introduce the Poisson bracket (classical) and commutator (quantum),
and derive some implications of the bracket + symmetry group laws.
Throughout, we will stick as close as possible to differential
equations, and ignore or tread around any functional analysis issues
that crop up.

Category Theory as an Organization Tool

Zack Garza | Jan 27, 2017

This talk will continue some of the ideas introduced in last
quarter’s talks on Category Theory. (We’ll also review some of the
basics for people that didn’t make the first one.) The focus this
time will be to see a number of examples of how Category Theory ties
together various branches of branches of Mathematics.

We’ll cover many of the structures seen in undergraduate classes
like Analysis, Algebra, Combinatorics, and Topology, and how they
are related to each other. I’ll also be explaining what the term
“isomorphism” means in all of these contexts.

Math Jokes by Physicists: fractal dimension, random walks, the Ising model, and the continuum limit

David Moore | Dec 2, 2016

Efforts in statistical field theory - a physics subject similar to quantum field theory - are plagued with infinities. In this talk I will outline the path from fractals and self-avoiding random walks, to the Ising model and its ilk. This involves hilarities like the scrubbing away of infinities, limits as integers approach zero, and the path integral.

Matroids and Greedy Algorithms

Abel Mariam | Dec 2, 2016

There are a class of algorithms referred to as Greedy algorithms
which are broadly characterized by the strategy of “picking what
looks best at the moment”. It turns out that the theory of matroids
provides the appropriate framework to partially characterize
problems that have a greedy solution. A matroid, first introduced in
1935 by Hassler Whitney, is a combinatorial structures that
generalizes the notion of independence common in many different
areas of mathematics.

In this talk, I will introduce some basic definitions and results
and discuss Matroid formulations of some classic problems in Graph
Theory (Minimum Spanning Tree, Vertex Matching, and Interval
Scheduling).

Haskell for Mathematics

Zack Garza | Sep 30, 2016

Basic of Haskell syntax and semantics, using examples from Mathematics. Topics: types, functions, lazy evaluation, partial application/currying, custom datatypes/records, common paradigms in functional programming (map/filter/reduce), overview of Prelude & Hackage.

Introduction to Category Theory, Part 1

Zack Garza | Sep 30, 2016

Motivation and history, basic definitions, examples of categories, overview of big ideas (commuting diagrams, universal properties, functors, naturality, duality, limits/colimits, Yoneda).

The Next Greatest Number

Matt Chan | Sep 30, 2016

Basics of point-free style programming (i.e. programming by function composition) and equational reasoning. How to derive a solution to a typical CS interview brain teaser by equational reasoning in Haskell, and how to transform the derived algorithm into an efficient one by calculation.