Point, Lines, and Preference Orders

Date: Apr 13

Speaker: Erin Delargy

Suppose there are a number of candidates running in an election. Each candidate can be represented as a point in the plane, where each point’s coordinates correspond to the candidate’s position on two topics. Now suppose you’re a voter who must rank these candidates according to how closely their views align with your own. Your preferences give you some position in the plane. The distance between you and each candidate’s point defines a preference order: the candidate whose point you’re closest to is the person you would rank first, the next closest candidate you would rank second, and so on. 

Now consider many voters. How many preference orders are possible among all voters? It turns out this depends not only on the number of points in the plane (the number of candidates running), but also on how the points are arranged (what positions the candidates take). So how many ways can the candidates arrange themselves? And how does this affect the number of possible preference orders (i.e. the number of possible ways people can vote?)

These questions are the basis of a project I worked on as part of an REU last summer. I’ll talk about the approaches my peers and I developed to answer these questions for a small number of candidates. It turns out that everything boils down to points and lines–  so no specialized background knowledge is required!

The Geometry of Hypercubes and Hyperspheres

Date: Feb 23

Speaker: David Biddle

The n-cube and n-sphere are higher dimensional analogues of the square and circle respectively. In this talk we explore the geometry of these objects by finding their diameters and hypervolumes. We'll see the strange result that the hypervolume of an n-sphere has a maximum when n is about 7 and then decreases to zero as n goes to infinity. If time permits, we'll explore some other interesting properties of higher dimensional and infinite dimensional spheres, including an introduction to the homotopy groups of spheres. Only a background in calculus is necessary for this talk.

An Intro to Category Theory

Date: Feb 9

Speaker: Ashton Keith

Cantor, Turing, Gödel, and the General Diagonal Argument

Date: Jan 26

Speaker: Levi Axelrod

Mathematical logic was designed to prevent self-reference from occurring, but a proof technique called the diagonal argument cleverly works around this restriction to self-refer anyway. The technique is common to the proofs of Cantor’s Theorem, the Halting Problem, and the Incompleteness Theorem, three fundamental results from their respective fields. This talk will compare these proofs to break down and understand this mind-bending method.

Hackenbush

Date: Nov 10

Speaker: Ashton Keith

The game of Hackenbush is a game that is ideal for studying. This is what John Conway, Elwyn Berlekamp, and Richard Guy studied in order to build up combinatorial game theory in their four-part series Winning Ways for Your Mathematical Plays. These books can be found below:

https://drive.proton.me/urls/FH83WWW6ER#FulrcjhEiuxG

The Octonions: An Alternative (Algebra) to the Reals

Date: Oct 13

Speaker: Prof. Quincy Loney

Have you ever wondered what kinds of numbers there are? We know there are real numbers and imaginary numbers… but are there others? In this talk we will discuss some of the history and the properties of the octonions, the 8-dimensional normed division algebra discovered by John T. Graves in 1843. We will begin with the real number system and use the Cayley-Dickson process to construct this exciting, alternative algebra.

Some Density Functional Theory

Date: Nov 28

Speaker: Maxwell Meyers

Optimization of atomic structures involves adjustment of structural parameters until an enthalpy minimum is found. It has been proposed that allowing atoms to move in extra dimensions can help locate the global enthalpy minimum. In this work I have implemented this algorithm in the MAISE simulation package and investigated its efficiency for finding global minima. The test systems included Lennard-Jones clusters, known to have complicated ground states, and Li-Sn crystalline alloys, known to have potential in battery applications. The preliminary results indicate that the hyper dimensional optimization can indeed lead to better convergence to lower enthalpy configurations.

In this talk, rather than emphasize the trial results of the software, I plan to focus more on the math behind structure relaxation, the math tricks I came up with to ensure our hyperdimensional structures return to the 3d world, and Behler-Parrinello symmetry functions and their application in higher dimensional systems. 

What Numbers Really Are

Date: Nov 15

Speaker: Levi Axelrod

Factoring Equations

Date: May 5

Speaker: Kevin Ning

Visualizing Objects in 3- and 4-space

Date: Apr 28

Speaker: Prof. José Cuevas

It is a known fact that there are no knots in 4-space. On the contrary, there exist embeddings of 2-dimensional spheres into R^4 that are "knotted." Personally, it is non-trivial to imagine (or even draw) a knotted sphere in dimension four. In this talk, we will introduce ways to represent such surfaces using ideas of knot theory in dimension three. This talk is aimed at undergraduate students: No background will be assumed.

Integration Techniques

Date: Mar 3

Speaker: Ashton Keith

Presentation can be found at:
https://drive.proton.me/urls/4JTHHNBN3C#E9Fj603Jkt36

The Axiom of Choice, Transfinite Induction, and the Well-Ordering Theorem

Date: Feb 24

Speaker: Levi Axelrod

The Axiom of Choice is a widely-recognized necessity of modern mathematics. However, a variety of unintuitive, non-constructive results can be derived from it, one of which is the Well-Ordering Theorem. In this talk, I will explain the Well-Ordering Theorem, give a proof for it using a method known as transfinite induction, and show off a few of the mysterious conclusions that can be drawn from it. This talk will be easier to follow for people that have taken MATH 330, but it is not at all necessary.

Some Interesting Results from Discrete Geometry and Graph Theory

Date: Feb 10

Speaker: William Jones

A common way to attack a mathematical problem is to express it in a space where you have more tools to work with. The friendlier the space, the better chance you have at solving it--- and hopefully gaining some insight through the method of solution. In my talk, I will give an elementary introduction to convex polytopes (a generalization of polyhedra), and present an application of discrete geometry by proving a purely combinatorial result of graph theory, among other things, taken from Matousek's Lectures on Discrete Geometry. The only prerequisites are experience with induction, and a willingness to dive into something new.

Propositional Calculus: Theorem and Metatheorems

Date: Dec 09

Speaker: Pluto Wang

We often hear that logic is the foundation of mathematics, but what does that mean and what does logic do? One of the most important functions of logic in math is to set the standards for proofs. What are proofs? Why should we consider some arguments to be valid while others not? In this talk I will introduce some concepts universal to all logical systems with the example of propositional calculus. To better examine these concepts, we will use truth tables for the usual connectives. For example, 

A B A implies B A or B   A and B

T T T   T T

T F F   T F

F T T   T F

F F T   F F

Reinforcement Learning

Date: Nov 18

Speaker: Nicholas Pellegrino

How do you teach a computer to solve a problem through trial and error?

Reinforcement learning is a type of data-driven machine learning, where the data is collected by an agent exploring some real (or simulated) environment.In this presentation, we will talk about:

- Markov Decision Processes (MDPs)

- Formatting problems as an MDP 

- Solving MDP-formatted problems with Q-Learning, a classic reinforcement learning technique

Some Very Cool Topics in Cryptography

Date: Nov 03

Speaker: Ezra Dyer

This talk will focus on asymmetric encryption systems, namely the Rivest Shamir Adleman (RSA) encryption system. It will cover the basic formulas used to encrypt and decrypt data, and then delve into how and why they work. There will also be a discussion of why in a practical sense these formulas are cryptographically strong due to the capabilities of computers and the problems they have a hard time solving quickly. Much of this talk will focus on the properties of prime numbers, prime factorizations, Euler’s phi function, and modular arithmetic. Using some tricks we can create a formula that is computationally easy to solve in one direction, but difficult to solve in the opposite direction without key pieces of information, yielding a perfect setup for asymmetric encryption. The importance of asymmetric encryption in today’s world can not be overstated, and understanding how it works can lead to some interesting insights into the patterns of prime numbers and semiprime numbers. 

Artificial Neural Networks

Date: Oct 07

Speaker: Adiel Felsen

An artificial neural network (ANN) is a collection of nodes loosely inspired by human brains. While artificial neural networks were first theorized in the 1940s, advances in computer hardware have allowed ANNs to become a core component of modern machine learning. 

Neural networks are everywhere today: from natural language processing (Google Translate, Siri), to autonomous driving, to medical research. The mathematics underlying these systems is a fascinating topic to study.

I plan on discussing:

• The mathematical theory behind neural networks

• Loss functions, activation functions, weights and biases

• A full forward and backward propagation of a simple neural network

• My current research under Professor Kenneth Chiu and Professor Frank Lu

Mathematical Art, A Personal Perspective

Date: Sept 23

Speaker: Prof. Alex Feingold

Art inspired by or related to mathematics goes far back in history (who drew the first circle? first square?) but I would like to concentrate in this talk on more modern contributions to what is now called mathematical art. M. C. Escher is famous for his artwork illustrating hyperbolic geometry in the Poincare disk (for example, Circle Limit IV). Sculptures based on advanced mathematical concepts like elliptic curves or knot theory have been made by Helaman Ferguson. Meeting him in 1978 and learning how he carved torus knots from stone inspired me to make mathematical art from wood, stone and cast bronze. I will show examples of what people have made using a variety of methods, including 3D printing.

The Hadwiger-Nelson Problem

Date: Sept 09

Speaker: William Jones

At the intersection of geometry and graph theory, spacial coloring problems are easily posed, yet many continue to elude the brightest minds. This presentation will discuss the history and progress on coloring problems in general, with a focus on the Hadwiger-Nelson problem (the chromatic number of the plane with respect to unit distance). Last, the speaker will discuss some research he conducted with Auburn University, introducing some open questions which attendees will now be able to work on themselves. The only prerequisites are basic proof methods.

Peano Axioms: A Starting Point

Date: Aug 26

Speaker: Levi Axelrod

Mathematics is based on axioms: statements that serve as the starting assumptions from which interesting conclusions can be drawn by means of logical proofs. This talk discusses the Peano Axioms, a system for axiomatizing the natural numbers, and by extension, the other commonly known number systems. It will primarily focus on the Principle of Mathematical Induction, one of the strongest tools in this field, and it will use this tool to prove basic facts about addition on the natural numbers, such as commutativity and associativity.

Peano Axioms and Number Systems

Date: May 17

Speaker: Pluto Wang

What exactly are the natural numbers? Can you explain to someone who doesn't know what numbers are what "3" is? In the late 19th century, Richard Dedekind came up with a way to do this seemingly impossible task, and Giuseppe Peano formulated a simple construction that captures Dedekind's method. The Peano Axioms are a set of statements about a set (N), an element (0), and an operation (S) on the set, with which we can define the natural numbers. With the Peano Axioms, we can show some important properties about the basic arithmetics; for example, have you considered why a+b is always equal to b+a? We will also discuss how to build other number systems based on these natural numbers.

What is the Riemann Hypothesis, and why does it matter?

Date: Mar 11

Speaker: Ken Ono

The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken  Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances.

A Peek at Hyperbolic Geometry (Part I)

Date: Jan 26 (Part I);   Feb 2 (Part II)

Speaker: David Biddle

Hyperbolic geometry was created in the first half of the nineteenth century in the midst of centuries long attempts to understand Euclid's axiomatic basis for geometry. Einstein, Minkowski and others found in hyperbolic geometry a geometric basis for the understanding of physical time and space. In this introductory talk we introduce a 'model' of hyperbolic space using complex numbers and a little calculus that will allow us to explore the fascinating and often surprising hyperbolic world. For instance we will see that there is an upper bound for the area of triangles, and hyperbolic triangles have angle sum strictly less than pi.

Date: Sept 8

Speaker: Prof. Vaidehee Thatte

I will speak briefly about my mathematical life as a student, teacher, mentor, and researcher. I will also discuss the goals we have as a future AWM chapter and how we will strive to achieve them. Students are strongly encouraged to ask questions; I hope that this will lead to a lively discussion during the event and an open dialogue going forward.

Mathematical Modeling in Virology

Date: Feb 12

Speaker: Dr Jonathan Forde

Mathematical modeling is the art and science of describing real-world phenomena in mathematical form.  When a model incorporates enough detail of current biological understanding, it can be used in conjunction with other sciences to help interpret experimental results and guide future experiments or policies.  In this presentation, I will present mathematical models of viral infections such as hepatitis B and HIV, and the immune responses that protect us from them.  We will look at the analytical and numerical methods used to design and validate the models.

Some of this work was developed as part of an REU program hosted at Hobart and William Smith Colleges.  I will also discuss the REU program, and how to find a research opportunity for this summer.

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Date: Nov 5

Speaker: David Biddle

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Date: Oct 3

Speaker: Matt Evans

The n-dimensional geometry of non-attacking chess pieces

Date: Mar 27

Speaker: Prof. Thomas Zaslavsky

It's actually 2n-dimensional geometry (it lives in R2n) if there are n pieces: two coordinates per piece. An attack by one piece on another is an equation involving the coordinates of the two pieces; that is, it's a subspace of R2n. Avoiding that attack means avoiding that subspace. I'll explain this and explain how it connects with the well-known Eight Queens Problem and similar problems that involve other pieces of chess, like the knight and bishop, and fairy chess, like the nightrider and vizier. The talk will be at 7pm in EB N24.

Estimating a population parameter with different sampling techniques

Date: Feb 27

Speaker: Robert Beblavy

In statistics, there are several ways of taking a sample. Each of these sampling methods have their pros and cons. A question one might wonder is, which method of sampling works “best?” Of course, the term “best” requires context. “Best” may have a different meaning depending on the objective one is trying to achieve by collecting the data. However, here we take “best” to mean providing us with the closest estimate to the desired population parameter. That is, we define “best” to be providing us with a result closest to the actual value of the statistic we are after. The aim of the project is to answer this question by comparing the accuracies with which the statistics generated by the different sampling methods estimate the population parameter. With the statistics tools available today, we can mathematically show how far off our estimates were for samples generated by each sampling technique and compare their accuracies. The talk will be at 7pm in EB N25.

An Introduction to Semigroups

Date: Feb 13

Speaker: Casey Donoven

A semigroup is a set with an associative operation, like real numbers with addition. Semigroups can come in a variety of sizes (empty to infinite) and range from familiar examples (matrices) to the bizarre (bicyclic monoid). In this talk, I'll describe semigroups while focusing on fundamental examples, like integers, words, and transformations. I will also describe one of my current research projects on decomposing semigroups into smaller subsemigroups. The talk will be at 7pm in EB N25.

Cantor's Diagonal Argument

Date: Oct 25

Speaker: Prof. Dikran Karagueuzian

We'll review Cantor's Diagonal Argument showing that the real numbers are uncountable and talk about two other facts that can be proved by the same method: the compactness of the p-adic integers (Don't worry if you don't know what these are.) and the unsolvability of the Halting Problem. (This is computer science, and will also be explained.) The talk will be at 8pm in UU 108.