Math Research Areas | Publications | Presentations

Mathematical Interest and Research Areas

• Automorphism groups of Riemann surfaces and
• tilings of the Riemann surfaces & hyperbolic plane.
  Two websites of interest are:
  • Automorphisms of Rieman Surfaces/Algebraic Curves Interest Group
  • Tilings of Surfaces
• Finite group theory
• Geometry in dimensions two and three
• The mathematics of signal and image processing
  Here is the second edition cowritten with my Rose colleague Kurt Bryan.
  Discrete Fourier Analysis and Wavelets – Applications to Signal and Image Processing
• Lie groups and Lie algebras
• Singularities of polynomials of several complex variables
• Using computer algebra systems in teaching and research in mathematics

Publications and Technical Reports (reverse chronological order)

Partial, but fairly complete, lists of my published papers, technical reports, and online presentations are given here. All links open up an entry from my BEpress Selected Works site in a new tab. The Selected Works site contains full publication data and links to my online publications and presentations.

Books and book chapters

1. Discrete Fourier Analysis and Wavelets: Applications to Signal and Image Processing, 2nd Edition, with Kurt M Bryan, Wiley (2018).
2. The Science of Nanotechnology: An Introductory Text, with Luanne Tilstra, Robin Tanke, Daniel Jelski, Guo Ping Zhang, Valentina French,
   Alexander Popov, Thomas George, and Arthur Western, Nova Science Publishers, (2007).

Published Papers

1. Equivalence of finite group actions on Riemann surfaces and algebraic curves, Contemporary Mathematics, accepted (2021).
2. Future directions in automorphisms of surfaces, graphs, and other related topics, with Jennifer Paulhus and Aaron Wootton, Contemporary Mathemtatics, accepted (2021).
3. One dimensional equisymmetric strata in moduli space, with Antonio F. Costa and Milagros Izquierdo, Contemporary Mathematics, accepted (2021).
4. The engaging symmetry of Riemann surfaces: a historical perspective,  with Gareth A. Jones and David Singerman,  Contemporary Mathematics, accepted (2021).
5. On Automorphisms of Algebraic Curves,  with Aaron Wootton and Tony Shaska, in Algebraic Curves and Their Applications, Contemporary Mathematics #724 (2019).
6. Using Strong Branching to Find Automorphisms of n-gonal Surfaces, with C. Camacho, J. Paulhus, R. Winarski and A. Wootton, Albanian Journal of Mathematics (2018).
7. Galois action on regular dessins d’enfant with simple group action, in Higher Genus Curves in Mathematical Physics and Arithmetic Geometry, Contemporary Mathematics #703 (2018).
8. Quasi-platonic PSL(2,q)-actions on Closed Riemann Surfaces, Albanian Journal of Mathematics (2015) .
9. Exceptional automorphisms of (generalized) super elliptic surfaces, with Aaron Wootton, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, Contemporary Mathematics #629 (2014).
10. Superelliptic surfaces as p-gonal surfaces, in Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces, Contemporary Mathematics #629 (2014).
11. Ellipses in translation surfaces, with Chris Judge, Geometriae Dedicata (2012).
12. Cyclic n-gonal Surfaces and their Automorphism Groups – UNED Geometry Seminar, with Aaron Wootton, Disertaciones del Seminario de Matematicas Fundamentales (2010)
13. Topologically Unique Maximal Elementary Abelian Group Actions on Compact Oriented Surfaces,  with Aaron Wootton,  Journal of Pure and Applied Algebra (2009).
14. Finite Abelian Subgroups of the Mapping Class Group, with Aaron Wootton, Algebraic & Geometric Topology 7 (2007).
15. Anharmonic Vibrational Motions in C₆₀ : A Potential Energy Surface Derived from Vibrational Self Consistent Field Calculations, with Daniel Jelski and Laszlo Nemes, Journal of Cluster Science, Vol 16, No 1, March (2005).
16. Divisible Tilings in the Hyperbolic Plane, with Dawn M. Haney, Lori T. McKeough,  and Brandy M. Smith, New York Journal of Mathematics 6 (2000).
17. Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane, American Math. Monthly, October (2000).
18. Symmetries of Accola – Machlaclan and Kulkarni Surfaces, with E. Bujalance, A.F. Costa, J.M. Gamboa, G. Gromadski, Proc. AMS, 127 (3) (1999).
19. Symmetries of Riemann surfaces on which PSL(2,q) acts as a Hurwitz Automorphism Group, with E. Bujalance, A.F. Costa, J.M. Gamboa, G. Gromadski, J. of Pure and Appl. Algebra, 106 (1996).
20. Simple Group Actions on Hyperbolic Surfaces of Least Area, Pacific J. of Math. 158 (1) (1993).
21. Normalizers and centralizers of elementary Abelian subgroups of the mapping class group, Topology ’90, Walter de Gruyter, New York (1992).
22. The Gottlieb group of finite linear quotients of odd-dimensional spheres, Proceedings of the AMS 111 (4) (1991).
23. Classifying finite group actions on surfaces of low genus, J. of Pure & Appl. Algebra 69 (1990).
24. The equisymmetric stratification of the moduli space and the Krull dimension of the mapping class group, Topology and its Applications 37 (1990).
25. Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math 92 (1988).
26. Volumes of subgroups of compact Lie groups, Algebras, Groups and Geometries 4 (1987).
27. The homology and higher representations of the automorphism group of a Riemann surface, Transactions AMS 300 (1) (1987).
28. The height of two-dimensional cohomology classes of complex flag manifolds, with M. Hoffman and W. Homer, Canadian Bull. Math. 26 (4) (1983).
29. On the topology of polynomial hypersurfaces, Proc. Symposia Pure Math., 40, Amer. Math. Soc. (1983).

MSTR Technical Reports

1. Topological and H^q Equivalence of Cyclic n-gonal Actions on Riemann Surfaces – Part II, Mathematical Sciences Technical Reports (MSTR) (2020).
2. Branching Matrices for the Automorphism Group Lattice of a Riemann Surface,  Mathematical Sciences Technical Reports (MSTR) (2018).
3. Topological and H^q Equivalence of Prime Cyclic p-gonal Actions on Riemann Surfaces (Corrected), Mathematical Sciences Technical Reports (MSTR) (2016).
4. Continuous Dependence of Solutions of Equations on Parameters, Mathematical Sciences Technical Reports (MSTR) (2014).
5. Analytical Solution of the Symmetric Circulant Tridiagonal Linear System, with Jeffery J. Leader, Mathematical Sciences Technical Reports (MSTR) (2014).
6. Calculation of the Killing Form of a Simple Lie Group, Mathematical Sciences Technical Reports (MSTR) (2014)
7. Flattening a Cone, Mathematical Sciences Technical Reports (MSTR) (2009).
8. The Barycenter of the Numerical Range of a Matrix, with Roger G. Lautzenheiser and Thomas Werne, Mathematical Sciences Technical Reports (MSTR) (2007).
9. The Birational Isomorphism Types of Smooth Real Elliptic Curves, Mathematical Sciences Technical Reports (MSTR) (2004).
10. Triangular Surface Tiling Groups for Low Genus, with Robert M. Dirks, Maria Sloughter, and C. Ryan Vinroot, Mathematical Sciences Technical Reports (MSTR) (2001) .
11. Splitting Tiled Surfaces with Abelian Conformal Tiling Group, Mathematical Sciences Technical Reports (MSTR) (1999).
12. Counting Ovals on a Symmetric Riemann Surface, Mathematical Sciences Technical Reports (MSTR) (1997).

Presentations

1. Signals, Bases of Waveforms, and Frames  Holy Cross Math/CS Faculty Seminar (2019).
2. Topological and H^q Equivalence of Prime Cyclic p-gonal Actions on Riemann Surface, AMS Spring Western Sectional Meeting – Automorphisms of Riemann Surfaces and Related Topics (2018).
3. Defining Equations for Riemann Surfaces, with Aaron Wootton, BIRS workshop on Symmetries of Surfaces, Maps and Dessins (2017)
4. Sabbatical Travelogue: Sweden, Spain, and China, Rose-Hulman Global Studies Faculty Seminar (2017).
5. Subgroups of the Mapping Class Group Corresponding to 1-Dimensional Strata in the Branch Locus of Moduli Space, with A. Costa and M. Izquierdo, AMS Annual Meeting (2017)
6. Moduli space – Geometry/Math with Letters, Rose Math Seminar (2016)
7. Scientitic Visualization and Trigonometry: Do They Mix? New College
   of Florida Data Science Seminar (2016)
8. Riemann Surfaces: A playground for analysis, topology, geometry, group theory, and Galois theory, MAI mathematics colloquium, LIU, Linkoping Sweden (2016).
9. Quasi-platonic actions of some simple groups on Riemann surfaces and their dessins d’enfant, Joint Mathematics Meetings, Seattle, WA – January 2016
10. Symmetric surfaces with quasi-platonic PSL(2,q) action, AMS Regional Conference, Loyola, Chicago IL – October 2015.
11. Quasi-platonic actions of PSL2(q) and their Dessins, AMS Regional Conference, Michigan State University, East Lansing MI – March 2015.
12. Framing the Image, Rose Math Seminar, February 2015.
13. Exceptional Automorphisms of (Generalized) Super-elliptic Curves, Riemann and Klein Surfaces, Symmetries and Moduli Spaces, Linkoping Sweden, June 2013.
14. Pairs of Pants and the Congruence Laws of Geometry, Rose Math Seminar, Spring 2013
15. Who painted this Painting? Rose Math Seminar, Spring 2012.
16. Galois actions on regular dessins and Fuchsian group covers, Conference on Riemann Surfaces and Dessins d’Enfants On the Occasion of Jürgen Wolfart’s 65’th Birthday – May 2010
17. Flat Surfaces, Teichmueller Discs, Veech Groups, and the Veech Tessellation, AMS Regional Conference, Pennsylvania State University, University Park, PA – October 2009.
18. Roll-ups and Differential Geometry, INMAA Section Meeting – Calumet, Fall 2009.
19. Roll-ups and Differential Geometry, Rose Math Seminar, Fall 2009.
20. Cyclic n-gonal surfaces – computational methods, joint with Aaron Wootton, 25th Nordic and 1st British-Nordic Congress of Mathematicians, June 2009.
21. Classification of Pairs of Fuchsian Groups, Second of two talks in the UNED Geometry Seminar, March 2009
22. Full Automorphism Groups of Cyclic n-gonal Surfaces, joint with Aaron Wootton, First of two talks in the UNED Geometry Seminar, February 2009
23. Voronoi Tesselations, Delaunay Tesselations and Flat Surfaces – Sabbatical Report Part II, Rose Math Seminar, Fall 2008.
24. Billiards and Flat Surfaces – Sabbatical Report Part I, Rose Math Seminar, Fall 2008.
25. The Barycenter of the Numerical Range of an Operator, ISU Math and CS Research Seminar, November 28, 2007.
26. Elementary Abelian Group Actions on Surfaces and the Geometry of Moduli Space, IU Geometry Seminar, November 8, 2007.
27. Classifying Pairs of Fuchsian Groups of Finite Type, AMS Regional Conference, Tucson, April 2007.
28. Geometry from Chemistry – Bucky Balls and Nanotubes INMAA Spring Meeting, 2007
29. Geometry from Chemistry II – The Geometry of Nanotubes, Rose Math Seminar, Fall 2006.
30. Geometry from Chemistry I – The Dynamics of Bucky Balls, Rose Math Seminar, Fall 2006.
31. Higher Genus Soccer Balls, ISU Math and CS Seminar, Fall 2006,
32. Enumeration of the Equisymmetric Strata of the Moduli Space of Surfaces of Low Genus, AMS Regional Conference, Santa Barbara, April 2005.
33. Fostering Undergraduate Research in Mathematics, Best Assessment Practices VII – Showcase, Rose-Hulman, April 2005, contributed presentation.
34. Equivalence of Real Elliptic Curves – Part II – Birational Equivalence, Rose Math Seminar, October 2004.
35. Equivalence of Real Elliptic Curves – Part I – Linear Equivalence, Rose Math Seminar, October 2004.
36. Are the Students Competent Users of Mathematics?  AMS-MAA Joint  Meetings, Phoenix, January 7, 2004.
37. Kaleidoscopic Tilings on Surfaces, Group Algebras and Separability, Rose Math Seminar, Spring 2003.
38. Kaleidoscopic Tilings on Surfaces, This Time with the Groups, Rose Math Seminar, Spring 2003.
39. Vanishing Cycles and Kaliedoscopic Quadrilateral Tilings, Rose Math Seminar, Winter 2002.
40. Signals, Images, …, What’s Next in Scientific Vizualzation, Sigma Xi, Mount Holyoke College, Spring 2001.
41. Higher Genus Soccer Balls, Mount Holyoke Math Club Spring 2001.
42. Transform Methods in Image Processing, Mathematics Faculty Seminar, Mount Holyoke College, Spring 2001.
43. Divisible Tilings of the Hyperbolic Plane, Rose Math Seminar, April 2000.
44. Higher genus Soccer Balls and Kaleidoscopic Tilings in the Hyperbolic Plane, Rose Math Seminar, April 2000.
45. The Unreasonable Effectiveness of Mathematics,  Rose-Hulman Symposium August 25, 2000.
46. Finite Groups, Tilings and Hyperbolic Geometry at the Rose-Hulman REU   Kanazawa Institute of Technology, July 29, 1999.
47. Mathematical Methods of Image Processing, A Progress Report on Course Development, AMS Meeting, San Antonio, January 16, 1999.
48. Wavelet Based Methods In Image Processing  Applied Math Seminar, Rose-Hulman, Winter Quarter 1998-99.
49. The Rose-Hulman Approach to Undergraduate Research – What Works for Us,  AMS Meeting, Baltimore, January 8, 1998.
50. Finite Groups, Tilings and Hyperbolic Geometry at the Rose-Hulman REU,  INMAA Meeting, Wabash College, October 18 1997.
51. The Role of Technology in Enhancing Learning in Various Disciplines,  IHETS Conference, Purdue University, October 15,1997.