Sze-Man Ngai

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 and Preprints

• Laplace operators related to self-similar measures on R^d (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.

 
• Topology of connected self-similar tiles in the plane with disconnected interiors (with T.-M. Tang), Topology Appl. 150 (2005), no. 1-3, 139 - 155.

 
• A technique in the topology of connected self-similar tiles (with T.-M. Tang), Fractals 12, no. 4 (2004), 389 - 403.

 
• Graph-directed iterated function systems with overlaps (with Manav. Das), Indiana Univ. Math. J. 59 (2004), no. 1, 109-134.

 
• Fractal Laplacians on the unit interval (with Erik J. Bird and Alexander Teplyaev), Ann. Sci. Math. Quebec 27 (2003), no. 2, 135-168.

 
• Dimensions of the boundaries of self-similar sets (with K.-S. Lau), Experiment. Math. 12 (2003), 13-26.

 
• The Heighway dragon revisited (with Nhu Nguyen), Discrete Comput. Geom. 29 (2003), 603-623.

 
• Hausdorff dimension of self-similar sets with overlaps (with Y. Wang), J. London Math. Soc. (2)

63 (2001), no. 3, 655-672.
 
• Iterated function systems with overlaps and self-similar measures (with K.-S. Lau and H. Rao),

J. London Math. Soc. (2) 63 (2001), no. 1, 99-116.

• Second-order self-similar identities and multifractal decompositions (with K.-S. Lau),

Indiana Univ. Math. J. 49 (2000), 925-972.
 
• Iterated function systems with overlaps (with A.-H. Fan and K.-S. Lau), Asian J. Math. 4 (2000), 527-552.

 
• On 2-reptiles in the plane (with V. Sirvent, J. J. P. Veerman, and  Y. Wang), Geom. Dedicata 82

(2000), 325-344.

• Multifractal measures and a weak separation  condition (with K.-S. Lau), Adv. Math. 141 (1999), 45-96.

 
• L^q-spectrum of Bernoulli convolutions  associated with P.V. numbers (with K.-S. Lau), Osaka J. Math. 36  (1999),  993 -1010.

• L^q-spectrum of the Bernoulli convolution associated with the golden ratio (with K.-S. Lau),

Studia Math. 131 (1998), 225-251.
 
• Multifractal decomposition for a family of overlapping  self-similar measures, Fractal frontiers

(Denver, CO, 1997), 151--161, World Sci. Publishing, River Edge, NJ, 1997.

• A dimension result arising from the L^q-spectrum of a measure, Proc. Amer. Math. Soc. 125 (1997), 2943-2951.

• A generalized finite type condition for iterated function systems (with K.-S. Lau), Adv. Math. (to appear).
  
  
• Singularity and L²-dimension of self-similar measures, preprint.

 
• On the finite type condition (with Q.-R. Deng and K.-S. Lau), preprint.

For reprints or preprints, please send me an e-mail: ngai@gsu.mat.GeorgiaSouthern.edu

Selected Presentations

• Multifractal structure of noncompactly supported measures, Special Session on Fractals and Tilings, 21^st 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 R^d. 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,

Fractal 97 --- The 4-th International Multidisciplinary Working Conference on Fractals, Denver Colorado, USA, April 8 - 11 1997.

• 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 $L^q$-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.

My Mathematical Geneology

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Web Author: Sze-man Ngai