Abstract: The Hopf fibration is a fundamental example in topology. It describes the 3-sphere as a bundle of circle fibers over a 2-sphere base. This presentation will introduce the Hopf fibration, formally define the linking number, and prove that the linking number of all fibers is 1.

Delivery: Powerpoint/slides 

Abstract: Symplectic reduction is a process that allows us to reduce the dimension of a symplectic manifold by eliminating symmetries while maintaining its symplectic structure. This presentation will discuss the basics of symplectic manifolds, outline the process of symplectic reduction alongside a proof of the Marsden-Weinstein-Meyer Theorem, and provide examples of its applications.

Delivery: Powerpoint/slides 

Abstract: I was (along with Hannah Shen) mentored by Baran Cetin and Yijie Pan this semester on Symplectic Geometry. My presentation will be an extension of Hannah's presentation, linking Symplectic Geometry with Algebraic Geometry. Namely, I will introduce the concept of Geometric Invariant Theory (GIT) in the affine and projective case. Then, via the statement of the Kempf-Ness Theorem, I will provide examples for which symplectic reduction is homeomorphic to the GIT quotient.

Delivery: Powerpoint/slides 

Abstract: We are going to give a survey on finite dimensional representations of the symmetric groups. We go over the Specht modules, the irreducible representations, and the tableaux combinatorics behind them. 

Delivery: Powerpoint/slides 

Abstract: This will be a presenation on the Borsuk Ulam Theorem and explains its interesting real world application. My presentation will explain what this theorem is, introduce some prerequisite theorems needed for the proof and its equivalent theory. I will then proof this theory by the end geometrically.

Delivery: Powerpoint/slides 

Abstract: The Steinitz Exchange Lemma establishes that in any vector space, a spanning set must contain at least as many vectors as any linearly independent set. This presentation provides the motivation for why this theorem is essential, presents a complete proof using matrix methods, and demonstrates its application through concrete example. The implications extend far beyond traditional vectors to all vector spaces, including polynomials, functions, and matrices.

Delivery: Powerpoint/slides 

Abstract: In this presentation I will introduce basic concepts in Linear Symplectic Geometry, a foundational part of Symplectic Geometry. I will then state and sketch out a proof for the Affine Non-squeezing theorem, a linear version of Gromov's Non-squeezing theorem that indicates that a ball can only be embedded into a symplectic cylinder by an affine symplectomorphism if the ball has a smaller radius than the cylinder.

Delivery: Powerpoint/slides