Faculty Research Interests
1. Anything cryptologic.
2. Implementing primality testing and factoring algorithms, including parallelizing.
3. Linear or Integer Programming models for various practical problems. This is kind of open-ended and the actual problem would be determined by the student.
Suppose we have a balance scale and an unknown quantity of objects with various weights. You are handed the objects one at a time. For each one, you must choose a side of the scale on which to put it, with the objective of getting the scale as close as possible to balanced when the last object is placed. You are not allowed to move any object once it has been placed, and you never know if the current object will be the last. This challenge is an example of an online problem. In contrast to traditional algorithmic problems in which you know all of the input in advance, an online algorithm must make decisions with limited information about the future. There are many scenarios in computer science and other fields that are inherently online: scheduling problems, network routing problems, auctions and trading problems, and games. Online algorithms are almost never able to devise optimal solutions due to their incomplete knowledge of the input. Therefore, we evaluate online algorithms with respect to their ability to construct a solution that is close to the optimal solution, a technique known as competitive analysis. In the past, I have worked with students on online algorithms for scheduling tasks on parallel machines and routing flows in networks.
I am also interested in interdisciplinary research projects, especially those related to computational biology and genomics.
My primary research interests are in artificial intelligence and machine learning. I am interested in studying how people learn to solve complex problems and then capturing that same behavior in a computer algorithm. I also dabble in games and game theory, complex optimization problems (related to AI), robotics, and evolutionary computation.
Take a length of rope. Tie a knot in it and then glue the ends together. This is a mathematical knot – a closed loop in three-dimensional space. The study of knot theory has been a tremendously active area for undergraduate research in the last ten years as the questions are easily stated and the material is very tangible – you can make knots out of rope! Moreover, in resent years there have been a number of applications of knot theory to other areas including the Genome Project and Molecular Chemistry.
Over the past six years I have worked with students in two particular areas of knot theory. The first area, spatial graphs, provides a nice mixture of graph theory and knot theory. This work was resulted in three publications, two with students. More recently, I have been working with knot mosaics. This is a way to represent knots with specific tile pieces to create a square mosaic that depicts a specific knot. While this work uses many concepts from knot theory, it is also conducive to algorithmic approaches that can be programmed. This work has produced two pieces that have been submitted for publication.
In addition to publications, I strongly encourage my students to present our work at national conferences. In the past six years my students have won 10 top awards at national meetings for the quality of results and presentation. Of my seven research students who graduated from Denison, six have pursued graduate degrees in mathematics.
I do research with undergraduates in various subfields of advanced Linear algebra. One current ongoing project is an attempt to connect graph theory to linear codes. A graph is a set of points (vertices) together with a set of connections between these points(edges). A linear code is simply a matrix that encodes vectors via matrix multiplication. It turns out that there is a rich interplay between linear codes that preserve the length of vectors and certain families of graphs. These codes are interesting because they are excellent at correcting errors in transmitting the coded vector. They are also interesting because of the rich Mathematical structures to which they give rise, in particular a link between eigenvalue analysis, graphs, and problems in geometry. To do research in this area, a student should take Math 210 and 231.
A second project is the study of Operator Spaces. This field grew out of mathematical questions arising in Quantum Mechanics. In my project, these operator spaces arise as certain subspaces of matrices which give rise to an interesting interplay between algebraic structures (think matrix multiplication) and topological structures (think distances between vectors). This project is particularly suitable for those students who wish to go to graduate school in mathematics, as it involves many advanced ideas in Abstract Algebra, Real Analysis, and Complex Analysis. To do research in this area, a student should take Math 210, Math 231, and either Math 321 or Math 332.
I am also interested in a new project on mathematical modeling of basketball performance. The goal is to create useful comprehensive statistics that measure a player's overall offensive and defensive performance and to create a method of testing the validity of such statistics. This involves using cutting edge methods used by real NBA teams. To do research in his area, a student should take Math 242.
Books Authored by Faculty
Quantum Processes Systems, and Information
Benjamin Schumacker and Michael Westmoreland, Cambridge Press, 2010.
Elementary Cryptanalysis: a Mathematical Approach. Abraham Sinkov and Todd Feil. Mathematical Association of American (MAA), 2009
Real Infinite Series
Daniel D. Bonar and Michael J. Khoury '03, Mathematical Association of America (MAA), 2006.
A First Course in Abstract Algebra: Rings, Groups, and Fields, 2nd edition
Todd Feil and Marlow Anderson, Chapman & Hall/CRC Press, 2005.
Todd Feil and Marlow Anderson, D. Reidel, 1988.
On Annular Functions
Daniel D. Bonar, 1971.
Recent Research Papers Authored by Faculty
Optimal Online Ring Routing
J. T. Havill and Kevin R. Hutson. Networks, 2010.
Online malleable job scheduling for m <= 3
J. T. Havill. Information Processing Letters, 2010.
Dowker Spaces Revisited
L. Ludwig, P. Nyikos and J.E. Porter. Tsukuba Journal of Mathematics, 2010.
Existence of contractive projections on preduals of JBW*-triples
M. Neal. Israeli Journal of Mathematics, 2010.
Metric characterizations of isometries and of unital perator space
M. Neal. Proceedings of the American Mathematical Society, 2010.
The Hodge Structure of the Coloring Complex of a Hypergraph
S. Rundell, Submitted, 2010.
An Algorithm for Detecting TPP Riboswitches in Archaea (poster)
Chinmoy I.S. Bhatyia, J. T. Havill and Jeffrey S. Thompson. Ohio Collaborative Conference on Bioinformatics (OCCBIO), 2009.
The Homology of the Cyclic Coloring Complex
S. Crown. Journal of Combinatorial Theory, Series A, submitted, 2008.
Competitive online scheduling of perfectly malleable jobs with setup times
J. T. Havill and W. Mao. European Journal of Operational Research 87(3), pp. 1126–1142, 2008.
When graph theory meets knot theory
L. Ludwig and J. Foisy. Contemporary Mathematics, to appear, 2008.
Spaces with a σ-point-discrete weak bases
L. Ludwig, C. Liu, and S. Lin. Tsukuba Journal of Mathematics 32(1), pp. 165-177, 2008.
Metric characterizations of isometries and of unital operator spaces and systems
M. Neal and D. Blecher. Submitted, 2008.
Contractively complemented subspaces of pre-symmetric spaces
M. Neal and B. Russo. Submitted, 2008.
Hereditary subalgebras of operator algebras
M. Neal, D. Blecher, and D. May. Journal of Operator Theory, 2008.
Strategic Network Formation with Structural Holes
T. Wexler, J. Kleinberg, S. Suri, and E. Tardos. In Proceedings of the 9th ACM Conference on Electronic Commerce, 2008.
Technically speaking: fostering the communication skills of computer science and mathematics students
J. T. Havill and L. D. Ludwig. In Proceedings of the ACM SIGCSE Technical Symposium on Computer Science Education, 2007.
Ordered involutive operator spaces
M. Neal, D. P. Blecher, K. Kirkpatrick, and W. Werner. Positivity 11 (2007), 497-510.
Open partial isometries and positivity in operator spaces
M. Neal and D. Blecher. Studia Math 182 (2007), 227-262.
A neighborhood search technique for the freeze-tag problem
D. Bucatanschi '06, B. Hoffman '05, K. Hutson, and R. M. Kretchmar. INFORMS Computing Society Conference, submitted, 2006.
Performance Analysis Based upon Complete Profiles
J. Krone, W. F. Ogden, and M. Sitaraman. Workshop on Specification and Verification of Component-based Systems, 2006.
Software verification is not dead, but it needs a new way to express mathematics
J. Krone and W. F. Ogden. Proceedings of the RESOLVE Workshop, 2006.
A project approach to programming language theory
J. Krone. Proceedings of the CCSC:MW Conference, 2006.
Open partial isometries and positivity in operator spaces
M. Neal and D. Blecher. submitted, 2006.
Hereditary subalgebras of operator algebras
M. Neal, D. Blecher and D. May. Journal of Operator Theory, in press, 2006.
Classification of contractively complemented Hilbertian operator spaces
M. Neal, B. Russo and E. Ricard. Journal of Functional Analysis, in press, 2006.
Quantum one-time keypad
M. Westmoreland and B. Schumacher. Physical Review A, in press, 2006.
Reverend Bayes takes the unexpected examination
M. Westmoreland and B. Schumacher. Math Horizons, in press, 2006.
Pushing the virtual envelope: Internet suspend/resume over HTTP(S)
T. C. Bressoud, M. Kozuch, and P. Nath. Proceedings of IASTED Conference on Web Technologies, Applications, and Services, pp. 69-76, 2005.
κ-Fréchect Urysohn spaces
C. Liu and L. D. Ludwig. Houston Journal of Mathematics 31(2), pp. 391-401, 2005.
Nagata-Smirnov revisited: spaces with σ-wHCP bases
C. Liu and L. D. Ludwig. Topology Proceedings 29(2), 2005.
Representation of contractively complemented Hilbertian operator spaces on the Fock space
M. Neal and B. Russo. Proceedings of the American Mathematical Society 134(2), pp. 475-485, 2005.
Reversibility of local transformations of multiparty entanglement
N. Linden, S. Popescu, B. Schumacher, and M. Westmoreland. Journal of Quantum Information Processing 4(3), pp. 241-250, 2005.
Locality and information transfer in quantum operations
B. Schumacher and M. Westmoreland. Journal of Quantum Information Processing 4(1), pp. 3-34, 2005.
OpenCAS: A flexible architecture for content addressable storage
T. C. Bressoud, M. Kozuch, C. Helfrich, and M. Satyanarayanan. ISCA International Conference on Parallel and Distributed Computing Systems, pp. 580-587, 2004.
Seamless mobile computing on fixed infrastructure
M. Kozuch, M. Satyanarayanan, T. C. Bressoud, C. Helfrich, and S. Sinnamohideen. IEEE Computer 37(7), pp. 65-72, 2004.
Robust reinforcement learning for heating, ventilation, and air conditioning control of buildings
C. W. Anderson, D. Hittle, R. M. Kretchmar, and P. Young. chapter in Handbook of Learning and Approximate Dynamic Programming, IEEE Press, 2004.
Robust reinforcement learning using integral quadratic constraints
C. W. Anderson, R. M. Kretchmar, P. Young, and D. Hittle. chapter in Handbook of Learning and Approximate Dynamic Programming, IEEE Press, 2004.
Keeping pointers or references under control: a component based approach to list based data structures
J. Krone. The Journal of Computing Sciences in Colleges 20(1), pp. 42-53, 2004.
State spaces of JB*-triples
M. Neal and B. Russo. Mathematische Annalen 328(4), pp. 585-624, 2004.
Normal contractive projections preserve type
M. Neal, C. H. Chu, and B. Russo. Journal of Operator Theory 51(2), pp. 281-301, 2004.
Engineering fault-tolerant TCP/IP servers using FT-TCP
D. Zagorodnov, K. Marzullo, L. Alvisi, and T. C. Bressoud. Proceedings of the International Conference on Dependable Systems and Networks, pp. 393-402, 2003.
Opportunistic use of content addressable storage for distributed file systems
N. Tolia, M. Kozuch, M. Satyanarayanan, B. Karp, T. C. Bressoud, and T. Perrig. Proceedings of the USENIX Annual Technical Conference, General Track, pp. 127-140, 2003.
Optimal configuration for BGP route selection
T. C. Bressoud, R. Rastogi, and M. Smith, Proceedings of INFOCOM, 2003.
Improved automatic discovery of subgoals for options in hierarcical reinforcement learning
T. Feil, R. M. Kretchmar, and Rohit Bansal '03. Journal of Computer Science and Technology 3(2), pp. 9-14, 2003.
Improved parallel job scheduling with overhead
J. T. Havill, W. Mao and Vesselin Dimitrov '03. Proceedings of the Seventh Joint Conference on Information Sciences, pp. 393-396, 2003.
Abstract OO big O
J. Krone and W. F. Ogden. Proceedings of the Workshop on Specification and Verification of Component-Based Systems, 2003.
Multiple implementations for component based software using Java interfaces
J. Krone. The Jounal of Computing Sciences in Colleges 19(1), 2003.
Contractive projections and operator spaces
M. Neal and B. Russo. Transactions of the American Mathematical Society 355(6), pp. 2223-2262, 2003.
Operator space characterizations of C*-algebras and ternary rings
M. Neal and B. Russo. Pacific Journal of Mathematics 209(2), 2003.