| Name | Research Interests |
| Prof Alfred M Bruckstein | Variational Methods in Image Analysis and Synthesis,
Multi A(ge)nt Robotics and Applied Geometry |
| Assoc Prof Andrew James Kricker | Prof. Kricker's most significant research interest lies in the mathematical ramifications of current developments in mathematical and theoretical physics. To be precise, he is interested in the ramifications of certain developments in quantum field theory and quantum gravity in the fields of topology, algebra, and combinatorics. Prof. Kricker's particular speciality is in so-called "quantum topological invariants". These are invariants of knots, 3-manifolds, and various other low-dimensional topological structures, that arise from Topological Quantum Field Theories. More generally, he has a considerable general interest in the fields that surround this topic: knot theory, the theory of low-dimensional manifolds, Lie algebras, Hopf algebras, representation theory, homological algebra, algebraic combinatorics, and so on. |
| Assoc Prof Ang Whye Teong | W. T. Ang's general research interest is in applied and engineering mathematics. Specific research topics that he has worked on include boundary integral equation methods, stress analysis around cracks, heat transfer with applications to modern engineering and biological systems, analyses of advanced materials (such as functionally graded materials and piezoelectric materials) and non-classical boundary value problems in physical and engineering sciences. |
| Assoc Prof Arindam Basu | Low-power Reconfigurable Mixed-signal design, Neural recording systems, Computational neuroscience, Nonlinear dynamics, Smart sensors for hearing-aids/ultrasound etc, Neuromorphic VLSI |
| Asst Prof Bei Xiaohui | Computational economics, social networks analysis and general algorithm design. |
| Assoc Prof Bernhard Schmidt | Finite Geometry
Coding Theory
Algebraic Number Theory
Computing |
| Assoc Prof Chan Song Heng | Dr Chan's areas of expertise are Number Theory, Combinatorics, and Special functions. His current research works focus on sums of squares formula, the partition function and various related functions, and q-series which includes modular forms, theta functions, and basic hypergeometric series. |
| Prof Chee Yeow Meng | 1) Combinatorics of Nanotechnology: low-power design, thermal-aware design, crosstalk issues, testing of deep submicron & nanometer-scale circuits, oligonucleotide sequence design for DNA computing, quantum error-correcting codes.
2) Designs, Codes & Cryptography: triple systems, block designs, pairwise balanced designs, group divisible designs, t-designs, Latin squares, error-correcting codes, erasure-resilient codes, codes for nonconventional channels, combinatorial cryptography, algorithms & computational methods, applications in computer science, engineering & biology.
3) Extremal Set Systems: Turan-type problems, packings & coverings, cover-free systems, nonadaptive group testing, applications in computer science, engineering & biology. |
| Assoc Prof Chen Ning | Algorithmic Game Theory and Computational Economics
Algorithmic and Economic aspects of the Internet
Algorithms and Combinatorial Optimization |
| Asst Prof Cheung Sai Hung | -Catastrophe risk modeling, analysis, mitigation and management due to natural disasters and man-made hazards
-Reliability, Risk engineering and science
-Stochastic dynamics
-Complexity science
-Earthquake engineering, Performance-based engineering
-Sustainable urban planning and development
-Climate Change Impact Studies
-Optimal decision making, design and control under uncertainty
-Uncertainty quantification, System identification
-Structural health monitoring |
| Assoc Prof Chew Lock Yue | Dr Chew research interest is to investigate and uncover the fundamental physical mechanisms and organization principles of complex systems. He explores and elucidates the underlying behaviour and dynamics of these systems through theoretical modelling based on statistical and nonlinear physics. His current research focus is on the topic of: quantum complexity; quantum chaos and entanglement; statistical physics of protein aggregation; critical phenomena in social and coupled social-ecological systems; complexity measure and characterization of complex systems. |
| Asst Prof (Adj) Chong Kai Fong Ernest | Dr. Chong's current research deals with combinatorics, commutative algebra, and machine learning. He is primarily interested in algebraic, geometric, topological, and enumerative combinatorics; combinatorial and computational commutative algebra; and the theoretical aspects of deep learning. A common theme in his research is the understanding of numerical or algebraic data (usually from topological objects) via some underlying combinatorial structure, and he finds it useful to draw ideas from other fields such as discrete geometry, algebraic geometry and algebraic topology. |
| Assoc Prof Chua Chek Beng | Prof. Chua studies the theory of continuous optimization, and develops efficient solution methods for several types of optimization models. He has designed and analyzed interior-point algorithms for semidefinite optimization, symmetric cone optimization and homogeneous cone optimization. He studies the possibility of applying homogeneous cone optimization on various problems where semidefinite optimization models are used. This study is partly driven by the possible reduction in the size when semidefinite optimization models are solved as homogeneous cone optimization problems, hence allowing large-scale problems to be solved via homogeneous cone optimization. He also investigated and proved several properties of the primal-dual central paths for semidefinite optimization and homogeneous cone optimization. These properties are useful in the study of local convergence behaviour of path-following algorithms. |
| Prof Deng Yuefan | Parallel Computing
Monte Carlo Methods
Computational Physics
Computational Applied Mathematics
More information about Prof. Deng can be found at https://en.wikipedia.org/wiki/Yuefan_Deng |
| Dr Erik Anders Mikael Gustavsson | Mathematics: Potential Theory and its application in rational and harmonic approximation.
Mathematics Education: Is Team Based Learning effective in teaching mathematical reasoning. |
| Assoc Prof Fan Hui | Prof. Fan has been conducting research in the area of solid mechanics for over 20 years. His publications touched topics: fracture mechanics, mechanics of composites, micromechanics of defects in the solids, and multi-physics. |
| Dr Fedor Duzhin | Dr Fedor Duzhin's research interest is pure mathematics, mostly topology. Recently he has been working in areas like mathematical billiards and Artin's braid group. |
| Assoc Prof Frederique Elise Oggier | Dr Oggier's research interests are in applications of algebra and number theory to coding and security. |
| Prof Fung Tat Ching | Prof Fung's areas of expertise are Computational Mechanics and Structural Engineering. His current research works focus on structural behaviour of tubular joints and the design of higher order time-step integration algorithms. |
| Dr Gary Royden Watson Greaves | Algebraic combinatorics and number theory |
| Asst Prof Hartmut Klauck | Theoretical Computer Science, Computational Complexity, Quantum Computing |
| Assoc Prof Ho Moon-Ho Ringo | Assoc Prof Ho Moon-ho's research interests are concerned with the development and application of quantitative methods, in particular, multilevel modeling, resampling methods, structural equation modeling, and time-series analysis in the neural and behavioral sciences. His current research work focuses on neuroinformatics research, in particular, the theoretical development and applications of multivariate time series analysis method for extracting meaningful information from complex brain imaging data. |
| Assoc Prof Huang Weihong | Dr. Huang has wide research interests ranging from microeconomics, industrial organization, financial economics, public economics to nonlinear economic dynamics. Recently, Dr Huang has devoted much time and effort to reexamine the economic behaviors from the perspective of ancient Chinese philosophy. In recent years, he has devoted his most effort in incorporating ancient philosophical wisdom to the analysis of the economic behaviors. |
| Assoc Prof Hui Siu Cheung | Assoc Prof Hui Siu Cheung's areas of expertise are data mining, Web mining, information retrieval, natural language processing, mathematical data mining, intelligent tutoring systems and Semantic Web. His current research works focus on information retrieval, text mining, natural query processing and Semantic Web. |
| Asst Prof Jitamitra Desai | Dr. Desai's research interests are primarily in the area of mathematical optimization and nonconvex analysis, with a focus on analyzing and designing algorithms for mixed-integer, nonlinear, nonconvex programs. He is also pursuing research ideas in stochastic decision analysis, network and graph problems, and computational optimization. Application areas include transportation systems, wireless communications, and emergency response management. |
| Prof John Allen Paulos | Originally mathematical logic, specifically model theory, then philosophy and probability, and then writing - books, articles, essays
for a variety of publications. Several books became bestsellers. They're described here: http://www.math.temple.edu/~paulos/books.html |
| Assoc Prof Justin Dauwels | His research interests are in Bayesian statistics, iterative signal processing, and computational neuroscience.
Some of the projects include:
- Mathematical modeling of the start and ending of epileptic seizures
- Diagnosis of Alzheimer's disease from EEG signals
- Machine learning techniques for guiding neurosurgery
- Detection of mental states from EEG signals
- Tracking and predicting traffic in dynamic urban networks
- Data-driven dynamical models of human behavior
- Tracking and control of synthetic cell tissue
- Copula-based modeling of extreme events
- Copula-based graphical models |
| Assoc Prof (Adj) Khoo Khoong Ming | My interest is on applications of Mathematics to Cryptography. My current focus is on Symmetric Cryptography, specifically: Cryptographic Boolean Functions, Stream Ciphers and Block Ciphers. |
| Dr Kiah Han Mao | Coding Theory
- Design and analysis of encoding and decoding schemes.
- Bounds on the size and constructions of optimal codes.
Combinatorics
- Construction of combinatorial designs with application to codes.
- Methods in enumerative and extremal combinatorics. |
| Dr Ku Cheng Yeaw | Dr Ku's research interests lie in combinatorics and graph theory. He is interested in matching polynomials, extremal problems for permutations and related combinatorial structures, probabilistic and algebraic methods used to understand these structures. |
| Assoc Prof Kwoh Chee Keong | Development of a Computer Prediction System For Rational Design Of HLA-Based Peptide Vaccine;
Data Mining and Analysis on Infectious Disease
Heterogeneous Multi-Core Systems For Bioinformatics
Constrained Optimzation for Bioinformatics
Protein Interaction Network Analysis Using Graph Mining Approaches |
| Dr Le Hai Khoi | Dr. Le Hai Khoi's areas of expertise are Complex Analysis in Several Variables, and Mathematical Foundations of Information Technology. His current research works focus on a theory of representing systems and Dirichlet series in several complex variables as well as applications to functional equations. He also works on some problems related to harmonic functions. |
| Dr Leon Chuen Hwa | Dr. Leon current research interests are in the areas of asset pricing, financial modeling, investment, portfolio and risk management. |
| Assoc Prof Lin Zhiping | Dr Zhiping Lin's research interests include multidimensional systems and signal processing, array and radar signal processing, and biomedical signal processing. |
| Prof Ling San | 1. Applications of algebra and number theory to combinatorial designs, coding theory, cryptography and sequences
2. Arithmetic of modular curves and Galois representations |
| Assoc Prof Low Chor Ping | Combinatorial Optimization, Design & Analysis of Algorithms, Network Performance Modeling & Analysis, Algorithmic Graph Theory. |
| Assoc Prof Luo Jun | - Wireless Networking
- Network Modeling and Performance Evaluation
- Applied Operations Research
- Network Security |
| Asst Prof Ng Keng Meng | Dr. Ng's main research interest is in computability theory, a branch of mathematical logic. He is interested in the theory of computation, particularly in descriptive and algorithmic complexity. He works in classical and applied computability theory, and in algorithmic information theory and randomness. He is also interested in reverse mathematics and the application of computability theory to combinatorics and analysis. |
| Assoc Prof Nicolas Privault | Stochastic analysis, probability, mathematical finance |
| Assoc Prof Pan Guangming | Assistant Professor Pan Guangming's areas of expertise are random matrices theory, limiting theorem and multivariate analysis. |
| Asst Prof Paterek Tomasz | I am interested in foundations and limits of quantum mechanics and in studies on the border between quantum physics and other fields of science such as information theory, computing, mathematics, biology or gravity. |
| Prof Peter M. A. Sloot | I try to understand how nature processes information. I study this 'natural information processing' in complex systems by computational modeling and simulation as well as through formal methods. My work is applied to a large variety of disciplines with a focus on -but not limited to- Biomedicine. Recent work is on modeling the virology and epidemiology of infectious diseases, notably HIV, through Complex Networks, Cellular Automata and Multi-Agents. Recently in my work I try to build bridges to socio-dynamics. Currently I lead two large EU projects: ViroLab and DynaNets and supervise research from various NIH, NSF and NWO and Royal Academy projects.
See: http://www.peter-sloot.com/ |
| Prof (Adj) Phua Kok Khoo | High Energy Physics |
| Assoc Prof Pina Marziliano | - Digital signal and image processing
- Sampling theory and applications
- Biomedical signal and image processing
- Watermarking and information security
- Perceptual quality metrics for multimedia |
| Prof (Adj) Poh Eng Kee | Assoc Prof (Adj.) Dr Poh Eng Kee specializes in guidance, navigation, control and signal processing. His areas of research focus on advanced navigation and control design for unmanned aerial vehicles and autonomous underwater vehicle, satellite formation control and communication, GPS receiver design and algorithms. |
| Asst Prof Pun Chi Seng | Financial Mathematics
Robust Stochastic Control
High-Dimensional Statistics
Derivatives Pricing |
| Prof Rajapakse Jagath Chandana | Professor Rajapakse's areas of expertise are machine learning, brain imaging, and computational and systems biology.
Professor Rajapakse has pioneered several techniques for analysis of anatomical and functional MR images. His team was the first to develop techniques to model brain connectivity in an exploratory manner, using functional MR images. Presently, his team is investigating brain connectivity patterns underlying higher-order brain functions such as language and memory, and brain disease such as Parkinson's disease. He is also working on potential applications of brain connectivity and constrained independent component analysis (cICA) in Brain Computater Interface applications, especially in identifying different mental states and extracting features robust to inter- and intra-subject variations.
Professor Rajapakse is presently working on identifying key targets in biological pathways. His research is centered on identifying co-regulated genes, building gene regulary networks, fusion of protein-interactions, and identifying key molecules and core networks in pathways. His team also develops techniques to segment cells and nuclei, identify protein subcellular localizations, and model spatiotemporal changes of cell morphologies from cellular images obtained from electron microscopy and high content screening. |
| Prof S Viswanathan | Inventory Management, Joint Replenishment Inventory Systems,, Sustainable Operations, Policies for regulating carbon emissions, Reverse Logistics & Remanufacturing, Transportation Scheduling & Timetabling, and Supply Chain & Service Operations.
CV and full list of publications can be found at this link: http://www3.ntu.edu.sg/home/asviswa/Vish_CV_Full_Web.pdf |
| Assoc Prof Shu Jian Jun | Associate Professor Jian-Jun SHU's areas of expertise are Applied Mathematics, Thermo-Fluid Mechanics and Biophysics. His current research works focus on Biomolecular Mediated Computer, Mathematical Strategame Theory and Nano/Micro Fluids.
My published works have inspired articles, interviews, editorials worldwide. See some selected links below:
The next step in DNA computing: GPS mapping? http://www.acs.org/content/acs/en/pressroom/presspacs/2015/acs-presspac-may-6-2015/the-next-step-in-dna-computing-gps-mapping.html
The next computer: Your genes http://www.physorg.com/news/2011-05-genes.html
Efficient DNA-based computing could replace silicon http://www.popsci.com/science/article/2011-05/faster-more-efficient-dna-based-computing-could-replace-silicon
Is DNA computing going to terminate Internet banking? http://arstechnica.com/science/news/2011/05/is-dna-computing-going-to-terminate-internet-banking.ars
Computing with DNA http://kemo-d7.livejournal.com/1190416.html
DNA计算机： 计算的未来 http://sztqb.sznews.com/html/2011-05/30/content_1593552.htm (in Chinese-中文)
DNA是计算的未来 http://www.mittrchinese.com/single.php?p=61864 (in Chinese-中文)
科学家拟造DNA环保电脑 http://newspaper.jfdaily.com/xwcb/html/2011-08/31/content_646298.htm (in Chinese-中文)
Computadoras del futuro estarán basadas en ADN http://www.tecnopc.org/noticias/computadoras-del-futuro-estaran-basadas-en-adn/ (in Spanish-Español)
Futuros computadores pueden ser a base de ADN http://buscandoladolaverdad.blogspot.com/2011/05/futuros-computadores-pueden-ser-base-en.html (in Spanish-Español)
La siguiente computadora - Tus genes http://www.neotroid.com/index.php/tecnologia/la-siguiente-computadora-tus-genes.html (in Spanish-Español)
Computadoras con base en ADN http://www.zonafranca.mx/computadoras-con-base-en-adn/ (in Galician-Galego)
Calculatoarele viitorului ar putea fi pe bază de ADN http://totb.ro/?p=12767 (in Romanian-România)
ДНК — Будущее вычислительной области http://compblog.ilc.edu.ru/blog/science/2108.html (in Russian-Русский язык)
ДНК-Компьютер идет на смену своему кремниевому собрату? http://globalscience.ru/article/read/19396/ (in Russian-Русский язык)
Der nächste computer- Unsere gene! http://www.denkmaschinen.ch/2011/05/17/der-nachste-computer-unsere-gene/ (in German-Deutsch) |
| Assoc Prof Tan Eng Leong | Computational electromagnetics/optics/acoustics - FDTD, ADI/LOD/SS, fundamental implicit schemes; RF/Microwave IC; SAW/BAW; Photonic crystals, phononic crystals |
| Dr Tan Geok Choo | Past Interests: Category Theory (Reflective Subcategory), P-localization of Groups; Henstock Integration.
Present Interests: Applying mathematics to other disciplines. |
| Dr Teo Chee Chong | Dr Teo's area of expertise are in the application of operations research to solve problems in supply chain, production and service operations.
Current research projects include the following:
1. REMANUFACTURING: Value of Production Smoothing in Recover-and-Assemble Remanufacturing Systems (with Seyed Mehdi Zahraei)
2. E-COMMERCE LOGISTICS: Overcoming the Challenges of Last-mile E-commerce Urban Logistics (with Yiik Diew Wong)
3. SUPPLY CHAIN VARIABILITY: Minimizing Propagation of Demand Variability in A Supply Chain by Coordinating Order and Production Smoothing (with Zhibin Zhang and Tat Ching Fung)
4. CONTAINER SHIPPING: Optimizing Hedging and Service Design for Container Shipping in Countering Bunker Fuel’s Price Volatility (with Xiaoyu Wang and Zhibin Zhang) |
| Asst Prof Tong Ping | Numerical methods, inverse problems, seismic imaging (migration+tomography), time-series analysis |
| Assoc Prof Troy Lee | Troy Lee joined the Division of Mathematical Sciences (School of Physical and Mathematical Sciences) in September 2013. He received his Bachelors degree with honors in mathematics from the California Institute of Technology in 2000. From there, he went to the University of Amsterdam, for both Masters (2001) and PhD (2006) degrees, writing a thesis on Kolmogorov Complexity and Formula Size Lower Bounds. He won a Rubicon Fellowship from the Netherlands Science Foundation to do a Postdoc at University of Orsay, Paris. Then he spent two years as a postdoc at Rutgers University with a National Science Foundation Research Fellowhip, and one year at Columbia University. In 2010 he joined the Centre for Quantum Technologies as a Senior Research Fellow and has a joint appointment with NTU and CQT. In 2013 he was awarded a NRF fellowship. |
| Assoc Prof Viet Ha Hoang | Assistant Professor Hoang Viet Ha ares of expertise is multiscale problems, multiscale computations, homogenization, random partial differential equations. Currently he is working on some computational problems for multiscale problems using sparse finite elements and wavelet. |
| Prof (Adj) Vladimir Kuznetsov | bioinformatics, Genomics, Systems Biolgy, Applied Mathematics&Statistics, Cancer Biology, Immunology |
| Assoc Prof Wang Huaxiong | Cryptography Information Security, Coding Theory, Combinatorics, Theoretical Computer Science |
| Assoc Prof Wang Li-Lian | Dr. Wang's research focuses are centered around the design, analysis and implementations of efficient computational methods for problems in fluid dynamics, electromagnetics and finances. His recent research interest is also with variational and PDE-based image processing. |
| Assoc Prof Wong Jia Yiing, Patricia | Dr Wong has vast research experience in differential equations, difference equations, integral equations, dynamic equations on time scales, as well as numerical mathematics and inequalities. In the earlier years, Dr Wong worked in the fields of numerical mathematics and inequalities, particularly in splines and error inequalities of polynomial interpolation. The work done is documented in the monograph [1]. Later she has been working on differential and difference equations, focusing on the oscillatory behaviour and also the existence of solutions under various boundary conditions. Recently, in addition to differential and difference equations, Dr Wong also works on integral equations, as well as dynamic equations on time scales, which include differential and difference equations as special cases. The results of the recent work, particularly on the existence of solutions of boundary value problems, have applications to a wide spectrum of physical problems. The work is ongoing, those completed till 2013 is documented in the monographs [2]--[4]. Dr Wong has published 4 monographs and more than 180 research papers in international refereed journals. Presently, Dr Wong serves on the editorial board of Advances in Difference Equations (USA), Boundary Value Problems (USA), Abstract and Applied Analysis (USA), Advances in Dynamical Systems and Applications (USA), and International Journal of Differential Equations (USA).
1. Agarwal, R.P. & Wong, P.J.Y. Error Inequalities in Polynomial Interpolation and their Applications. Kluwer Academic Publishers, Dordrecht (1993), 365 pp.
2. Agarwal, R.P. & Wong, P.J.Y. Advanced Topics in Difference Equations. Kluwer Academic Publishers, Dordrecht (1997), 507 pp.
3. Agarwal, R.P., O'Regan, D. & Wong, P.J.Y. Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (1999), 417 pp.
4. Agarwal, R.P., O'Regan, D. & Wong, P.J.Y. Constant-Sign Solutions of Systems of Integral Equations. Springer, New York(2013), 648 pp.
5. Wong, P.J.Y. Three fixed-sign solutions of system model with Sturm-Liouville type conditions. Journal of Mathematical Analysis and Applications, USA (2004), 298: 120 - 145.
6. Wong, P.J.Y. Multiple fixed-sign solutions for a system of difference equations with Sturm-Liouville conditions. Journal of Computational and Applied Mathematics , Netherlands (2005), 183: 108 - 132. |
| Assoc Prof Wu Guohua | Theory and Models of Computation, Recursion Theory, Complexity Theory, Mathematical Logic |
| Assoc Prof Wu Hongjun | Cryptography, cryptanalysis, computer security |
| Assoc Prof Wu Yuan | Prof Wu's research interests are in Actuarial Science, Data Mining, Application of Statistics to Business and Finance, and Traffic Accidents Analysis. |
| Assoc Prof Xavier Bresson | Data Science
Deep Learning
Artificial Intelligence
Computer Vision
Sparse Convex Optimization
Spectral Graph Theory |
| Asst Prof Xia Kelin | Dr. Kelin Xia's research focuses on three aspects: mathematical modeling of biomolecule, topological data analysis and elliptic interface problem. The first subject of his research is to construct multiscale geometric models based on the variational multiscale theory. With these models, the features of biomolecular surface and structure can be characterized and quantitatively analyzed. Further, these models can be used in the study of surface visualization, protein-protein interaction, drug design, protein flexibility, and protein folding.
Topological data analysis is his recent interest. For the first time, he applies persistent homology in the analysis of fullerene molecule properties, i.e., topological structure, heat formation energy, and isomer stability. He utilizes persistent homology for protein rigidity, flexibility and folding analysis. He proposes a multidimensional persistence scheme and a multiresolution topological simplification method, and further applied both in excessively large biomolecular data analysis.
The last subject belongs to scientific computation. A major emphasis of his research is to design robust and highly accurate schemes specifically for the elliptic interface problem from the realistic biological setting. In general, challenges come from the complex geometry originated from complicated biomolecular structure. |
| Assoc Prof Xiang Liming | Dr. Xiang Liming's areas of expertise are clustered/longitudinal data analysis, survival analysis, robust estimation, statistical diagnostics, biostatistics and statistical quality control. Her current research work focuses on robust approaches for finite mixture with random effects and survival analysis. |
| Prof Xing Chaoping | Algebraic curves over finite fields; application of algebraic geometry and number theory to block coding, quantum coding, space-time coding, cryptograpgy and quasi-Monte Carlo methods; lattice packings. |
| Asst Prof Yong Ee Hou | My general research interests lies at understanding patterning and growth in nature.
1) Brain and vision
2) Biomechanics of membrane biogenesis
3) Pattern formation in biological systems
My goal is to discover, explore and understand these complex biological systems using the language of physics and mathematics. |
| Asst Prof Zhang Lei | Prof Lei Zhang's area of expertise is in empirical corporate finance and asset pricing. His current research works focus on mergers and acquisitions, capital structure, IPOs and financial intermediaries. |
| Assoc Prof Zheng Jianmin | Dr Zheng's areas of expertise are computer aided geometric design and computer graphics. His current research work focuses on T-splines, subdivision surfaces, interactive digital media processing, and sketch-based modeling. |
| Asst Prof Zheng Jie | Computational Biology and Bioinformatics
Genomics and Epigenomics
Systems Biology
Molecular Evolution, Comparative & Population Genomics
Algorithm Design, Scientific Data Mining and Knowledge Discovery |
