Current Organizers

Sung Kim; Jonathan Michala; Zijian Rong

• 7:00-7:15: Nathaniel Lee & Raja Somou, Exploring Diffusion Probabilistic Models (Mentor: Zixiang Zhou)

Abstract: This presentation surrounds the diffusion probabilistic model, a class of generative models that has recently gained significant attention in the fields of machine learning. At its core, the model mimics the real-world natural process of diffusion by leveraging a gradual, stepwise procedure of adding and then removing random noise to generate data.

We will begin with an introduction to the mechanism of diffusion models, highlighting how they transform distributions through a series of forward and reverse diffusion steps, while also covering the unique loss function from its derivation to how it effectively guides the learning process. A large portion of the presentation will surround the "U-net" model architecture used in the learning for diffusion and provide insights into practical applications and advantages of this type of model.

Attendees will leave with a solid understanding of the foundational principles of diffusion models.

• 7:00-7:15: Nathaniel Lee & Raja Somou, Exploring Diffusion Probabilistic Models (Mentor: Zixiang Zhou)

Abstract: This presentation surrounds the diffusion probabilistic model, a class of generative models that has recently gained significant attention in the fields of machine learning. At its core, the model mimics the real-world natural process of diffusion by leveraging a gradual, stepwise procedure of adding and then removing random noise to generate data.

We will begin with an introduction to the mechanism of diffusion models, highlighting how they transform distributions through a series of forward and reverse diffusion steps, while also covering the unique loss function from its derivation to how it effectively guides the learning process. A large portion of the presentation will surround the "U-net" model architecture used in the learning for diffusion and provide insights into practical applications and advantages of this type of model.

Attendees will leave with a solid understanding of the foundational principles of diffusion models.

• 7:15-7:30: Anika Gupta, Stochastic processes and its applications in financial markets (Mentor: Thejani Gamage)

Abstract: This presentation studies Stochastic processes and their applications in financial markets, in particular in option pricing. We begin by studying the Brownian motion, which can be viewed as a continuous version of the standard random walk. We also define stochastic integrals and stochastic differential equations (SDE). An SDE models a stochastic process, which is a function of time and random noise. Finally, we study the Euler Maruyama method as a tool to simulate a stochastic process given by an SDE, and study the use of the Euler Maruyama method in finding the price of an European and Asian call option.

Break: 7:30-7:35

• 7:35-7:50: Jason Liu, Free modules of finite rank over principal ideal domain (Mentor: Haoyang Liu)

I plan to go over the concepts of rings, ideals, principal ideal domain, modules and free modules, etc., and introduce a famous theorem about the classification of finitely generated free modules and their submodules over principal ideal domain. I will introduce some results of the theorem specifically in linear algebra or group theory and the brief philosophy of how to prove the theorem if time is enough.

• 7:50-8:05: Leonard Badt & Richard Chen, Visualizing and understanding Hilbert Nullstellensatz theorem (Mentor: Yash Somaiya)

Using various methods from algebraic geometry, we will discuss an approach to the Hilbert Nullstellensatz theorem and examine some useful applications!

Break: 8:05-8:10

• 8:10-8:25: Gary Yu, Convexity (Mentor: Siyang Liu)

Abstract: I will try to prove Proposition 3.8 in Wendell Fleming’s Functions of Several Variables

• 8:25-8:40: Junii Choi & Odin Schor, Diffeomorphism and Hamiltonian Mechanics (Mentor: Boxi Hao)

I will talk about how a level set (n-fucntions in involution) 2n-dimensional phase space (simplectic manifold) is diffeomorphic to n-dimensional torus parametrized by n-angle variables.