# Mathematical Sciences Research Institute

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# MSRI-UP

MSRI-UP 2014: Arithmetic Aspects of Elementary Functions June 21, 2014 - August 03, 2014
Location: MSRI: Baker Board Room, Commons Room, Atrium
Organizers Duane Cooper (Morehouse College), Ricardo Cortez (Tulane University), LEAD Herbert Medina (Loyola Marymount University), Ivelisse M. Rubio (University of Puerto Rico), Suzanne Weekes (Worcester Polytechnic Institute)
Speaker(s) No Speakers Assigned Yet.
Description
The MSRI Undergraduate Program (MSRI--UP) is a comprehensive summer program designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences.  The main objective of the MSRI-UP is to identify talented students, especially those from underrepresented groups, who are interested in mathematics and make available to them meaningful research opportunities, the necessary skills and knowledge to participate in successful collaborations, and a community of academic peers and mentors who can advise, encourage and support them through a successful graduate program.

The academic and research portion of the 2014 MSRI-UP will be led by Prof. Victor Moll from Tulane University.

MSRI-UP 2014: Arithmetic Aspects of Elementary Functions

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The question of evaluation of finite sums with entries in a reasonable largeclass (of hypergeometric type) has been settled by the algorithms developed by H. Wilf, D. Zeilberger and collaborators.  On the other hand, arithmetic properties of these sums offer interesting challenges. For instance, it is an elementary result that the central binomial coefficient is always even. This motivates the natural question: what is the exact power of $$2$$ that divides these coefficients?  Is there a closed-form formula for this?

The fact that binomial coefficients satisfy certain recurrences, for example in the formation of Pascal's triangle, has been used to analyze their arithmetic properties. What can be said about sequences generated by similar recurrences? For example, factorials $$n!$$ satisfy $$x_{n} = n\, x_{n-1}$$. Is it possible to describe arithmetic properties for $$y_{n} = P(n)\,y_{n-1}$$ with a polynomial $$P$$? Very few results are known.

Graphical representations offers some indication of the complexity involved. For example, there is a marked difference between the power of two that divides $$n^{2}+1$$ and $$n^{2}+7$$. What is the reason behind this? The second graph looks almost random. Is there a way to quantify this phenomena?

Some sequences with surprising arithmetical properties include Stirling numbers, Catalan numbers that count legal typing words using parenthesis, the ASM numbers that count the number of matrices with entries from $$\{ 0, \, \pm 1 \}$$ satisfying an ordering condition and many other coming from Combinatorics. Recent symbolic experiments include sequences such as the harmonic numbers $$H_{n} = 1 + \tfrac{1}{2} + \cdots + \tfrac{1}{n}$$ and the sequence of formed by partial sums of the exponential function.

These type of problems are ideal for introduction to undergraduates: they can be explained with a minimal amount of background, data can be obtained by using symbolic languages and partial results are available in the literature. Thus, this REU is accessible to students who have had three semesters of calculus, linear algebra, and a course in which they have had to write proofs.

### General Description

During the summer, each of the 18 student participants will:

• participate in the mathematics research program under the direction of Dr. Victor Moll, Tulane University, a post-doc and two graduate students
• complete a research project done in collaboration with other MSRI-UP students
• give a presentation and write a technical report on his/her research project
• attend a series of colloquium talks given by leading researches in their fields
• attend workshops aimed at developing skills and techniques needed for research careers in the mathematical sciences and
• learn techniques that will maximize a student's likelihood of admissions to graduate programs as well as the likelihood of winning fellowships
• receive a \$3100 stipend, lodging, meals and round trip travel to Berkeley, CA.

After the summer, each student will:

• have an opportunity to attend a national mathematics or science conference where students will present their research
• be part of a network of mentors that will provide continuous advice in the long term as the student makes progress in his/her studies
• be contacted regarding future research opportunities

### Application Materials

Applications for MSRI-UP 2014 should be submitted via the MathPrograms listing, which lists the required application materials.  Due to funding restrictions, only U.S. citizens and permanent residents are eligible to apply, and the proram cannot accept foreign students regardless of funding.  In addition, students who have already graduated or will have graduated with a bachelor's degree by June 16, 2014 are not eligible to apply.  The application link will be available on this page starting November 15, 2013. Applications submitted by February 15, 2014 will receive full consideration. (Applications submitted after February 15, 2014 but by March 1, 2014 may still be considered in a second round of acceptances.) We hope to begin making offers for participation in late February or early March.

Email: msriup@msri.org (Primary)

The directors of MSRI-UP are:

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