Check back for our next GSS in Fall 2024!
Next year's GSS will be looking a little different from those past. Until we reveal more, be sure you're up to date on our current events!
Past Talks
Wednesday, November 29th, 2023
Speaker: Phuong Hoang
Topic: Optimal transport: from static to dynamic problems
Description: In this talk, I firstly present the original static version of optimal transport - the Kantorovich problem. After that, the next part will be about the formulation of optimal transport for stationary Markov chains, which can capture dynamics structures of these systems. Then, I will bring my recent work in which I develop a new kind of transport problem called optimal reversible transition coupling (ORTC). It enables us to couple two reversible Markov chains such that the reversibility and dynamical properties are preserved. Finally, if time permits, the application of ORTC in detecting graph isomorphisms will also be mentioned.
Monday, December 4th, 2023
Speaker: TBA
Topic: Qualifying Exam Prep: Fundamental Theorems of Calculus
Description: A presentation of solutions to past qualifying exam questions pertaining to the fundamental theorems of calculus, bounded variation, and absolute continuity.
Wednesday, November 8th, 2023
Speaker: Xingnan Zhang
Topic: Multivariate Dickman Distribution and Its Application
Description: A multivariate extension of the Dickman distribution was recently introduced, but very few properties have been studied. We discuss several properties with an emphasis on simulation. Further, we introduce and study a multivariate extension of the more general class of Vervaat perpetuities and derive a number of properties and representations. Then, we provide the algorithm for simulating the whole Levy process. At last, we extend our findings to the stochastic integral processes.
Monday, November 6th, 2023
Speaker: Elaine Gorom-Alexander
Topic: A comprehensive study on multiscale models in material science
Description: This talk is a brief overview of multiscale modeling methods in materials science. Multiscale models are impressive methods to capture material phenomena at multiple time and length scales. An in-depth description of the work done by the author is provided with examples of the math and coding done for this. Static and dynamic problems are discussed for these local and nonlocal methods.
Wednesday, November 1st, 2023
Speaker: Wai-Lun Lam
Topic: Introduction to Stochastic Calculus
Description: This talk serves as a brief introduction to stochastic analysis. It first summarizes the evolution of integration theory from Riemann integral to Lebesgue integral to stochastic integral and then establishes connection between the random world and the deterministic world. Lastly, the topic will lead to the connection between stochastic differential equations and partial differential equations in finance and mathematical physics.
Monday, October 30th, 2023
Speaker: Jade Raymond and other senior grad students
Topic: Qualifying Exam Prep: Lebesgue measurable functions
Description: A presentation of solutions to past qualifying exam questions pertaining to Lebesgue measurable functions.
Wednesday, October 25th, 2023
Speaker: Gang Cheng
Topic: TBA
Description: TBA
Wednesday, October 18th, 2023
Speaker: Jade Raymond
Topic: Qualifying Exam Prep: Lebesgue measurable sets: Part 2
Description: As a continuation from Monday October 16th, there will be a presentation of solutions to past qualifying exam type questions pertaining to Lebesgue measurable sets.
Monday, October 16th, 2023
Speaker: Jade Raymond
Topic: Qualifying Exam Prep: Lebesgue measurable sets: Part 1
Description: A presentation of solutions to past qualifying exam type questions pertaining to Lebesgue measurable sets.
Wednesday, October 11th, 2023
Speaker: Hayden Pecoraro
Topic: Open Problem Session
Description: The problem session will be geared towards Real Analysis, with the idea being that students can bring whatever problems or topics they'd like to discuss, and senior graduate students will be there to assist them. This can be especially helpful for those taking the qualifying exam within the next year.
Monday, October 9th, 2023
Speaker: Harrison Latimer
Topic: Modeling Coevolution, a New Perspective in Mathematical Epidemiology
Description: Following the groundbreaking insights in ecology brought forth by the Lotka-Volterra equations, epidemiologists A.G. McKendrick and W. O. Kermack applied a similar mathematical lens to the study of disease. Since then, the latter half of the 20th century saw many mathematicians bring about more and more sophisticated techniques which built upon the original models these men proposed in the 1920's and 30's. With the change of millennium, however, a new and more dynamic perspective emerged: an emphasis on the study of host-pathogen coevolution. I will briefly discuss the history of thought behind mathematical modeling for diseases, and then follow this discussion with a focus on this newer idea of coevolution. I will further introduce the even more recent concept of what is called "pathogen perspective" modeling and conclude my talk with a proposal for a way to bridge these two ideas. Pathogen mutation is the single most important threat for the rise of disease epidemics, and these mutations are the primary focus of coevolution modeling. In the wake of the Covid-19 outbreaks witnessed in the last three years, there is more of a need than ever to comprehend these mechanics. The new techniques in mathematical modeling for epidemiology that I will be speaking about provide a rich and powerful lens with which we hope that such an understanding can be achieved.
Monday, October 2nd, 2023
Speaker: Sergei Miles
Topic: Dynamics on the shift action on linear sequence spaces over groups beyond Z
Wednesday, September 27th, 2023
Speaker: Hannah Powell
Topic: An Introduction to Number Theory
Description: An exploration of the Riemann-Zeta function!
Monday, September 25th, 2023
Speaker: Hayden Pecoraro
Topic: Some Introductory Topology
Description: This will be a survey of several ideas from general topology relevant to measure theory and functional analysis. First we will introduce the basic tools of topology using examples of metric spaces. From here, the topics discussed may include general open sets, closed sets, and sets of (slightly) higher Borel complexity, topologies as algebras of sets, characterizations of continuous functions, and product and quotient spaces as well as other constructions. We will motivate each topic through concrete examples of interesting topological spaces and omit proofs unless they may be particularly enlightening for the audience. Deeper tools and results will be covered as time allows.
Wednesday, September 20th, 2023
Speaker: Sarah H. Murphy
Topic: Stacking PINNs for Dynamical Systems
Description: Physics-informed neural networks have demonstrated vast potential for effectively solving systems of equations for modeling physical systems. However, for some dynamical systems, PINNs can be difficult or impossible to train. This work is focused on modeling the behavior of chaotic dynamical systems using physics-informed neural networks. We consider a novel multifidelity framework for stacking physics-informed neural networks that will allow for modeling systems such as the Lorenz system. We iteratively train a multifidelity PINN for a number of steps, where the low-fidelity model at each step takes the output of the previous step as input. This method allows for fast computations of the dynamics of complex systems of equations, without the need for computationally expensive codes. The proposed method is particularly useful for dynamical systems where standard PINNs fail to train, such as the pendulum equation, the wave equation, and the Lorenz system.
March 30th, 2022
Speaker: Van Pham
Topic: Link Invariants
Description:
The word “writhe” means twist in Old English. In knot theory, though not a general invariant, it gives the net sign of an oriented link diagram. In my dissertation, several “writhe-like” quantities were defined and shown invariants for a large family of links.
April 6th, 2022
Speaker: Xi Ning
Topic: Statistical Inference for Semiparametric Cox-Aalen Transformation Models with Failure Time Data
Description:
A class of transformation models has been used intensively to characterize the association between the event of interest and the covariates in the literature. However, the assumption of a common baseline intensity/hazard is violated in many applications, e.g., the existence of categorical variables forms natural candidates for stratification of the baseline intensity/hazard. To bridge the gap, we propose a broad class of so-called Cox-Aalen transformation models that incorporate both multiplicative and additive covariate effects on the baseline intensity/hazards. The proposed model provides more flexibility and encompasses a versatile and useful class of semiparametric models. By considering a system of complete-data estimating equations, we devise an ES-type algorithm, which performs fast and robustly in calculations. Moreover, the ES algorithm yields a computationally simple method for estimating the variance of the resulting parameter estimates. Finally, we demonstrate the performance of our procedures through extensive simulation studies.
March 18th, 2022
Speaker: Fay Xia
Topic: Estimation and Simulation of Tempered Stable Distribution
Description:
In this thesis, we introduce a methodology for the simulation and parameter estimation of multivariate tempered stable distributions. Our approach is based on an approximation due to a discretization of the Lévy measure. We derive this discretization in general and give an explicit construction of the discretization in the bivariate case. Also, our approximation results hold for a wide class of multivariate infinitely divisible distributions.
Based on our main approximation, we develop a method for simulations, which we call the discretization and simulation (DS) method. We apply our methodology to two bivariate financial datasets related to exchange rates and perform goodness- of-fit tests to show that the multivariate tempered stable model does a good job fitting the model.
Further, we apply our model for the pricing of the bivariate basket option. To- ward this end, we provide theoretical results on the existence of equivalent martin- gale measures. Then combing this with our model for parameter estimation and the DS method for simulation, we develop a Monte Carlo based method for option pricing. We apply it to the pricing of European call options with different strikes and the pricing of the Multi-asset rainbow option.
March 2nd, 2022
Speaker: Jade Raymond
Description:
I will talk about various properties of measures, with special emphasis on the Lebesgue measure. I will also work through some past qualifying exam questions as examples.
February 23rd, 2022
Speaker: Hayden Pecoraro
Topic: On various essential objects and proofs
Description:
From some uses of geometric and harmonic series, to a broad overview of hierarchy of continuity, I will talk about various small, but important topics, as well as present proofs of some useful facts that are important to know, but that you may not know well.
February 16th, 2022
Speaker: Jade Raymond
Topic: The importance of Bases in Finite Dimensional Vector Spaces
Description:
Starting from nothing, I will define abstract Vector Spaces (over the reals), and move towards defining what a basis is, and how you can think about them. Then, I will discuss various theorems from Linear Algebra relating to bases, and how they essentially characterize entire vector spaces as well as transformations on them.
February 9th, 2022
Speaker: Robert Bland
Topic: Series
Description:
Starting with an overview of some popular series examples, we will look at a few problems from analysis and adjacent fields which involve series and the various tricks to manipulate and analyze them.
February 2nd, 2022
Speaker: Jade Raymond
Topic: The variety of techniques in solving problems in Dynamical Systems
Description:
In a more interactive session, I will present an introductory problem from Dynamical Systems. Attendees will discuss potential directions for solving the problem, before I ultimately present my solution to the problem as well as some directions I looked at for solving the problem which did not pan out, but are interesting nevertheless. This is all in pursuit of demonstrating the variety of techniques which may be used in solving problems in Dynamical Systems.