Abstracts:

Speaker: Matthew Allen

Title: An Introduction to p-adic Number Theory

Abstract: In this talk, I will introduce the topic of p-adic number theory by introducing the p-adic valuation, norm, and metric where p is a prime number. This essentially creates a notion of distance on the rational numbers where a number is small if it is divisible by a higher power of p. I will then introduce the p-adic numbers and briefly discuss an example of why this is useful in number theory with the Hasse-Minkowski theorem.

Speaker: Abby Perryman

Title: An Overview of Noether's Theorem

Abstract: In physics, we are interested in which quantities remain the same throughout the evolution of a system. Noether’s theorem, proven by Emmy Noether in 1915, gives us a way to find these quantities. It states that every continuous symmetry of the action of a system has a corresponding conservation law. This can be used to derive conservation of momentum, conservation of energy, and conservation of angular momentum.

Speaker: Ryan Charette

Title: Merten's Function and the Distribution of Primes

Abstract: In this talk, I will introduce the idea of arithmetic functions and Dirichlet convolution, which I will use to discuss the Mobius function. Then, I will show some of the properties of the Mobius function and explain why it is of interest to number theorists.

Speaker: Michael Panner

Title: The Fundamental Group and Covering Spaces

Abstract: In this talk, I will introduce the concept of the fundamental group as loops on a space up to homotopy. I will then introduce the concept of a covering space and its relation to the fundamental groups. Finally, I will introduce a theorem that nice enough spaces have a universal covering space and use the rest of the time to give examples.

Speaker: Anand Bhardwaj

Title: Sophie Germain Primes and Cunningham Chains

Abstract: A prime p is a Sophie Germain prime if 2p + 1 is also prime. Named in honor of French mathematician Marie-Sophie Germain and her work towards proving Fermat's Last Theorem, these primes spark interest through their mysterious properties, as well as their application in areas such as cryptography and primality testing. In this talk, we focus primarily on Cunningham chains, recursive sequences of Sophie Germain primes, also referred to as "chains of nearly doubled primes". After an initial examination of their fascinating behavior, we present deeper connections with other areas of number theory, including an amazing result in relation to primitive roots and Artin's conjecture.

Speaker: Lohit Jagarapu

Title: The Pre-Image Theorem and consequences

Abstract: The Pre-Image theorem is a result from differential topology that allows one to construct manifolds easily. It also has some cool partial converses. Come check it out!

Speaker: Darryl E Berry Jr

Title: A Short Biography of Srinivasa Ramanujan

Abstract: I will give a talk on the mathematician Srinivasa Ramanujan. I intend to start by discussing his humble beginnings, in having no formal mathematical training. Through self-study he developed a mathematical understanding in various areas. He even added his own contributions to mathematics. 

LAND ACKNOWLEDGEMENT

We would like to acknowledge that we are meeting on the Indigenous lands of Turtle Island, the ancestral name for what now is called North America. Moreover, we would like to acknowledge the Alabama-Coushatta, Caddo, Carrizo/Comecrudo, Coahuiltecan, Comanche, Kickapoo, Lipan Apache, Tonkawa and Ysleta Del Sur Pueblo, and all the American Indian and Indigenous Peoples and communities who have been or have become a part of these lands and territories in Texas.

To learn more about the importance and purpose of Land Acknowledgments, as well as the Land Engagements and/or Commitments that the NAIS has called upon The University of Texas at Austin, please visit the NAIS website: https://liberalarts.utexas.edu/nais/land-acknowledgement/index.php