Abstract: Analysis of Variance (ANOVA) is a statistical method used to determine whether the means of three or more groups are significantly different. It works by comparing the variance between group means to the variance within groups using the F-statistic, calculated as the ratio of Mean Square for Treatments (MST) to Mean Square for Error (MSE). 

Delivery: Powerpoint/slides 

Abstract: Householder transformations are a numerical technique used to perform QR decomposition of a matrix by successively reflecting vectors to introduce zeros below the diagonal. This method is numerically stable and efficient, making it well-suited for solving linear systems.

Delivery: Powerpoint/slides 

Abstract: Algebraic topology is the study of topology under the scope of algebraic techniques and concepts. During my DRP meetings, I was able to learn fundamental concepts in algebraic topology such as the homotopy groups, homology groups, and cohomology groups under the guidance of my mentor, Yijie Pan. This presentation will illustrate the power of algebraic topology by tackling a famous theorem in algebra proved by John F. Adams in 1960. One among many implications of the theorem is the limitation of defining division algebras due to the Hopf invariant one (Only n = 1(R), 2(C), 4(H), and 8(O) acceptable). Another implication is that the only spheres admitting trivial tangent bundles are S^1, S^3, S^7. Overall, the theorem utilizes various concepts in algebraic topology such as the homotopy groups, cohomology rings, Hopf invariants, tangent bundles, and division algebras. I will be introducing and defining such concepts in my presentation. In addition, the theorem will be stated and its applications will be explored. 

Delivery: Paper/Handwritten

Abstract: This will be an analysis of the first five chapters of "A Modern Approach to Quantum Mechanics" by Townsend, going into detail on Heisenberg’s uncertainty principle, as well as Schrödinger's equation in the Heisenberg picture. There will especially be a focus on EPR (Einstein Podolsky Rosen), which will include an introduction to Einstein’s commentary of “spooky action at a distance”, as well as Bell’s contradiction to Local Hidden Variable Theory. This will demonstrate the nature of entanglement and some of the surrounding historical debate. Lastly, there will be discussion on the ramifications of this “spooky action at a distance” such as Quantum Teleportation and Superdense coding. 

Delivery: Powerpoint/slides

Abstract: This presentation delves into a proof that every positive integer can be expressed as a sum of four squares. The proof uses concepts from abstract algebra, including an exploration of a subring of the quaternions, to derive this result. 

Delivery: Powerpoint/slides

Abstract: Differential Privacy (DP) is a widely used mathematically rigorous framework for data privacy. In this brief talk we cover its definition, properties, examples, and a few differentially private algorithms in machine learning.

Delivery: Paper/Handwritten

Abstract: In this presentation we will introduce the motivations for Topological Quantum Computing and construct Modular Tensor Categories, which act as the language that dictates how particles can be swapped or fused. Lastly, we will provide examples of some Modular Tensor Categories and introduce the N, F, and R matrices, which encode the algebraic data critical to defining a Modular Tensor Category. 

Delivery: Powerpoint/slides

Abstract: The Hamburger Moment Problem asks: “For what real sequences (mn) can we construct a positive measure µ, such that mn is the nth moment of µ for all n?” The answer eventually involves concepts from linear algebra, namely Hankel matrices, orthogonal polynomials, and linear functionals. In this talk, we introduce these concepts and connect them with the Hamburger Moment Problem by sketching the proof. 

Delivery: Powerpoint/slides

Abstract: The axiom of completeness is a fundamental principle in real analysis that ensures the real number system is "complete," meaning every nonempty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers. This axiom distinguishes the real numbers from the rationals and underpins many key results in analysis, including convergence theorems and continuity properties. Its formulation provides a rigorous foundation for limits, sequences, and functions, making it central to the structure of calculus and higher mathematics. The completeness axiom not only supports the intuitive notion of no "gaps" in the real line but also enables precise mathematical treatments of concepts such as compactness, connectedness, and metric space theory. 

Delivery: Powerpoint/slides