Deterministic Solutions to the Radiative Transport Equation and Their Application to Spatial Frequency Domain Imaging

Sean Horan

December 5th, 2018 - 3:00 - 3:50pm - NSII 1201

Abstract

On the micro and millimeter length scale, the behavior of light in turbid media is governed by the Radiative Transport Equation. We will discuss a deterministic method for approximating a solution to this integro-differential equation, compare it to the current "gold standard" of monte carlo based solutions and show its use in recovering optical properties from layered media using data gathered by spatial frequency domain spectroscopy. The properties and layer thickness of the media used represent medically and biologically relevant cases and have previously been challenging to recover.

About the Speaker

Sean is a fifth year PhD student in the math department. His research is focused around the use of mathematical models in medical imaging. He holds a BS in Mathematics and a BA in Philosophy from the University of Missouri, St. Louis. Before returning to school, Sean was a professional fencing coach and continues to coach and compete while at UCI.

Advisor and Collaborators

Sean's advisors are John Lowengrub and Vasan Venugopalan.

Average Cyclicity for Elliptic Curves with Nontrivial Torsion

Luke Fredericks

December 3rd, 2018 - 4:00 - 4:50pm - TBD

Abstract

Given an elliptic curve E over ℚ , we can reduce (mod p ) for all but finitely many primes p to obtain an elliptic curve over the finite field F _p . There are many natural questions we can ask about the group E ( F _p ) of F _p -rational points. For example: how often is E ( F_p) a cyclic group? We will discuss the prime counting function \pi_E^{cyc} = #{p \leq x : E(F_p) is a cyclic group}.

This talk will summarize what is known about this function; in particular, we address the average growth of this function where we average over curves with specified torsion over ℚ .

About the Speaker

TBD

Advisor and Collaborators

Luke's advisor is Nathan Kaplan.

Some Aspects of Parametrized Furstenberg Theorem on Products of Random Matrices

Fernando Quintino

October 25th, 2018 - 4:00 - 4:50pm - RH 340P

Abstract

We will be talking about the norm growth of products of matrices and their application to the Schrodinger operator

About the Speaker

Fernando is a graduate student with mathematical interests in dynamic and math physics.

Advisor and Collaborators

Fernando's advisor is Anton Gorodetski.

Random Butterfly Matrices

John Peca-Medlin

October 19th, 2018 - 1:00 - 1:50pm - RH 340P

Abstract

Random Butterfly Matrices are a recursively defined ensemble of random orthogonal matrices, first introduced by D. Stott Parker in 1995. The recurrent structure significantly decreases the complexity of typical matrix operators using these matrices. For instance, matrix-vector multiplication can be carried out in O(Nlog 2 N) operations rather than the typical O(N 2 ). Random Butterfly matrices can also be used to remove pivoting in Gaussian elimination. I will highlight some spectral and numerical properties of these matrices, along with performance comparisons to other random transformations.

About the Speaker

I am a 4th year focusing on Random Matrix Theory (in Probability). My advisors are Tom Trogdon and Mike Cranston. (What else do you want? My favorite movie is Blade Runner. My favorite color is "Carolina Blue". )

Advisor and Collaborators

John's advisors are Tom Trogdon and Mike Cranston.

TBD

Ali Kassir

May 31st, 2018 - 4:00 - 4:50pm - RH 440R

Abstract

In this presentation, we explore Markov chains with random transition matrices. Such chains are a development on classic Markov chains where the transition matrix is taken to be random. The intuition for this is that we may be interested in modeling phenomenas where the homogeneity assumption of classic Markov chains is invalid. We first use such chains to model credit risk. The randomness of the transition matrix is used to represent the randomness of the economy that underlies credit risk. With this, we model a portfolio of loans and the risk due to having a shared economy. We then proceed to explore theoretical properties of such chains with a focus on their asymptotic behavior. In the case of absorbing chains, we show that the the infinite product of independent and identically distributed random matrices must converge almost surely. We also introduce perturbed Markov chains as a special form of Markov chains with random transition matrices and look at some applications.

About the Speaker

Ali has been a student for 9 years at UCI and is glad to graduate. He is currently working as a data scientist at iHerb Inc. He is the father of two beautiful children. His wife has only known him as a graduate student and is excited to discover her non-graduate student husband.

Advisor and Collaborators

Ali's advisors are Dr. Patrick Guidotti and Dr. Knut Solna.

Deformation Quantization of Vector Bundles on Lagrangian Subvarieties

Taiji Chen

May 29th, 2018 - 4:00 - 4:50pm - RH 440R

Abstract

We consider a smooth subvariety Y in a smooth algebraic variety X with an algebraic symplectic form ω. Assume that there exists a deformation quantization Oh of the structure sheaf OX. When Y is Lagrangian, for a vector bundle E on Y, we establish necessary and sufficient conditions for the existence of the deformation quantization of E. If the necessary conditions hold, we describe the set of equivalence classes of such quantizations. In the more general situation when Y is coisotropic, we reformulate the deformation problem into the lifting problem of torsors. We expect a deformation quantization of line bundles on coisotropic subvarieties is equivalent to a solution of curved Maurer-Cartan equation of a curved L-infinity algebra.

About the Speaker

Taiji is a sixth year PhD student.

Advisor and Collaborators

Taiji’s advisor is Vladimir Baranovsky.

Color Categorization: Quantitative Methods and Applications

Nikki Fider

May 17th, 2018 - 4:00 - 5:00pm - RH 440R

Abstract

Color categorization in humans has been actively studied in the behavioral and social sciences for many decades. Key ideas in this field are basic color terms (BCTs) and corresponding basic color categories (BCCs, each made up of a basic color term and the colors it describes); the identification of BCTs and BCCs has historically required knowledge of the underlying culture and language, and therefore involved subjective analysis. In this dissertation, we present a quantitative data driven method of identifying the BCTs of a language using a category strength function CS. By setting a threshold, the CS function identifies which color terms are BCTs, and which are not. We obtain results which are consistent with the classical method of color categorization, but achieve better consistency and avoid subjectivity. We also analyze several methods of identifying the color foci, or best exemplars. We further present a quantitative method of identifying the boundary of a BCT with a (color) stimulus strength function StS and show that the relationship between BCC hearts and BCC boundaries follows a square-root pattern. Finally, we apply our methods to the study of category scheme evolution and the study of male/female color categorization behaviors. We find that the differences observed between male and female categorization schemes are statistically significant. Throughout this work, we use data provided by the World Color Survey Data Archives.

About the Speaker

Nikki is a sixth year PhD student.

Advisor and Collaborators

Nikki's advisor is Natalia Komarova.

Random Matrix Models for Datasets with Fixed Time Horizons

Greg Zitelli

April 30th, 2018 - 10:00 - 10:30am - RH 440R

Abstract

Students in Math 133B learn about building stock portfolios using Markowitz mean-variance portfolio theory, which relies on knowledge about the covariance of stock price fluctuations. In reality, this information (if it exists) is unknown, so it must be estimated by repeatedly observing stock prices. When the number of stocks and number of observations are both large, results in random matrix theory have been used to give an idea of the bias of some of these estimators. Although these results are universal (i.e. they do not depend on the distributions of the stock returns), they may not be accurate for highly non-Gaussian returns over modest time periods. I will introduce new types of random matrix models designed with these issues in mind, and talk about their eigenvalues.

No background in advanced probability or finance is required.

About the Speaker

Greg is a 4th year graduate student specializing in probability. His work is focused on applications of random matrix theory to finance and wireless applications.

Advisor and Collaborators

Greg's advisors are Patrick Guidotti and Knut Solna.

How to Make Music by Drawing

Kiyoshi Okada

February 22nd, 2018 - 12:00 - 12:50pm - RH 440R

Abstract

I will talk about some of the ways that we visualize sound before presenting a way to sonify images. Then I will present some additional tools that allow even greater control over the resulting sound. Attendants will have the opportunity make a drawing that will be converted.

About the Speaker

I am a first year graduate student that enjoys using math and programming to make art. After a few years of making generative gifs over at enchantedconsole.tumblr.com I have shifted to the world of sound, focusing on using and devising accessible and intuitive methods to make music.

Mathematical Models of Virus Infections

Jesse Kreger

January 11th, 2018 - 12:00 - 12:50pm - RH 440R

Abstract

Motivated by recent experimental data, this talk will investigate mathematical models regarding the evolutionary outcomes of viral infections, specifically human immunodeficiency virus (HIV), in humans. The presentation will analyze how the interplay between multiplicity of infection, synaptic cell-to-cell transmission of the virus, and free virus transfer can affect the dynamics of an infection taking place. We consider models with competition between virus strains, characterized by different mutations, to see how each strain’s infection strategy can affect outcome. Finally, we will discuss how recombination between virus strains can change the evolutionary outcomes of infection and influence the course of disease. The overall goal of the project is to better understand the dynamics of viral infections, specifically HIV, and to help design more effective healthcare and vaccination approaches.

About the Speaker

Jesse is a 2nd year PhD student studying mathematical biology.

Advisor and Collaborators

Jesse's advisors are Natalia Komarova and Dominik Wodarz.