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Mieczyslaw K Dabkowski
Associate Professor-Mathematical Sciences Department
Office MailstopMail Box: EC35, Room No.: EN3.914 
Email Address  mdab@utdallas.edu    Primary Phone Number 972.883.4435    URL UTD Webpage    Media Contact
 Professional Preparation
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 DegreeMajorInstitutionYear
 Ph.D.MathematicsGeorge Washington University2003
 M.A.MathematicsGdansk University1998
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Overview
Knot invariants and 3-manifold invariants, applications of topology to biology, recursion theory.

Research Interest

A major focus of my current research is to develop new techniques for studying deformations of tangle replacement moves and skein modules of three-dimensional manifolds. Development of such techniques is not only important for knot theory, but has significance in the problem of determining the full three-dimensional structure of recombination systems. Various mathematical approaches could be used, including techniques used in knot theory, in order to study DNA recombination systems. In particular, Sumners and Ernst, in their work, proposed a theoretical model for studying DNA recombination. Using the methods of 3-D topology techniques (cyclic surgery method), Sumners and Ernst derived first results concerning the three-dimensional structure of such systems. Their pioneering work opened series of research investigations concerning applications of topology, in particular knot theory, to problems concerning DNA. The applications of topology to biology were, for example, discussed recently during U.S.-Mexico Workshop, Knots in Biological Sciences. The workshop was sponsored by CIMAT, UT Dallas, and UT Southwestern Medical Center and held at UTD on April 29, 2005.

The recombination model, proposed by Sumners and Ernst, could be understood in terms of tangles and the problem of determining the full three-dimensional structure of recombination systems could be viewed as an instance of a tangle embedding problem. This problem leads to the necessity of developing new techniques for studying tangle replacement moves on links, which is the major focus of my current research. My main contributions to this area include defining and developing a new family of link invariants - n-Burnside groups of links. Various applications of the new invariants allow us to solve some of the open problems in this area. In particular, in my published work the new invariants were used to solve the following problems: Montesinos-Nakanishi 3-move conjecture (1981), Harikae-Nakanishi (2,2)-move conjecture (1992) and to answer Kawauchi's question concerning Nakanishi's 4-move conjecture (1985). The new techniques and some of their extensions to non-associative invariants could be successfully used to derive criteria for the tangle embedding problem and, in this way, have important applications in study of DNA recombination mechanisms.

Another area of my mathematical research investigations concerns properties of known algebraic invariants of 3-manifolds called fundamental groups. In my research work, I investigated the property of the spaces of orders on 3-manifold groups. This topic of mathematical research has recently gained a considerable attention due to its applications to other areas of 3-dimensional topology. My main contributions to this area include results about existence of left orders on important classes of 3-manifold groups. The results have applications to the important problem of the existence of foliations.

In my current research I study computational properties of the spaces of orders. We showed, in particular, that many classes of 3-manifolds groups admit infinitely many orders that are arbitrarily computationally complex. That is, there are examples of 3-manifold groups that admit an order of arbitrary Turing degree. The spaces of orders on such groups also admit an embedding of the Cantor set, which establishes new connections between topology and recursion theory.

The topic that I find very interesting for my future research is the applications of knots invariants to protein structures similarity problem. Assessing similarity between two protein structures is among one of the most challenging and important problems in computational biology. Since the number of known structures is constantly growing, the need for faster and more accurate methods persists. The problem has been approached using a variety of methods from almost all areas of sciences. In particular, some recent results in this area have been obtained by using knot invariants. This suggests that ideas coming from knot theory have potential for another important range of applications.

Collapse Section Expand Section Publications
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 1  2  
  YearPublication  Type
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Turing Degrees of the Isomorphism Types of Orderable Groups, Malgorzata A. Dabkowska, Mieczyslaw K. Dabkowski, Valentina S. Harizanov, Adam S. Sikora (preprint.)
 
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Spaces of orders and their Turing degree spectra, Malgorzata A. Dabkowska, Mieczyslaw K. Dabkowski, Valentina S. Harizanov, and Amir A. Togha (preprint.)
 
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Khovanov Homology for Signed Graphs, Mieczyslaw K. Dabkowski, Ramanjit K. Sahi (work in progress).
 
2005
Rational moves and tangle embeddings, Mieczyslaw K. Dabkowski, Makiko Ishiwata, Jozef H. Przytycki, preprint.
Other
2005
Non-Left-Orderable 3-Manifold Groups, Mieczyslaw K. Dabkowski, Jozef H. Przytycki and Ataollah Togha, Canadian Mathematical Bulletin, vol. 48X (2005), no. 1, 32-40.
Category: Canadian Mathematical Bulletin
Peer reviewed
Collapse Section Expand Section Presentations and Projects
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  1  2 
Start DateEnd DatePresentation/Project
Nov-02 Nov-02 Rational moves and p-th Burnside groups of links
Rational moves and p-th Burnside groups of links, 2002 Fall Southeastern Section AMS Meeting, Special Session on Invariants of Knots and Low-Dimensional Manifolds, University of Central Florida
May-04 May-04 Burnside groups in knot theory
Burnside groups in knot theory, Knots in Washington XVI, Conference on Knot theory and its Ramifications, The George Washington University
Mar-03 Mar-03 Rational moves on links and Burnside Groups
Rational moves on links and Burnside Groups, Conference on Recent Progress in Homotopy Theory, Special Session on Analogies between Primes and Knots, Japan-U.S. Mathematics Institute, Johns Hopkins University
Jun-02 Jun-02 Burnside group of a link as an obstruction to the Montesinos-Nakanishi 3-move conjecture
Burnside group of a link as an obstruction to the Montesinos-Nakanishi 3-move conjecture, 2002 Spring Western Section AMS Meeting, Special Session on Quantum Topology, Portland State University
Jan-03 Jan-03 Moves on links and Burnside group obstructions
Moves on links and Burnside group obstructions, 2003 Joint Mathematics Meeting, Special Session on Algebraic Topology Based on Knots, Baltimore
Collapse Section Expand Section Appointments
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DurationRankDepartment / SchoolCollege / OfficeUniversity / Company
2003-presentAssistant Professor  The University of Texas at Dallas
1998-2003Graduate Teaching Assistant  The George Washington University
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 Additional Information
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Conferences and Workshops
  • The Fourth Andrzej Jankowski Memorial Lecture, University of Gdansk, Poland, May 2002.
  • Knots in Montreal; Workshop on hyperbolic volume conjecture, UQAM, Montreal, Canada, April 2002.
  • Topology in Biology, University of Florida, Gainesville, March 2002.
  • Frontiers in Mathematical and Computational Biology, University of Texas at Dallas, June 2001.
  • Knots, Links and Manifolds, 4th International Siegen Topology Symposium, University of Siegen, Germany, January 2001.

Educational Presentations
  • Non-left-orderable 3-manifolds groups, Seminar Series, UTD Department of Mathematical Sciences, October 2003.
  • Rational moves and tangle embedding, Seminar Series, UTD Department of Mathematical Sciences, February 2005.

Honors and Awards
  • Nominated for the Natural Sciences and Mathematics Teacher Award The University of Texas at Dallas, April 2005. Nominated by students, stuff and faculty for the effort to fulfill one of the principal missions of the school.
  • 2002 - 03 Distinguished Teaching Assistant Award, The George Washington University, May 2003. Cited for enthusiasm and innovation in the teaching/learning process, an ability to engage and inspire students, high standards of professionalism and collegiality, and a commitment to the educational enterprise.
  • James H. Taylor Graduate Mathematics Prize, Department of Mathematics, The George Washington University, April 2002. Awarded to an outstanding mathematics graduate student.
  • Marvin Green Prize, Department of Mathematics, The George Washington University, May 2001.
  • Supplementary Scholarship, Stefan Batory Foundation, Poland, July 1999.
  • The George Washington University Fellowship, September 1998 - May 2003.
  • Research Scholarship, Ministry of Education, Poland, October 1997.
  • President's Scholarship Award, Gdansk University, October 1996, October 1995.

The University of Texas at Dallas, Instructor: Courses taught
  • Math 6311 Abstract Algebra, Spring 2006
  • Math 3379 Complex Variable, Spring 2006
  • Math 4341 Topology, Fall 2005
  • Math 6306 Topology and Geometry, Fall 2005
  • Math 6390 Topics in Mathematics, Spring 2005
  • Math 3379 Complex Variables, Spring 2005
  • Math 4341 Topology, Fall 2004
  • Math 6306 Topology and Geometry, Fall 2004
  • Math 6343 Computational Biology, Spring 2004
  • Math 5390 Topics in Mathematics, Fall 2003
  • Math 4341 Topology, Fall 2003

The University of Texas at Dallas, Senior Honors Theses Supervisor:
  • Arin R. Bratt (current, Spring 2006)
  • Nicholas A. Reynolds, Fall 2005
  • Travis Thompson, Spring 2005
  • Bradley R. Duesler, Spring 2005
  • John K. Gallagher, Spring 2004
  • Elizabeth A. Gan, Spring 2004

The University of Texas at Dallas, Ph. D. Thesis Supervisor
  • Ramanjit Sahi, Fall 2004- present

The George Washington University, Teaching Assistant: Courses taught:
  • Calculus with Precalculus I, Fall 2002, Spring 2002, Fall 2000, Spring 2000.
  • Calculus with Precalculus II, Spring 2001.
  • Single-Variable Calculus I, Fall 2001.
  • Finite Mathematics for Social and Management Sciences, Fall 2001, Spring 1999.
  • Calculus for Social and Management Sciences, Fall 1999

The George Washington University, Instructor: Courses taught
  • Finite Mathematics for Social and Management Sciences, Summer 2002.
  • Calculus for Social and Management Sciences, Summer 2001.

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