Department of Mathematics and Statistics

# Colloquia for Fall 2018

## Friday, Sept. 7 -- Holt 175, 4:00pm

Isaiah Lankham (University of California Office of the President)
Propensity Score Matching: A Cautionary Tale

Abstract: Imagine you've been hired as a data analyst for a pre-college outreach program called 2&Thru, and you're asked to show how the program is effective.  In other words, are high-school graduates who participate more likely to enroll in, persist at, and graduate from college than if they hadn't participated?

From a statistical standpoint, this sounds like an ideal setup for randomized controlled trials (RCTs): Randomly assign students to either a treatment group (those who participate in 2&Thru) or a control group (those who don't), and compare the two groups' outcomes. However, random assignment would be unethical since the outcomes of 2&Thru participation are potentially life-altering. We can't intentionally deny program benefits, so we'll need a better way of estimating the counterfactual, meaning the results 2&Thru participants would have had if they hadn't participated.

Fortunately, there's a clever workaround called propensity score matching (PSM), which allows us to mimic the effects of randomized group assignments. By identifying the factors (aka covariates) most important in determining a student's propensity to self-select for program participation, we can match 2&Thru participants to comparable non-participants using logistic regression and build a quasi-control group. Because the entire technique hinges on selection of the "right" set of covariates, the "wrong" set will quickly lead to invalid conclusions, as will be demonstrated.

## Friday, Sept. 14 -- Holt 175, 3:30pm

John Lind (California State University, Chico)
The algebra of spheres

Abstract: A thread can wind around a peg any number of times—and to a topologist this counting of the winding number is what the integers are! In this talk I will explore a generalization of this idea to higher dimensions. We can ask in a similar way how many times a sphere of a given dimension can wrap around another sphere. Contemporary methods that attempt to answer this question are complex and abstract, so I will use pictures to harness our geometric intuition. By the end, I will try to convince you that the patterns we detect in the spheres are shadows of a fundamental object of algebra.

## Friday, Sept. 21 -- Holt 175, 3:30pm

Jonathan Sands (University of Vermont)
A potential game-changer for speedy factoring: Shor's algorithm in quantum computing

Abstract: In 1994, Peter Shor showed that factoring could be done in polynomial time if significant quantum computing becomes a reality. This would imply that standard cryptosystems such as RSA are no longer secure. We present Shor's algorithm for a general mathematical audience, focusing on the number theory and requiring no previous knowledge of quantum mechanics.

## Friday, Oct. 19 -- Holt 175, 3:30pm

Thomas Mattman (California State University, Chico)
There are too many knotted graphs!

Abstract: (Joint with Goldberg and Naimi) The powerful Graph Minor Theorem of Robertson and Seymour ensures that, for any graph property, whatsoever, there is an associated finite list of graphs that are minor minimal with respect to that property. For example, with Thomas, they show that the seven graphs in the Petersen family are exactly the minor minimal intrinsic linked (MMIL) graphs.

A graph is intrinsically linked (knotted) if every embedding in R^3 has a pair of non-trivially linked cycles (a non-trivially knotted cycle). Through 2003, 41 MMIK (minor minimal intrinsically knotted) graphs were known. We have 220 new examples and our methods suggest there are likely "many, many more."

The talk will begin with a gentle introduction to graph theory. Undergraduate students are encouraged to attend.

## Friday, Oct. 26 -- Holt 175, 3:30pm

Christine Herrera (California State University, Chico)
The struggle is real!

Abstract: This talk will discuss a study that examined the development of preservice teachers (PSTs) understanding of productive struggle using video episodes which PSTs analyzed through the lens of professional teacher noticing. The goal of the study was to give PSTs opportunities to observe students struggling with the course content the PSTs were studying and to enact their specialized content knowledge and pedagogical knowledge for teaching simultaneously. Findings suggest that the PSTs develop the ability to attend to and interpret the mathematics underlying the student struggles through video analyses. They also begin to identify teaching strategies and practices that appear potentially useful for supporting productive struggle. One implication for this study is to carefully weave opportunities to develop teaching practices such as support of productive struggle into the content course for teaching. The PSTs may better bridge the stance between being a student and becoming a teacher in what productive struggle looks like in learning mathematics.

## Thursday, Dec. 13 -- Holt 352, 4pm

Enrique Treviño (Lake Forest College)
Playing with triangular numbers

Abstract:  A number m is said to be triangular if it can be written as 1+2+3+...+n for some integer n. The first triangular numbers are 1,3,6,10,15. The number 10 is triangular and it is the sum of 3 consecutive triangular numbers. Let k be a positive integer. In this talk we'll explore the following question: Is there a triangular number that can be written as the sum of k consecutive triangular numbers? We will show that for infinitely many k, the answer is YES, but that that set has density zero. In our route to this proof we'll travel through different areas of number theory: Pell equations, the Cohen-Lenstra heuristics for class numbers, and sieve methods.

## Friday, Dec. 14 -- Holt 175, 3:30pm

Matt Krauel (California State University, Sacramento)
Vertex operator algebras: an intersection of number theory, group theory, and physics.

Abstract:  This talk centers around an intersection between number theory, group theory, and theoretical physics. In the 1980s these three areas were found to connect via functions associated to algebraic structures called vertex operator algebras. The study of vertex operator algebras, and their influence in number theory and physics, has been studied heavily since. However, while problems are being solved, many more questions are being raised. In this talk I will explain the connection mentioned above, and briefly describe some areas of active research.