I work in the fields of Discrete Mathematics and Operations Research.
My research interests lie in Graph Theory, Discrete Optimization, Discrete Geometry, Probabilistic Discrete Structures, and their interdisciplinary applications in Civil Engineering (Transportation Networks), ECE (Renewable Energy distribution), Computer Science (Network Security), Homeland Security, etc.
Discrete optimization models arise naturally in all branches of discrete mathematics and their applications. My motivation is to develop the understanding of hard optimization problems through a variety of approaches - using probabilistic and combinatorial tools to tackle problems, developing and analyzing the complexity of exact and approximation algorithms, developing both practical and theoretical aspects of heuristics, and understanding local search algorithms and multi-objective optimization. I work with collaborators in Computer Science and in Engineering to apply these methodologies to interdisciplinary applications.
My past research has included topics in Stochastic Combinatorial Optimization, Network Optimization, Exact Combinatorial Algorithms, Meta-Heuristics, Local Search Landscapes, Multi-objective Discrete Optimization.
Graph Theory is the language used to describe and study binary relations, which explains its numerous and varied applications in engineering, social sciences, and natural sciences. For nontrivial applications of graph theory, it is essential to understand the substructures (and corresponding properties) in graphs and the parameters that imply their existence. My aims in graph theory are two-fold - to deepen the theoretical understanding of graphs as discrete structures, and to apply graph-theoretic techniques to problems in other disciplines.
My past research has included topics in Graph Packing, Graph Coloring, MAX-CUT or Largest Bipartite Subgraph problem, Maximum Independent (Stable) Set problem, Maximum Planar subgraph problem, Graph Layout problems.
Math 332: Elementary Linear Algebra:Syllabus; Discussion and Course.
Math 410: Number Theory: Syllabus; Discussion and Course.
Math 435: Linear Optimization:Syllabus; Discussion and Course.
Math 380; Math 486/522: Mathematical Modeling:Syllabus; Discussion and Course..
Math 454: Graph Theory and Applications: revised existing joint undergrad-grad course to
make it an appropriate independent course for undergraduate students from various backgrounds - Syllabus.
Math 535: Linear Optimization: Syllabus; Discussion and Course.
Math 553: Graduate Graph Theory: revised existing joint undergrad-grad course
to make it an appropriate core-course for graduate students with advanced topics and modern techniques - Syllabus.
Math 554: Discrete Applied Math II (Modern Methods in Combinatorics): Syllabus; Discussion and Course.
At UIUC :
Topics in Probabilistic Methods for Discrete Mathematics: Proposal and Course.
Discrete and Convex Geometry : Proposal.
Fall 2009 Math 332: Elementary Linear Algebra (Matrices)
Math 591: Research with graduate student on 'computational art gallery problem'.
Math 491: Independent reading with undergrad student on advanced topics in 'Extremal Graph Theory'.
Independent reading with undergrad student on introduction to 'Stochastic Optimization'.
Math 100: Lectures and problems on 'Visual Combinatorial Proofs', 'Planar Graphs and Graph Drawing', and 'Graph Coloring'.
Fall 2008 Math 332: Elementary Linear Algebra (Matrices)
Math 491: Research with undergrad on 'Guarding Orthogonal Art gallery with holes'.
Math 100: Lecture on 'Graph Coloring'; Project (expository report and computer implementation) on 'Theory and Algorithms for Stable Matchings' with 2 students.
Math 593: Graduate Student Seminar coordinator.
Math 491: Research with undergrads on 'Guarding Orthogonal Art gallery with holes' and on 'Fall coloring of Graphs'.
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