# Research

Research in the GW Department of Mathematics explores topics within the core focus areas of 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

### 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**:**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

### 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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