## Undergraduate Mathematics Seminar

The Johns Hopkins Undergraduate Mathematics Seminar hosts regular talks given by speakers of all academic levels about advanced topics in mathematics, geared to an undergraduate audience. The Spring 2024 semester marks the seventh semester of the undergraduate seminar. The talks will be held in **Krieger 413** on **Mondays at 3:30**.

If you are interested in giving a talk, please submit *this form*.

Below is a list of talks that will be updated through the semester. For a list of talks given in previous semesters, please take a look at our *archive of talks.*

## Schedule of Talks - Spring 2024

**April 22nd**

*Yash Lal*:

**TBD**

**Abstract:** TBD

**April 15th**

*Yoyo Jiang*:

**TBD**

**Abstract:** TBD

**April 8th**

*Ben Marlin*:

**TBD**

**Abstract:** TBD

**April 1st**

*Sean Owen*:

**TBD**

**Abstract:** TBD

**March 4th**

*Kailee Lin*:

**Functional Graph Theory**

**Abstract:** In my talk we will see some fundamental ideas of functional graph theory. A functional graph is a directed graph where each vertex has out-degree = 1. We will explore what taking a functional approach to graph theory can reveal, and what polynomials have to do with any of this? We will use this understanding to explore problems relating to graph coloring, graceful labeling, and more! This talk will be accessible to folks of any math background.
**Prerequisites**: None

**February 26th**

*Jonathan Lin*:

**The Kissing Problem and Linear Programming bounds**

**Abstract:** The Kissing Problem (in dimension n) asks how many blue
spheres of radius one can touch a red sphere of the same size. Using
what are known as "Linear Programming Bounds" we will illustrate how
this problem can be elegantly resolved in dimensions 8 and 24.
**Prerequisites**: None

**February 19th**

*Simone Masserini*:

**Topological methods for solutions to semilinear equations.**

**Abstract:** Introduction to topological degree theory in finite and infinite dimensional spaces; Brouwer and Leray-Shauder fixed point theorems and existence of nontrivial solutions to Partial Differential Equations.
**Prerequisites**: None

**February 12th**

*Anish Chedalavada*:

**Simplicial Homology, starting from V - E + F.**

**Abstract:** We will discuss some of the history behind the Euler characteristic, the Betti numbers, the homology groups and ultimately the definition of simplicial homology. This story is an incredibly exciting snapshot into how mathematical intuitions come about and build on each other, and if time permits I would like to highlight how this might inform your own experiences with the field.
**Prerequisites**: None

**February 5th**

*Sofia Taylor*:

**Classification Using Support Vector Machines with Uncertainty Quantification**

**Abstract:** Binary classification using machine learning is needed to address engineering problems such as identifying passing/failing parts based on measured features from aging hardware. In these classifications, providing the uncertainty of each prediction is essential to support engineering decision making. One popular classifier is the support vector machine (SVM). There are many variations, with the simplest being a linear division between two classes with a hyperplane. Kernel methods can be implemented for data that is not linearly separable. Adding the use of a loss function allows for control over misclassification in the training data. The many variants of SVMs and their tunable model parameters readily enable ensembling, which we use to represents model (epistemic) uncertainty. SVMs also can be used to depict intrinsic uncertainty in the data, or aleatoric uncertainty. One of the major shortcomings of SVMs is that they make predictions indiscriminately over the entire prediction space, resulting in seemingly high-confidence predictions where there is no data. In this talk, we present one way around this by combining ideas from outlier-detection methods and SVMs, with the goal being to surround each class while simultaneously separating it from the other class. A unique representation of UQ results from this “single-class” SVM strategy is presented.
**Prerequisites**: None

**February 2nd**

*Prof. Su Ji Hong*:

**Integer Decomposition Property of Lattice Polytopes**

**Abstract:** Given a polytope (higher dimensional polygon), P, and k \geq 0, we can multiply every point of P by k to obtain another polytope kP. One of the properties P can have is the integer decomposition property: if x is a point in kP, then there exist k points in P such that they add up to x. Not all polytopes have this property. At the Graduate Research Workshop in Combinatorics, we showed that certain lattice polytopes have this property.
**Prerequisites**: None

**January 26th**

*Prof. Xavier Ramos Olivé*:

**The role of curvature in the production of sound: Music, Geometry and Beyond**

**Abstract:** When a string vibrates, it produces different
pitches depending on its length. This is how guitarists can
play several notes using only one string. But what happens
when we have surfaces or 3D objects that vibrate, like
when playing a flute, a digeridoo, or a handpan? Beyond
their size, does their shape affect the pitch? We will discuss
the role that curvature plays in this question, introducing
some intuition for what curvature is, and presenting some
recent developments regarding how sound changes under
small perturbations of the geometry of an instrument.
More precisely, we will revisit classical theorems in
Geometric Analysis related to the eigenvalues of the
Laplacian on Riemannian manifolds, and we will introduce
newer versions of these results that use integral Ricci
curvature conditions. If time permits, we will also discuss
new related directions of research that connect Differential
Geometry, Graph Theory and Data Science.
**Prerequisites**: None

**January 24th**

*Dr. Kayla Wright*:

**Frieze Patterns**

**Abstract:** As we are in the midst of a cold, dark winter... there is no better time to learn about frieze patterns. Frieze patterns were first defined by mathematicians to classify certain symmetries in ancient art and architure. In this talk, we will explore arrays of mathematical objects that obey
these classified frieze pattern symmetries. We will
start by looking at frieze patterns of integers and
then show that these integers can be
interpreted both geometrically and algebraically.
More specifically, we will connect frieze patterns of
integers to lengths of diagonals in polygons as well
as generators for a combinatorially defined algebra
called a cluster algebra. At the end of the talk, I will
give a loose definition for these algebras and
mention some important questions in cluster
theory.
**Prerequisites**: None

**January 22nd**

*Prof. Boya Wen*:

**Ways to write a positive integer into sums of positive integers**

**Abstract:** Take a positive integer, say 4; we can write it as a sum of positive integers in the following five ways: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Among these five ways, how many have only odd numbers as summands? How many of them have no repeating summands? Do the two numbers agree? Do the odd-summand ones have more summands in total, or do the distinct-summand ones? What if we replace 4, by 5, or 6, or any positive integer n? In this talk, we will explore these questions, methods to study them, and their generalizations. Time permitting, I will talk about my recent joint work with Cristina Ballantine, Hannah Burson, William Craig, and Amanda Folsom, answering a generalization of these questions.
**Prerequisites**: None

## Schedule of Talks - Fall 2023

**November 27th**

*Yash Lal*:

**Size and Topology**

**Abstract:** In this talk, we will begin by motivating the definition of a topology on a set through geometric considerations. Then, we will be talking about a fundamental concept from topology: the notion of size. We will discuss it from two distinct viewpoints, that of compactness and that of filters, and combine the two to prove a hard theorem. In the end, we will touch on the more abstract implications of this theory.
**Prerequisites**: an informal understanding of continuity of a function and the extreme value theorem (covered in calculus 1)

**November 6th**

*Daniel Pezzi*:

**An introduction to the Fourier Transform and the Restriction Conjecture**

**Abstract:** This talk will motivate the definition of the Fourier Transform and establish its connection to PDEs. From a PDE perspective we will state an equivalent form of the restriction conjecture, one of the most prominent outstanding problems in harmonic analysis.

**October 23rd**

*Prof. Yiannis Sakellaridis*:

**The mathematics of quantum mechanics**

**Abstract:** The development of quantum mechanics in the early 20th century was a revolution in physics, and had a transformational impact on mathematics, as well. Even as we still grapple with the interpretations of the theory, its mathematical foundations are rigorous, and rooted in simple concepts of linear algebra. I will discuss these foundations from scratch, assuming only familiarity with vector spaces, linear operators, and eigenvalues.
**Prerequisites**: None

**October 9th**

*Prof. Nathan Pennington*:

**Why you should take Differential Equations**

**Abstract:** A first encounter with differential equations often leaves the impression that differential equation is just integrating with extra steps. In this talk we'll show that what it means to "solve" a differential equation is more subtle (and mathematically interesting) than it initially appears. We'll also describe how these different types of "solutions" contribute to the wide applicability of differential equations.
**Prerequisites**: Calc 2, Some Linear Algebra, Calc 3

**September 25th**

*Prof. Ashwin Iyengar*:

**Elliptic Curves and Modular Forms**

**Abstract:** If you’ve heard of Fermat’s last theorem, you might have heard that its proof involves some sophisticated geometric objects that seemingly have nothing to do with whether a^n + b^n = c^n for n > 2. The objects in question are elliptic curves and modular forms. I will attempt to explain what these are, and how despite their apparent unrelatedness (both from each other and from Fermat’s “theorem”) they are related in intricate and unexpected ways. The prerequisites for the talk will be as minimal as possible.
**Prerequisites**: None

**September 11th**

*Prof. Emily Riehl*:

**A Reintroduction To Proofs**

**Abstract:** An introduction to proofs course aims to teach how to write proofs informally in the language of set theory and classical logic. In this talk, I'll explore the alternate possibility of learning instead to write proofs informally in the language of dependent type theory. I'll argue that the intuitions suggested by this formal system are closer to the intuitions mathematicians have about their praxis. Furthermore, dependent type theory is the formal system used by many computer proof assistants both "under the hood" to verify the correctness of proofs and in the vernacular language with which they interact with the user. Thus, there is an opportunity to practice writing proofs in this formal system by interacting with computer proof assistants such as Coq and Lean.
**Prerequisites**: None

## Schedule of Talks - Spring 2023

**March 13th**

*Prof. Victoria Akin*:

**Harry Houdini’s Magic Math Trick**

**Abstract:** What theorem is so mystifying that Harry Houdini was writing about it in his 1922 book, Paper Magic? In this talk, we’ll hint at the proof of a surprising theorem and demonstrate some magic of our own! Along the way, we’ll develop some mathematical machinery for paper folding and paper flattening. We’ll also explore some cutting, unfolding, and flattening of polyhedra. With any luck, we’ll state an open question or two.
**Prerequisites**: None

**April 14th**

*Dr. Ben Dees*:

**Buffon's Needle**

**Abstract:**Suppose that I draw a bunch of lines parallel to each other, spaced one inch apart, and drop my standard-issue one-inch needle onto them at random. How likely is it that the needle will cross one of the lines? More generally, what if the needle is longer, shorter, or is actually a squiggly piece of uncooked pasta? There are, broadly speaking, two approaches to this. One of them involves setting up some number of integrals to figure out these probabilities. This approach is entirely valid and will work, but there’s another approach that just relies on probability theory. In this talk, we will compute no integrals, and use no calculus. Instead, we will see a marvelous display of the glorious power called “linearity of expectation,” and that’s all we’ll need.
**Prerequisites**: None

**April 28th**

*Yoyo Jiang*:

**Introduction to Monoidal Categories**

**Abstract:** In this talk, we will introduce categories, look at some examples, and see how category theory can help us describe mathematical objects. We will attempt to reconstruct the integers with the familiar properties that we know and love using categorical language, and use that to motivate the definition of a monoidal category. We will finish by seeing some fun examples of monoidal categories. No background is required, although some exposure to linear algebra or abstract algebra will be helpful.
**Prerequisites**: None