Grad STEM for All 2019



Overview: The goal of Grad Stem For All is to empower and further inspire all interested students from colleges and universities in Western New York area to pursue advanced degrees in STEM fields, particularly mathematics and statistics. We are especially looking to involve groups that have traditionally been underrepresented in STEM, including women and underrepresented minorities. Our goal is to provide these students with the information, skills and training needed to succeed in graduate school, academic careers, and industry. The three components of the program are coursework, research and mentoring. During the summer, faculty from local universities will be teaching a series of mini-courses that will enhance the undergraduate education the students receive at their liberal arts institution. These mini-courses are meant to be a preview of graduate coursework and provide students with a deeper background in mathematics and/or statistics. Each student participant will also be paired with an experienced faculty mentor from an area college or university, who can provide support and guidance throughout the student's college career and beyond. By having an identified mentor, students will more readily be able to get involved in research projects, learn about graduate schools, get advice on the application process, and learn more about life in academia in general. We intend for the mentor/student relationship to be long lasting, inspiring students to pursue advanced coursework and careers in STEM field. After the mini-courses, the program is going to enter the research phase where the students are going to be guided through a series of accessible open problems.

Interested individuals are encouraged to contact us at urstemforall2019@gmail.com

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Organizers: This program was launched in 2018 by Drs. Joseph Ciminelli and Alex Iosevich from the Department of Mathematics at the University of Rochester and by Dr. Sally Thurston from the Department Biostatistics and Computational Biology. You can find the description of last year's program by clicking on this link. This year the program is organized by Alex Iosevich and Vitaly Lorman from the Department of Mathematics. Outreach for the program is organized by Dr. Cheri Boyd from the Department of Mathematics at Nazareth College. Critical to the success of these efforts are the cooperation from faculty from the School of Mathematics from Rochester Institute of Technology, and other faculty and graduate students from these departments. We hope to expand this list to cover every college and university in the Rochester area that teaches mathematics and statistics courses.

Mentoring: Every participant in our program will be assigned to a mentor, who will assist them with course selection, advise them on additional readings and help them seek out graduate and professional opportunities. The participant will be able to maintain phone and email contact with the mentor, in addition to regular meetings.

Mini-courses: The first wave of summer courses will be offered in July, 2019. Students will be responsible for their housing and transportation to the program. Each mini-course will run Monday through Friday for a two-week period.

Each program will be followed by an informal two-week followup where the interested students can explore the topics in more depth with the instructor based on the background they have gained during the initial phase of the program. Long term research opportunity plans can be made for students who wish to continue the process once the program ends.

A detailed description of the mini-courses follows below.

Description of the mini-courses:

Mini-course 1 July 15-19, 22-26, 2019 with the research phase running the following two weeks.
Instructor: Ayla Gafni
Graduate assistants: To be announced
Time: 9:30-11:30 a.m. and 12:30-1:45 p.m. (problem solving session MWF and mentoring on TTh)
Location: To be announced
Title: Integer partitions
Prerequisites: First year calculus.

Description:  A partition of an integer n is a multiset of positive integers whose sum is equal to n.  For example, there are five partitions of 4, namely, 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.  In this course we will explore the various properties of integer partitions.  Topics include partition identities and bijections, Ferrer's diagrams and Durfee squares, partition generating functions, and partition congruences.

Mini-course 2 July 15-19, 22-26, 2019
with the research phase running the following two weeks.
 Instructor: Alex Iosevich and Yakun Xi
Graduate assistants: To be announced
Time: 2:00-4:00 p.m. and
12:30-1:45 p.m. (problem solving session MWF and mentoring on TTh)
Location: To be announced
Title: Sums, products and beyond
Prerequisites: High school geometry and first year calculus.

Description
: We are going to focus on the sum-product phenomenon for subsets of the integers and subsets of cyclic groups of prime order. The question is, can the set of pairwise sums and the set of pairwise products of a finite sum be simultaneously small. The idea is that addition and multiplication are very different operations, so it is difficult for a set to both resemble an arithmetic progression and a geometric progression. Remarkably, this problem can be studied from a geometric point of view using rather elementary ideas that do not require much background to absorb.


Mini-course 3 July 15-19, 22-26, 2019 with the research phase running the following two weeks.
Instructor: Steven Kleene and Ian Alevy
Graduate assistants: To be announced.
Time: 2:00-4:00 p.m. and 12:30-1:45 p.m. (problem solving session MWF and mentoring on TTh)
Location: To be announced
Title: From elementary geometry to cutting edge mathematics
Prerequisites: two semesters of first year calculus and high school geometry

Description: Euclid's axiomatic treatment of geometry in The Elements was, among other things, remarkable for its amazing and unparalleled  precision and rigour, neither of which were surpassed until the founding of truly modern mathematics in the 1800's. Nonetheless, the original work is somewhat poorly understood by many for what it does and does not do. In particular, for all the rigour with which it treats its subject, a truly axiomatic treatment of geometry had to wait until certain modern mathematical concepts, such as continuity, could be completely formulated. In this two week course, we will carefully examine the foundations of Euclidean geometry from the point of view of Euclid's original axioms from the Elements. We will discuss the limitations of these axioms and the modern notions which quite literally filled gaps in several important classical results. The second part of the course, we will discuss how Euclid's formal treatment of geometry, namely the Parallel Postulate, led to the discovery of  so called non-euclidean geometries and ushered in a new era of modern mathematics and physics. We will conclude the class with a discussion of some modern open problems in geometry

Mini-course 4 July 15-19, 22-26, 2019 with the research phase running the following two weeks.
Instructor: Vitaly Lorman
Graduate assistants: To be announced
Time: 9:30-11:30 a.m. and 12:30-1:45 p.m. (problem solving session MWF and mentoring on TTh)
Location: To be announced
Title: Knots and polynomials
Prerequisites: Two semesters of calculus (required).

Description: Knot theory is an accessible and exciting area of math with connections to geometry, topology, physics, and abstract algebra, among other fields. It has also seen great progress in the last few decades with the invention of powerful new ways to study knots, such as the Jones and HOMFLY polynomials. We will start with a hands on introduction to knot theory and work through lots of examples. We will then focus on polynomial knot invariants and investigate exactly how much these polynomials can tell us about knots.

Additional mini-courses may be added to the list in the coming weeks.