# The Undergraduate Experience in Mathematics

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We discuss the unique challenges facing undergraduate education in mathematics and discuss ideas for moving forward.

Nicholas Hoell

University of Toronto

Thursday, February 14, 2019 - 3:00pm to 4:00pm

RH 440R

We discuss the unique challenges facing undergraduate education in mathematics and discuss ideas for moving forward.

Nicole Fider

University of California, Irvine

Friday, February 8, 2019 - 3:00pm to 4:00pm

RH 306

Active learning is, in essence, any teaching strategy that encourages students to take an active part in the learning process. Research suggests that active learning is a beneficial addition to the STEM curriculum—it promotes a deeper level of understanding and may help improve the retention rates of STEM majors. To demonstrate how active learning components can be utilized in UCI mathematics courses, I present materials I developed and implemented for upper and lower division courses at UCI. These materials span an assortment of media and are designed to facilitate student engagement in a variety of ways. I also discuss ideas for future active learning opportunities in UCI mathematics courses, and I summarize potential undergraduate projects, which will allow students to explore advanced mathematics and applications (including work with real world data) in a hands-on exploratory fashion.

Roberto Pelayo

University of Hawaii at Hilo

Monday, February 4, 2019 - 3:00pm to 4:00pm

RH 306

Crafting mathematics curriculum and creating research opportunities for undergraduates have been central to my academic career. In this talk, I discuss my experiences in these areas while at the University of Hawai`i at Hilo and how I envision incorporating them at UC Irvine. I will begin by describing my role in creating and directing the UH Hilo Data Science program and generating an interdisciplinary curriculum that serves the needs of students with interdisciplinary interests. Then, I will describe my work as undergraduate research advisor and principal investigator for the Pacific Undergraduate Research Experience in Mathematics (PURE Math), a 5-year REU housed at UH Hilo targeting underrepresented mathematics majors. I will end by describing much of my educational outreach work, which centers on developing a high school curriculum for the state of Hawai`i and for the College Board, as well as providing professional learning and mentoring for in-service teachers around the country.

Shuhao Cao

University of California, Irvine

Tuesday, January 29, 2019 - 3:00pm to 4:00pm

RH 440R

Adaptive methods are used in almost all disciplines of applied and computational mathematics, where the problem solving procedure adapts to the feedback from the quantities of interest. Inspired by this methodology, we will discuss the practice of the adaptive education, and the adaptive design of classes for the data science specialization. With the help of the smart classroom technologies at UC Irvine, we can better execute the instruction of programming in mathematics and data science to make our students learn to be more competent for both industry and graduate school. Lastly, we share some ideas of future plans to further strengthen our strong undergraduate program and serve our community.

Lucia Simonelli

International Center for Theoretical Physics

Friday, January 25, 2019 - 3:00pm to 4:00pm

RH 306

I will describe my experiences learning, teaching, and communicating mathematics in various settings and capacities.

Elina Robeva

MIT

Thursday, January 10, 2019 - 4:00pm to 5:00pm

RH 306

My research focuses on studying models that depict complex dependencies between random variables. Such models include directed graphical models with hidden variables, discrete mixture models (which give rise to low rank tensors), and models that impose strong positive dependence called total positivity. In this talk I will give a brief overview of my work in these areas, and will particularly focus on the problem of density estimation under total positivity.

Nonparametric density estimation is a challenging statistical problem -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints such as total positivity allows a path forward. Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.

Asaf Ferber

MIT

Friday, January 11, 2019 - 4:00pm to 5:00pm

RH 306

Given an integer vector $a = (a_1,\dots,a_n)$, let $\rho(a)$ be the number of solutions to $a\cdot x=0$, with $x\in \{\pm 1\}^n$. In 1945, Erdos gave a beautiful combinatorial solution to the following problem that was posed by Littlewood and Offord: how large can $\rho(a)$ be if all the entries of $a$ are non-zero?

Following his breakthrough result, several extensions of this problem have been intensively studied by various researchers. In classical works, Erdos-Moser, Sarkozy-Szemeredi, and Halasz obtained better bounds on $\rho(a)$ under additional assumptions on $a$, while Kleitman, Frankl-Furedi, Esseen, Halasz and many others studied generalizations to higher dimensions. In recent years, motivated by deep Freiman-type inverse theorems from additive combinatorics, Tao and Vu brought a new view to this problem by asking for the underlying structural reason for $\rho(a)$ to be large --this is known as the Inverse Littlewood-Offord problem, which is a cornerstone of modern random matrix theory.

In this talk, we will discuss further extensions and improvements for both forward and inverse Littlewood-Offord problems where combinatorial tools and insights have proved to be especially powerful. We also present several applications in (discrete) matrix theory such as: a `resilience' version of it, a non-trivial upper bound on the number of Hadamard matrices, an upper bound on the number of $\pm 1$ normal matrices, and a unified approach for counting the number of singular $\pm1$ matrices from various popular models.

Nick Cook

UCLA

Monday, January 14, 2019 - 3:00pm to 4:00pm

RH 306

Let $G=G(N,p)$ be an Erdos--Renyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles (triples of vertices with all three pairs connected) as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of nonlinear large deviations, and also connect with issues in extending the theory of graph limits to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

Farbod Shokrieh

University of Copenhagen

Tuesday, January 22, 2019 - 4:00pm to 5:00pm

RH 306

Graphs can be viewed as (non-archimedean) analogues of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks. I will explain the interplay between these points of view, as well as some recent application in arithmetic geometry.

Ying Cui

University of Southern California

Wednesday, January 23, 2019 - 4:00pm to 5:00pm

RH 306

Classical continuous optimization starts with linear and nonlinear programming. In the past two decades, convex optimization (e.g., sparse linear regressions with convex regularizers) has been a very effective computational tool in applications to statistical estimations and machine learning. However, many modern data-science problems involve some basic ``non’’-properties that are ignored by the convex approach for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-(Clarke) regularity. In this talk, we present a rigorous computational treatment for solving two non-problems: the piecewise affine regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations. This is based on joint work with Jong-Shi Pang, Bodhisattva Sen and Ziyu He.