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2015–2016 Mathematics Seminar

Date: October 30, 2015, 3:30 p.m.
Speaker: Steve Costenoble, Hofstra University
Title: Calculations in Equivariant Ordinary Cohomology
Abstract: Ordinary cohomology is one of the calculational workhorses of algebraic topology. However, when we add in the consideration of symmetries, to get equivariant algebraic topology, the analogous cohomology theory is poorly understood and calculations are few and far between. Stefan Waner and I have worked to try to rectify these problems. I'll describe equivariant ordinary cohomology and the version we've developed with "extended grading," some of the calculations that have been done, and the hope for the future.

Date: November 18, 2015, 11:30 a.m.
Speaker: Barbara Gonzalez, Hofstra University
Title: Students as Partners in Curricular Design
Abstract: In order to increase student interest in mathematics and its connections to real applications, Roosevelt University began incorporating semester-long projects into its Calculus II course. Different project topics are used each semester. Project creation has lead to opportunities for student involvement, including work as embedded tutors, undergraduate research projects, and opportunities for students to present posters and talks to a broader audience. The student project designers are often mathematically early in their careers, and so this provides them with an opportunity to create and explore new mathematics while giving faculty the ability to involve students of all levels in research projects.

Evidence was gathered from interviews, surveys, and observation of student research work and its implementation in the classroom. We found that embedded tutors reported more confidence in their knowledge of calculus and insights into teaching it, and project designers experienced similar benefits to those of a traditional research experience.

(joint work with S. Cohen and M. Pivarski)

Date: March 18, 2016, 4:00 p.m.
Speaker: Patrick Dragon, Bard College at Simon's Rock
Title: The Grandmama De Bruijn Sequence for Binary Strings
Abstract: A de Bruijn sequence is a binary string of length 2n which, when viewed cyclically, contains every binary string of length n exactly once as a substring. For example, 00010111 suffices for n=3. Knuth refers to the lexicographically least de Bruijn sequence for each n as the "Grandaddy" sequence due to its venerable origin. Martin originally constructed these sequences greedily and later it was shown by Fredericksen et al. that the Grandaddy sequences can also be constructed by concatenating the aperiodic prefixes of the binary necklaces of length n in lexicographic order. It was recently proven that the Grandaddy has a lexicographic partner. The "Grandmama" sequence is constructed by concatenating the aperiodic prefixes of necklaces in co-lexicographic order. We will discuss the construction and some interesting properties of both the Grandaddy and Grandmama de Bruijn sequences.

Based off a similarly-titled paper by Williams, Hernandez, and Dragon (November 2015).

Date: April 13, 2016, 11:30 a.m.
Speaker: Scott Jeffreys, Hofstra University
Title: Computational Finance, Applications of Calculus, and R Programming
Abstract: Computational finance forges together applied mathematics, economics, computer science, and finance to solve problems in algorithmic and high-frequency trading, quantitative investing, and portfolio management. Financial engineers, or "quants," create models against which markets can be measured for returns, hedging opportunities, or risk controls.

During the Spring 2016 term following 18 months of research, the Hofstra University School of Engineering and Applied Sciences launched our first course in computational finance using algorithms developed in R to solve otherwise untenable problems. In this seminar, we will look at three complex problems from finance and offer elegant solutions drawn from our applied mathematics and computer science tools:

[1] How are coupon bonds efficiently priced?

[2] How are options priced using the Black-Scholes model?

[3] Why do heavy tails warn us about portfolio risk?

Date: April 22, 2016, 4:00 p.m.
Speaker: Mutiara Sondjaja, New York University
Title: A New Proof of Tucker's Lemma with a Volume Argument
Abstract: Tucker's lemma is a combinatorial theorem about labeled triangulations of the d-dimensional sphere. It states that if the vertices of such triangulations receive a label from {±1, ±2, ..., ±d} with the labels of antipodal vertices summing to zero, then there must exists a pair of adjacent vertices whose labels sum to zero. The lemma is known to be equivalent to the Borsuk-Ulam theorem in topology.

In this talk, we discuss an application of Tucker's lemma in constructing a solution to a fair division problem (Su and Simmons (2002)). Then we present a new proof of Tucker's lemma based on a volume argument. This is joint work with F. E. Su and undergraduates at the 2015 MSRI Undergraduate Program (B. Kutture, O. Leong, C. Loa).

Date: April 27, 2016, 11:30 a.m.
Speaker: Stephanie Nagel, Hofstra University
Title: Generating on the N-Sphere: An Undergrad Research Experience
Abstract: Before taking my departmental honors research course, I had some knowledge of what math research entailed, but I have learned so much through this experience. We will discuss what it is like to do research, and how doing research is different from taking a regular class. We will also talk about the research that I have been doing with Dr. Gonzalez, which is on the methods to generate uniform random variables on the d-dimensional unit spherical shell. These methods include scaling vectors of normal random variables and an algorithm involving spherical coordinates. We will look at the code for these methods and discuss further research that I plan to do with these methods.

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