Yongwu Rong - 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.

Abstract

Award: DMS-0555648
Principal Investigator: Jozef H. Przytycki

This award provides partial support for participants of the 21st
conference in the very successful ``Knots in Washington''
series. This meeting is devoted to Skein Modules, Khovanov
Homology and its relation to Hochschild Homology. ``Khovanov
homology'' is a new development in quantum topology that emerged
in the last several years. It is a generalization of the Jones
and Homflypt polynomials of links to homologies of certain chain
complexes. These homologies turn out to be significantly
stronger than the original invariants. Khovanov's ideas were also
applied to the polynomial invariants of graphs, as well as
Kauffman bracket skein module of some 3-manifolds. One of the
most recent developments is Przytycki's observation that Khovanov
homology (or its comultiplication-free variant developed by
Helme-Guizon and Rong) can be interpreted as Hochschild homology
of underlying algebras. This shows how seemingly distant branches
of mathematics can be put together.

Low dimensional topology studies shapes of three and four
dimensional spaces. These dimensions are of particular interest
to us because they are the dimensions of our space and our
space-time. Knot theory is a subfield of low dimensional
topology, which studies the knottedness in our space. One of the
main goals of knot theory is to distinguish knotted objects. This
is often done by means of the so called "knot invariants,"
functions that replace geometric objects, such as knots, with
those that are easier to compare. 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 from homological
algebra, has sparked a great deal of interest recently. The
purpose of the Knots in Washington Conferences is to bring
together the researchers of the field, including established
mathematicians as well as graduate students and recent PhDs, to
discuss the state of the art of the subject.

Abstract

Award: DMS-0817858
Principal Investigator: Jozef H. Przytycki, Yongwu Rong,
Alexander N. Shumakovitch, Hao Wu

A series of conferences devoted to knot theory and its
ramifications will be held in 2008-2010, continuing the "Knots in
Washington" conferences held every semester in the greater
Washington, D.C. area since fall 1995. Plenary talks will
survey the state of knowledge in areas such as Khovanov homology,
Heegard Floer homology, and categorification of existing
invariants such as skein modules. Contributed talks and periods
devoted to interaction among participants provide opportunities
for graduate students and recent PhDs to be a part of "Knots in
Washington," and the organizers encourage students and junior
researchers to attend these meetings.

Low dimensional topology studies shapes of three and four
dimensional spaces. These dimensions are of particular interest
to us because they are the dimensions of our space and our
space-time. Knot theory is a subfield of low dimensional
topology, which studies the knottedness in our space. One of the
main goals of knot theory is to distinguish knotted objects. This
is often done by means of the so called "knot invariants,"
functions that replace geometric objects, such as knots, with
those that are easier to compare. 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. The purpose of the "Knots in
Washington Conferences" is to bring together the researchers of
the field, including established mathematicians as well as
graduate students and recent PhDs, to discuss the state of the
art of the subject. The conference web page is
http://home.gwu.edu/~przytyck/knots/index.html.

This project focuses on one of the most popular models of complex networks, Boolean models, and proposes a new approach based on a process-viewpoint. In contrast to the standard attractor-basin portrait of a discrete dynamical system, the process-viewpoint starts with a single sequence of states and addresses the types of networks that might produce that sequence. The sequence of states, a process, corresponds to a time-course in biological terms and is often the only dynamical data available for real systems. The types of research questions include: what networks produce a given biological process? How do those networks differ and what do they have in common? This project will also consider the space of processes and the types of questions that arise from characterizing the space: Are some processes harder to build networks for? Do some processes need different network properties than others? The approach used is based on mathematical logic and results in a single expression characterizing the space of networks for a given process.

The importance of this research project is threefold. First, the research has the potential of increasing our understanding of complex biological networks, whose functioning is the basis of living systems. Second, the approach might result in a practical algorithm for inferring network structure from the types of data available today; this is significant because the underlying network structure is very difficult to infer with current technologies. The precise network structure and dynamics is essential to understanding how particular reactions within the cell work. Finally, the project will help increase our understanding of complex systems in general. Biocomplexity is merely one type of complexity; to the extent we make headway in building computational tools to help understanding this type of complexity, similar tools are likely to help with other types of complexity. Certainly, this has proved to be the case for other complexity-tools such as small-world networks, scale-free graphs and phase transitions.

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.

The George Washington University Mathematics and Statistics Training, Education, and Research (MASTER) program is designed to excite students about the mathematical questions and techniques underlying the computational challenges of analyzing large-scale data. The program will be run by the Departments of Mathematics and Statistics, with support from faculty in Computer Science. Computational and Data-enabled Science and Engineering (CDS&E) has now emerged as a distinct intellectual and technological discipline lying at the intersection of applied mathematics, statistics, computer science, core science and engineering disciplines. The program will help educate mathematics and statistics undergraduate students who are prepared to confront new challenges in CDS&E. Training students to think mathematically while working with big data will prove useful for students' successful careers in many application areas.

The project will feature curricular enhancements, research mentoring, and faculty development. The curricular enhancements will add new courses centered about the theme of CDS&E; the undergraduate research mentoring program will provide research experience for talented undergraduates majors in mathematics or statistics. The research projects will themselves involve CDS&E research questions arising from network dynamics, biological data analysis, clustering, topology of large data sets, and compressive sensing. Among the new courses to be offered are mathematics of networks, the statistics of data exploration, and CDS&E mathematical modeling; the courses will be focused on the variety of topics, including network data analysis, biological network data, graphical techniques for data exploration, statistical computing, dimension analysis, and modeling of large data sets. Summer workshops for faculty will enhance the skills of college faculty in CDS&E.