# Research

Research in the GW Department of Mathematics explores topics within the core focus areas of algebra, analysis, applied mathematics, combinatorics and graph theory, computational biology, logic and topology. Our well-published faculty regularly secure grant funding from institutions such as the National Science Foundation, the National Security Agency and NASA.

Undergraduate mathematics students earn a range of departmental prizes and academic honors in recognition of their research, and graduate students learn to conduct sophisticated, scholarly research suitable for publication during their degree programs.

## Faculty by Core Research Area

### Algebra

**William Schmitt**: Algebraic combinatorics, Hopf Algebras, Category Theory**Robert Won**: Noncommutative Algebra, Quantum Symmetries, Weak Hopf Algebras

### Analysis

**John Conway**(Emeritus): Function theoretic operator, Non-abelian approximation**Hugo D. Junghenn**: Functional Analysis, Semigroups, Financial Math**E. Arthur Robinson**: Ergodic Theory, Functional Analysis, Aperiodic Order, Computational Math**Lien-Yung (Nyima) Kao**: Dynamic Systems, Thermodynamic Formalism, Ergodic Geometry

### Applied Mathematics

**Frank E. Baginski**: Applied Mathematics, Partial Differential Equations**Murli M. Gupta**: Numerical Analysis, Computational Fluid Dynamics, Mathematical Education**Xiaofeng Ren**: Partial Differential Equations, Applied Mathematics**Yanxiang Zhao**: Numerical Analysis, Numerical Methods for differential equations

### Combinatorics and Graph Theory

**Joseph E. Bonin**: Combinatorics, Matroid Theory**Daniel Ullman**: Graph Theory, Combinatorics**Joel B. Lewis**: Enumerative and Algebraic Combinatorics**William Schmitt**: Algebraic combinatorics, Hopf Algebras, Category Theory

### Computational Biology

**Max Alekseyev**: Computational Molecular Biology/Bioinformatics, Comparative Genomics, Graph Theory, Combinatorics, Discrete Algorithms**Yanxiang Zhao**: Numerical Analysis, Numerical Methods for differential equations

### Logic

**Valentina S. Harizanov**: Logic, Recursion Theory, Recursive Model Theory**Michael Moses**: Logic, Recursive Model Theory, Computational Complexity

### Topology

**Lowell Abrams**: Low Dimensional Combinatorial Topology,Topological Graph Theory, Combinatorial Algebra**Jozef H. Przytycki**: Topology, Knot Theory, Algebraic Topology based on knots**Hao Wu**: Geometric Topology, Especially 3-dimensional Contact Topology, Heegaard-Floer Theory, Khovanov-Rozansky Theory

## Faculty Grant Funding

- EXTREEMS-QED: GW Mathematics and Statistics Training, Education, & Research (MASTER), 2014-2018, Yongwu Rong (PI), Maria Gualdani, Murli Gupta, Yinglei Lai, and Rahul Simha (Co-PI),
- Analysis of nonlocal effects in nonlinear parabolic partial differential equations, 2014-2017, Maria Gualdani.
- Building Capacity for Disciplinary Experts in Math and Science Teaching, 2014-2015, PI: Curtis Pyke, Co-PI: Daniel Ullman, Larry Medsker, Hartmut G. Doebel.
- Multi-Constituent inhibitory systems with self-organizing properties, 2013-2016, Ren, Xiaofeng
- WIDER: Planning: GW Reform and Advancement of STEM-education Practices (GRASP), 2013-2018, PI: Rahul Simha, Yongwu Rong (Co-PI).
- Jump: Joint Undergraduate Mathematics and Physics Scholarships at GW 2013-2018, Rong, Yongwu
- JUMP: Joint Undergraduate Mathematics and Physics Scholarships at GW, Y. Rong(PI), D. Ulmann(co-PI), L. Medsker(co-PI, Physics), G. Fieldman(co-PI, Physics), S. Roudenko(co-PI); 8/1/13- 7/31/16
- Combinatorial Link Homologies and Their Applications, Wu, Hao
- Topics in Computable Structure Theory, Harizanov, Valentina
- Topics in Computable Mathematics, Harizanov, Valentina
- Knots in Washington: Conferences on Knot Theory and its Ramifications, including Quantum Computing, 2011-2014, DMS-1137422 Jozef Przytycki (PI), Valentina Harizanov (Co-PI), Alexander Shumakovitch (Co-PI), Hao Wu (Co-PI), 9/1/11-8/31/14.
- Dispersive PDE CAREER: Nonlinear Phenomena in Evolution PDE, Roudenko, Svetlana
- Topics in Global Behavior of Solutions to nonlinear dispersive PDE, DMS-1103274, Svetlana Roudenko (PI), 9/1/11-8/31/14
- Singular limits, saturation, and defects in block copolymer morphology , DMS-0907777, Xiaofeng Ren (PI), 7/1/09-6/30/12
- Topics in Computable Mathematics, DMS-0904101, Valentina Harizanov (PI), 7/1/09-6/30/12
- Knots in Poland III; the conference on Knot Theory and its Ramifications, DMS-1034753, Jozef Przytycki (PI), Mikhail Khovanov (Co-PI), 7/1/10-6/30/11
- Knots in Washington: Conferences on Knot Theory and its Ramifications 2008-2010, DMS-0817858, Jozef Przytycki (PI), Alexander Shumakovitch (Co-PI), Hao Wu (Co-PI), Yongwu Rong (Co-PI), 5/15/08-4/30/11
- CDI-type II: Understanding Complex Biological Networks: A Process Viewpoint, Chen Zeng (PI), Yongwu Rong, Rahul Simha (CO-PI), NSF, Cyber-enabled Discovery and Innovation (CDI) program, CMMI-0941228, 2009 – 2013.

- ExaVolt Antenna: Supporting Technology for Suborbital Ultra-high Energy Particle Accelerators, George Washington University, NASA Goddard Space Flight Center, Frank E Baginski, 01/10/2012 - 01/09/2015.

- Summer Program for Women in Mathematics, Murli M. Gupta
- Investigating Algebraic Structures Motivated by Knot Theory: homology of quandles, Burnside groups, skein modules and Khovanov homology, Jozef Przytycki
- Structure of Combinatorial Hopf algebras, William R. Schmitt

- Singularity Formation in Evolution Equations and Pedagogical Connections of Math, Biology and Medical Imaging, Svetlana Roudenko
- Quantum Computations, Complexity and Information, V. Harizanov (PI) co-PIs: A. Eskandarian, (Physics), W. Parke (Physics) and J. Przytycki (Mathematics)
- Properties and Applications of Link Homology Theories, Alexander Shumakovitch
- Combinatorial link homologies and 3-manifold invariants, Hao Wu

- Algebraic structures motivated by and applied to Knot Theory, Simons Foundation, granted for 5 years, 2014-2019, Jozef Przytyck.
- Khovanov Homology Theories and their Applications, 2013-2018, Simons Foundation, Shumakovitch, Alexander
- MAA grant RUMC (Research Undergraduate Math Conference), joint conferences of DC-area universities, Roudenko(PI 2013), Rong(PI 2012)
- Undergraduate Math Conferance, 2013, Mathematical Association of America, Roudenko, Svetlana
- Dc FAME II 2011, DC Office of the State Superintendent of Education, Ullman, Daniel H.
- Geometric Structures in Self Organizing Physical and Biological Systems, Simons Foundation, Ren, Xiaofeng
- Dynamical Systems and Aperiodic Orderopics in Global Behavior of Solutions to Nonlinear, Simons Foundation, Robinson, E. Arthur
- Studies in Global Behavior of Solutions in Nonlinear Dispersive PDE, Simons Foundation, Roudenko, Svetlana
- Link Homologies and Their Applications, Simons Foundation, Wu, Hao
- District of Columbia Fellows Advancing Mathematics Education (DC FAME), DC Gov, Daniel Ullman (PI), 2010-12
- Collaboration in Computability, NSF research bi-national (with Russia/Kazakhstan) grant DMS-1101123, 5/1/011-4/30/2014, J. Knight (PI), V. Harizanov (senior personnel).
- Nonlinear Dispersive PDE, Simons Foundation, Roudenko Svetlana
- Polish Scientific Grant: Nr. N N201387034; Jozef Przytycki (Co-PI), 4/29/08-4/29/11.

## Spotlight on Student Research

### Computer Science for Social Good

Samsara Counts, a senior majoring in computer science and mathematics, used deep learning methods to develop tools that help patients with eating disorders avoid triggering images online.

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