Current Organizers

Zijian (Robin) Rong, contact: zijianro@usc.edu

• 6:45-7:00: Qianlin Ye, Square Fibonacci Numbers (Mentor: Wenhao Zhang)

Abstract: First introduce what is Fibonacci numbers and give a brief proof using the Generating Function. Then briefly introduce the main focus of the presentation, which is on square Fibonacci numbers. Lastly, the majority of the presentation will be proving that for n>0, 1 and 144 are the only square Fibonacci numbers.

• 7:00-7:15:  Mengchu Yue, Hilbert's Nullstellensatz (Mentor: Yash Somaiya)

Abstract: The proof of Hilbert's Nullstellensatz and relevant objects; serve as a introduction to what Kai Cai would present

• 7:00-7:15:  Xinkai Cai, Bézout's Theorem for Plane Curves (Mentor: Yash Somaiya)

Abstract: Introduction to projective space, short proof of bezout's theorem, and examples

• 7:15-7:30: Break

• 7:30-7:45:  Steven Shi & Ryan Wang, Represent Probability Independence using Directed Graphs (Mentor: Inga Girshfeld)

Abstract: Many real-world events can be modeled as a joint probability distribution. However, the naive parameterization of the joint as a table is typically inefficient as it requires exponentially many parameters. In our presentation, we will explore the Bayesian Network (B-Net), which is a tool for representing a set of conditional independence assumptions among variables using directed graphs. It can use the independence assumptions to represent joint distribution in a complete. Specifically, we will be presenting the concepts of I-Map (how the fact that a probability distribution is able to factorize according to a B-Net is equivalent to that the B-Net captures a subset of independence assumptions in the distribution) and I-Equivalence (how the graphical representation of a particular set of independence assumptions is not unique) with an intuitive example about college application.

• 7:45-8:00:  Daniel Gottschalk, Publish or Perish (Mentor: Chenchen Zhao)

Abstract: This talk will center around the Tenure Game, a two-player pusher-chooser game where the Dean (chooser) attempts to get a candidate to tenure against the Provost (Pusher). I will employ probabilistic methods to identify a perfect winning strategy for each player and demonstrate the application of Zermelo’s theorem in the process.

• 7:45-8:00:  Dung Pham, Bounding the Number of Empty Triangles with the Probabilistic Method (Mentor: Chenchen Zhao)

Abstract: In this talk, we explore the concept of empty triangles within a set S of n points in the Euclidean plane, where an empty triangle is defined as a triangle formed by vertices from S that encloses no other points of S. Our focus will be on Theorem 14.2.1 from Alon and Spencer's "The Probabilistic Method," which applies a probabilistic framework to demonstrate that the expected number of empty triangles is upper-bounded by (1+o(1))2n^2. This presentation will illuminate how probabilistic techniques can effectively address and solve geometric problems, providing insights into the dynamics between geometry and probability theory.

• 8:00-8:15: Break / Group Photo

• 8:15-8:30:  Anthony Winney and Tuojin Yin, Minimal Surfaces in R^3 (Mentor: Brian Fan)

Abstract: In this talk, we present definitions and conditions of minimal surfaces in R^3. Such minimal surfaces are impactful in differential geometry and occur in nature, such as soap films changing shape. We define minimal surfaces as surfaces in space which locally minimize the area on a given small boundary. We first show the equation of minimal graphs, then use mean curvature to give it a geometric meaning, then discuss when a conformal immersion yields a minimal surface. We use analytic methods that don't require differential geometry.

• 8:30-8:45: Scott Sobolewski, An Introduction to A-infinity Algebras and Fukaya Categories (Mentor: Yasin Uskuplu)

Abstract: After defining an A-infinity algebra, I give examples including a Fukaya category. Then, I discuss how an A-infinity algebra can fail to be associative, but can become associative by moving to the cohomology level. Finally, I explain the morphisms between A-infinity algebras along with some research applications for their theory.