E-mails: smngai@geoergiasouthern.edu
My main research interests are fractal geometry
and the theory of multifractal measures.
I am also interested in their interactions with
the theories of wavelets, self-affine tilings,
and fractal differential equations.
Publications
Spectral asymptotics of Laplacians associated to one-dimensional iterated function systems with overlaps, Canad. J. Math. (to appear).
Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps (with J. Chen), J. Math. Anal. Appl. (to appear).
Multifractal formalism for self-affine measures with overlaps (with Q. Deng), Arch. Math. 92 (2009), 514 - 625.
Separation conditions for conformal iterated function systems (with K.-S.
Lau and X.-Y. Wang), Monatsh.
Math. 156 (2009), no. 4, 325 - 355.
Singularity and L2-dimension
of self-similar measures, Proceedings of the Conference in Honor of
Professor Man-Keung
Siu (to appear).
Multifractal structure of noncompactly supported measures, Fractals, 16 (2008), 209 - 226.
A generalized finite type condition for iterated function systems (with K.-S. Lau), Adv. Math. 208 (2007), 647 - 671.
Laplace operators related to self-similar measures on Rd (with J. Hu and K.-S. Lau), J. Funct. Anal. 239 (2006), 542 - 565.
Vertices
of self-similar tiles (with D.-W. Deng), Illinois J. Math. 49
(2005), no. 3, 857 - 872.
Reptiles with holes (with F. Jordan), Proc. Edinb. Math. Soc. (2) 48 (2005), issue 03, 651 - 671.
Self-similar measures associated with IFS with non-uniform contraction ratios (with Y. Wang), Asian J. Math. 9 (2005), no. 2, 227 - 244.
Selected Presentations
Conformal iterated function systems with overlaps,Workshop on Fractals and Tilings 2009, Strobl, Austria, July 6 - 10, 2009
Fractal Laplacians defined by iterated function systems with overlaps, Workshop on fractal geometry and related topics, Zhuhai Campus, Zhongshan University, China, December 2008.
Fractal Laplacians defined by iterated function systems with overlaps and their spectral asymptotics, Workshop on fractal geometry and the theory of tilings, Zhongshan University, China, July 2008.
Spectral dimension of fractal Laplacians defined by iterated function systems with overlaps, Workshop on Functional Analysis and Optimization Theory, The Chinese University of Hong Kong, December 2006.
Multifractal structure of noncompactly supported measures, Special Session on Fractals and Tilings, 21st Summer Conference on Topology and its Applications, Georgia Southern Univ. Statesboro, GA, July 2006.
On a class of Laplacians defined by fractal measures. Conference on Differential & Difference Equations and Applications. Florida Institute of Technology, Melbourne, FL. August 2005.
Fractal Laplace operators on bounded open subsets of Euclidean spaces. 2nd Conference on Analysis and Probability on Fractals, Cornell University, Ithaca, NY, May -- June 2005.
Fractal Laplace operators on open subsets of Rd. International Conference on Applicable Harmonic Analysis, Hangzhou, China, May 2005.
Iterated function systems of generalized finite type. The Third International Congress of Chinese Mathematicians (ICCM 2004), December, 2004.
A generalized finite type condition for iterated function systems, 2004 AMS Central Sectional Meeting, Northwestern University, Evanston, IL, October 23, 2004.
Topological structure of connected self-similar tiles in the plane, Department of Mathematics, the Chinese University of Hong Kong, June 26, 2003.
Hausdorff dimension of the boundaries of a class of self-similar sets, International Conference on Fractal Geometry & Stochastics III, Friedrichroda, Germany, March 20, 2003.
Topological structure of reptiles and self-affine tiles, AMS Southeasten Sectional Meeting, University of Central Florida, Orlando, FL, November 10, 2002.
Fractal Laplacians on an interval, Conference on Analysis and Probability on Fractals, Cornell University, Ithaca, N.Y., June 16, 2002.
Fractal Fourier Series, Department of Mathematics, The Chinese University of Hong Kong, May 15, 2002.
Eigenvalues and eigenfunctions of a class of fractal Laplacians on the unit interval, Special Session on Dynamic Equations on Time Scales, AMS-MAA Joint Meeting, Atlanta, Georgia, March 10, 2002.
Hausdorff dimension of boundaries of self-similar sets, Department of Mathematics, The Chinese University of Hong Kong, December 22, 2001.
Graph-directed iterated function systems with overlaps, Special Session on Fractals of the American Mathematical Society Meeting, Columbus, OH, September 22, 2001.
Self-similar measures associated to iterated function systems with different contraction ratios, Department of Mathematics, The Chinese University of Hong Kong, June 23, 2001.
Hausdorff dimension of graph self-similar sets with overlaps, Department of Mathematics, The Chinese University of Hong Kong, June 9, 2001.
Self-similar measures and dilation equations, Colloquium, Department of Mathematics, University of Louisville, November 10, 2000.
Fractal measures and dilation equations, Georgia Institute of Technology, March 15, 2000.
Laplacians on fractals I, II, III, lecture series given at the Chinese University of Hong Kong, August 9-11, 1999.
Absolute Continuity of Self-Similar Measures, Young Analysts Meeting of the Southeast, Furman University, Greenville, SC, July 6-10, 1999.
Hausdorff Dimension of Overlapping Self-Similar Sets, Southeastern Analysis Meeting, Vanderbilt Univ., Nashville, TN, May 20-23, 1999.
Dilation equations and iterated function systems with overlaps, Minisymposium on ``Wavelets and Their Applications", SIAM Annual Meeting, Atlanta, GA, May 9-15, 1999.
Hausdorff dimension of overlapping self-similar fractals, Georgia Institute of Technology, January 13, 1999.
Second-order self-similar identities, AMS special session meeting on Fractal Geometry and Related Topics, Louisville, KY, March 20-21, 1998.
Self-similarity in fractals, wavelets and tilings, Cornell University, March 1998.
Multifractal formalism for overlapping self-similar measures, Cornell University, February 1998.
Some Applications of the Renewal Theorem in Fractal Geometry, Conference on Dynamical Systems and Fractal Geometry, ZhongshanUniv., Dec 23-26, 1997.
Using Mathematical Softwares in Education and Research, South China Normal Univ., Dec 22, 1997.
Dilation equations with scaling factor equal to the golden ratio, Workshop on Wavelets and Their Applications, Hong Kong, May5 - 9, 1997.
Multifractal decomposition for a family of overlapping self-similar measures,
Multifractal decomposition for certain iterated function systems with overlaps, Hong Kong Polytechnic University, Dec 1996.
Multifractal decomposition for certain overlapping self-similar measures, Zhongshan Univ., China, and Guangdong Univ. of Technology, China, Nov 1996.
A dimension result arising from the Lq-spectrum of a measure, The South China Normal University, May 1996.
Multifractal formalism and a dimension formula, Workshop on nonlinear dynamics, chaos and complexity, Hong Kong Baptist Univ., March 1996.
Iterated function systems and multifractal decomposition, Hong Kong University of Science and Technology, April 1995.
L^q-dimension of the Bernoulli convolution associated with the P.V. numbers, Conference in Wavelets and Fractals, University of Pittsburgh, May 1994.