Daniel H. Ullman - US grants

Over the past two decades, low dimensional topology has seen a great
deal of studies in two types of invariants: gauge theory type invariants
in dimension four and combinatorial type invariants in dimension three.
While both sides have deep connections with physics, they share little
common techniques and have rather different flavors. This picture could
change though, with recent work due to Khovanov and Ozsvath-Szabo.
In 1999, Khovanov introduced a graded homology theory for knots,
and proved that its graded Euler characteristic is the Jones polynomial.
This has turned out to be a far reaching generalization of the Jones
polynomial. Furthermore, there is strong evidence that Khovanov theory,
along with the Ozsvath-Szabo theory, could bridge the connection between
gauge theory type and combinatorial type invariants.
Motivated by Khovanov's work, the PI, with his student Laure
Helme-Guizon, has established a graded homology theory for graphs which
yields the chromatic polynomial when taking Euler characteristic.
The PI intends to further his investigation on these homology theories,
both for knots and for graphs. Some of the specific problems are:
understanding their geometric meanings, studying their behavior under
various cut and paste operations, constructing homology theories for
various other polynomials of knots and graphs, and investigating
relations with other invariants in low dimensional topology.

Low dimensional topology studies the shapes of three and four dimensional
spaces. These dimensions are of particular interests to mankind because
of the dimensions of our space and our space-time. A specific subfield
in low dimensional topology is knot theory, which studies the knottedness
in our three dimensional space. Knots are worthwhile to study not only
because they are fundamental in 3-dimensional spatial structure, but
also because of its connection to areas outside mathematics. For example,
biochemists have discovered knotted DNA molecule (1980s) and knotted
proteins (2004). It is also intimately related to the study of graph
theory, an area interesting to mathematicians, computer scientists, and
others. Over the past two decades, there have been a flourish of new
invariants in low dimensional topology, boosted by ideas from gauge theory,
quantum algebras, and mathematical physics. In particular, a new
invariant for knots, developed by Khovanov using ideas in homological
algebra, has sparked a great deal of interest recently. An analogous
theory for graphs has since been developed by the PI and his student.
This project aims to investigate these new invariants, with a particular
emphasis on the homological algebra methods for knots and graphs.

The JUMP Scholarship program is supporting students beginning in their sophomore year through a comprehensive set of activities involving financial aid, outreach, recruiting, cohort activities, and mentoring by faculty, student peers, and alumni from industry and government. The objectives of the project are to: (1) enhance the educational opportunities for 30 talented students in mathematics or physics with documented financial need by providing scholarships for each student for up to three years; (2) mentor and support these students through an integrated cohort program to complete their degrees in a timely manner and prepare them to enter the science, technology, engineering, or mathematics (STEM) workforce or graduate school; (3) expand the overall student support programs in mathematics and physics at the university to benefit all STEM majors; and (4) create a model for educational collaborations between the Mathematics and Physics Departments that can be adopted by other STEM departments at the university and other institutions.

Novel features of the JUMP program include: (1) a focus on great ideas at the intersection of physics and mathematics to motivate cohorts of students to stay in these majors, (2) an advisory committee of successful STEM professionals to mentor the JUMP Scholars toward STEM careers, (3) the use of best practices in Physics Education Research to give JUMP Scholars the latest in active-learning experiences.

Intellectual Merit: The JUMP program is based on joint efforts in mathematics and physics to enhance the educational opportunities and intellectual life for students. The scientific interaction between cohorts of mathematics and physics students provides additional motivation and role modeling for students about the exciting problems in the fields and the relationships to the current body of knowledge. Cohorts of mathematics and physics majors are learning about the important issues and problems in the fields and engaging in intellectual discussions through seminars and special courses. Students are strongly encouraged to participate in the active research areas of faculty.

Broader Impacts: The institution is working to significantly increase STEM interest by undergraduates. Both the Mathematics and Physics Departments have devoted considerable effort in recent years to high school outreach programs, and these efforts have led to more freshmen interested in STEM fields. This project is reaching out to the large pool of talented students currently in non-STEM fields, but with great potential to excel in mathematics and physics. The program is helping to increase the diversity of STEM students, building on (a) the fact that 60% of the students in the College of Arts and Sciences are women and (b) the close connections of the university to several high schools and community colleges with high percentages of minority students.

This project addresses the Nation's growing need to recruit, prepare, and provide effective induction support for new STEM middle and high school teachers with degrees in STEM fields and strong content-specific pedagogical preparation. In particular the project will create an innovative pathway to teaching in Washington, DC for STEM professionals and STEM majors. This population represents an untapped supply of future teachers committed to teaching in high needs schools. Upon completion, the project will have validated a strategy to tap into this pool of STEM disciplinary experts, and will design and pilot a comprehensive set of learning experiences tailored to them. The project work will feature education and STEM faculty working with expert mathematics and science teachers in developing and piloting new methods to support teacher candidate development. Museum internships will be established and infused throughout the program, because museums are places in the urban landscape where teacher candidates can participate in authentic learning experiences and practice teaching a diverse population of students.

The long-term goal of this collaborative effort is to greatly enhance the DC area's ability to prepare "disciplinary experts" for urban mathematics and science teaching. There are three project outcomes. First, the project will discover how to increase the number of STEM professionals and undergraduate STEM majors preparing to teach in the DC area. Second, the project will institutionalize a sustainable nexus of collaboration among George Washington University's (GWU) education and STEM faculty, the District of Columbia Public and Public Charter Schools, the Smithsonian National Museum of Natural History, Carnegie Academy for Science Education, and GWU's Center for Civic Engagement, to improve the mathematics and science teacher education pipeline in Washington, DC. Third, a robust program of study will be produced featuring redesigned teacher preparation coursework, integrated and innovative field experiences, and early career mentoring. A systematic analysis of pilot data collected from selected student work and video from courses and field placements will facilitate the development of new curriculum emphasizing methods that teach candidates to use their knowledge in mathematics and science, not just to judge student reasoning, but to recognize and use school age students' good thinking (i.e. good scientific and mathematical ideas) as building blocks for learning. Finally, the project is designed to respond to the need to extend the scope of teacher training beyond what is traditionally done within the boundary of a university-based teacher education program, and therefore will also produce a specialized induction and early career support strategy that will link candidates with mentors who will follow them in their early years of teaching in the Washington, DC metropolitan area.

As the nation faces an increasingly critical need for a strong technology workforce, secondary schools continue to be challenged with fulfilling the shortage of licensed teachers with strong content knowledge in STEM disciplines. High-need students are an insufficiently tapped population for helping meet these challenges. This George Washington University (GW) Track 1: Robert Noyce Teacher Scholarships and Stipends (S&S) project, in partnership with Loudoun County Community College (LCCC), will recruit, license, and retain 27 high-achieving undergraduate majors in biology, chemistry, mathematics, and physics to become grades 6-12 STEM teachers in high-need schools. This Noyce project will identify and educate new math and science teachers who had not previously chosen to become teachers. A pre-Noyce phase of the project will provide opportunities for incoming students at GW and LCCC to learn about high-need communities through courses, seminars, and related first-hand experiences in service learning projects focused on high-need populations. Noyce students will deepen their content knowledge through participation in undergraduate research projects and internships with established external partners, including Smithsonian, National Institute of Health, and high-tech companies. Additionally, Noyce students will receive intensive mentoring while in the Noyce program, and Noyce graduates will receive mentoring in their first years of teaching in high-need schools. Thousands of secondary students in the DC metropolitan area, and across the nation, will benefit from improved science and math instruction as a result of the GW Noyce program. Producing high-caliber secondary math and science teachers in high-need schools is essential to support our nation's increasingly STEM-driven economy.

The GW Noyce project is informed by GW's rigorous, research-based undergraduate STEM initiatives that will enrich the STEM content knowledge of new GW teachers, assuring that Noyce STEM teachers will have content knowledge at the same level of rigor as other STEM professionals. The pre-Noyce phase cohort, comprised of first and second year students from GW and LCCC, will be invited in their sophomore years to join a Learning Assistants program to develop teaching skills and learn about the teaching profession. Students from the pre-Noyce group will apply for admission into the Noyce Scholars program, which provides scholarships for their junior and senior years. The GW Noyce project will advance knowledge about Noyce program impacts on the production of highly qualified STEM teachers in high-need schools. Dissemination of project results will occur through conference presentations and publications in STEM education journals.