**Fall 2023’s talks:** (This list will be updated as talks are scheduled.)

**September 12:** Dan Margalit

**Title: **Stuffing Mobius Bands in Space

**Abstract.** How many Mobius bands can you put in 3-dimensional space? Answer: not so many. We’ll discuss this question and the answer, while touching on topics such as topology, set theory, and combinatorics. The talk will be accessible to anyone with an interest in mathematics.

**September 19: **Fabian Salinas

**Title: **Graph Pebblings and A Meta-Fibonacci Sequence

**Abstract. **Pebble domination is a mathematical game played on graphs. While the rules and objective of the game can be quickly grasped, solving for the optimal pebble strategy is surprisingly nontrivial. For the case of Perfect Binary Trees, we discuss our method on finding an optimal pebbling strategy. This method leads to strange connections to a Meta-Fibonacci sequence (OEIS A046699).

**September 26: **Junhwi Lim

**Title: **Shor’s Factoring Algorithm

**Abstract. **Given an integer N, how do you find its prime factors? There are several algorithms you can run on computers to get the answer. If N is a 100,000-digit integer, a modern supercomputer will give you the answer in over 10^15 times the age of the universe. However, if you run Shor’s Factoring Algorithm on a quantum computer, you can find the answer in less than a second. In this talk, we will explain what Shor’s Factoring Algorithm is. First, we start by introducing linear algebra, which is a mathematical theory used in quantum computation. Then we will see how general quantum computations and Shor’s factoring algorithm works. If time permits, we will go through the number theoretic details of the algorithm.** **

**October 10: **Fabian** **Circelli

**Title: ** Infinities, Foundations of Mathematics, and Set Theory

**Abstract. **Infinity is not an entirely intuitive concept: there are just as many whole numbers as there are even numbers, even though the set of whole numbers contains the set of even numbers. There are as many numbers in the interval (0, 1) as there are in the entire real line. We will see how mathematicians navigate infinities, and how the foundations of mathematics are designed to prevent paradoxes. This will lead us into set theory, the bedrock of mathematics, and its interesting history, which is full of debate about sets, infinities, and what is mathematically valid.

**October 17: **Glen Webb

**Title: **Models of COVID-19 Epidemics

**Abstract. **Mathematical models are developed to provide predictions for COVID-19 pandemics. The models incorporate asymptomatic and symptomatic transmission. The models incorporate reported and unreported cases. Reported case data is used to parameterize the models. The models are used to project the epidemic forward with varying public health measures.

**October 24: **Andreas Mono

**Title: **Who cares about numbers? A (gentle) introduction to (analytic) number theory

**Abstract. **The aim of this informal talk is to introduce the area of analytic number theory while requiring as little background as possible. The relationship between numbers and mathematics is somewhat similar to the one between molecules and chemistry: The former serves as a central building block of the latter. In this analogy, prime numbers are the atoms of mathematics in the sense that every natural number is a product of prime numbers. Moreover, this factorization is unique up to reordering factors like a molecule consists of atoms and it does not matter in which order we count its atoms. Number theory is devoted to the investigation of the behavior of prime numbers, and analytic number theory utilizes the tools from calculus to this end. We present some (famous) results as well as objects in the field, with beauty and elegance guaranteed. The knowledge of complex numbers is beneficial, as we will get started from there.

**November 7: ** Anna Vinnedge

**Title: **Intro to Number Theory, Abstract Algebra, and Applications in Cryptography

**Abstract. **In this talk, we will cover some basic concepts in number theory such as modular arithmetic, the Euclidean algorithm, and properties of prime numbers. We will then see how these concepts are utilized in concurrence with basic abstract algebra concepts to construct various types of encryption methods, as well as define terms relevant to the fields of cryptography, cryptology, and cryptanalysis. We will define groups and see how cyclic groups played a foundational role in the early cryptosystems, dating back as early as 100 B.C. We will further discuss how cryptography has progressed since its creation and give summaries of more modern cryptosystems. We will conclude with a brief overview of the state of cryptography today by introducing quantum cryptography and DNA-based cryptography.

**November 14: **Madi Sousa

**Title:** Frame Theory and Phase Retrieval

**Abstract. **Since its introduction in the early 1950s, Hilbert space frame theory has become an active area of research due to its applications in engineering and physics, including speech recognition, signal processing, and X-ray crystallography. Frames, like orthonormal bases, provide a stable way of representing a signal. However, frames allow for redundancy and flexibility in design, which makes frames much more adaptable in both theory and applications. Frame theory partly belongs to fields such as harmonic analysis, functional analysis, and numerical linear algebra, among others. Phase retrieval is one application of frame theory in which only the intensity of each linear measurement of a signal is available and phase information is lost. In 2006, Balan, Casazza, and Edidin introduced a more powerful notion of phase retrieval using the magnitude of frame coefficients. In this talk, we’ll give a brief introduction to frame theory and phase retrieval and discuss some of the open problems in the field. This talk will be accessible to undergraduates who have taken calculus and basic linear algebra.