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Ashay Dharwadker

Distinguished Professor of Mathematics & Natural Sciences

Endowed Chair

Institute of Mathematics
H-501 Palam Vihar
District Gurgaon
Haryana 122017
India

ashay@dharwadker.org

Honours:

$1 Million Endowed Chair of the Institute of Mathematics, Gurgaon

Distinguished Professor of Mathematics & Natural Sciences

Honorary Professor, University Euroacademy, Tallinn

Institute Lectures:

Ceremonial Lecture at the Institute of Mathematics, Gurgaon

The Hilbert Lectures at the Institute of Mathematics, Gurgaon

The Turing Lectures at the Institute of Mathematics, Gurgaon

The Einstein Lectures at the Institute of Mathematics, Gurgaon

The Ramanujan Lectures at the Institute of Mathematics, Gurgaon

Research:

Fundamental research in Mathematics, Computer Science and Theoretical Physics.

Mathematical Genealogy
 
Google Scholar Citations
 
Google Books Catalogue
 
Amazon Books Catalogue
 
Mathematician's Biography
 
Institute of Mathematics
 

The Theory of Everything
Endowment Lecture at the Institute of Mathematics, Gurgaon, 2012
A brief summary of how the mathematical proof of the four color theorem leads to the grand unification of the standard model of particle physics with Einstein's theory of gravity and the nature of dark matter and energy in big bang cosmology. Our exact predictions of the mass of the Higgs boson and Einstein's cosmological constant have since been expermentally confirmed, thus establishing our theory.

The Cosmological Constant
Proceedings of the Institute of Mathematics, 2011 :: ISBN 1466272317
We show how to calculate Einstein's cosmological constant Λ using the Grand Unified Theory. Using the topological properties of the gauge, we calculate the exact percentages of ordinary baryonic matter, dark matter and dark energy in the universe. These values are in perfect agreement with the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations.

Space, Time and Matter
Proceedings of the Institute of Mathematics, 2010 :: ISBN 1466403926
We show how the grand unified theory based on the proof of the four color theorem, can be obtained entirely in terms of the Poincaré group of isometries of space and time. All the particles of the standard model now correspond to irreducible representations according to Wigner's classification. The Steiner system of fermions shows how the Mathieu group acts as the group of symmetries of the fundamental building blocks of matter.

Higgs Boson Mass predicted by the Four Color Theorem
Proceedings of the Institute of Mathematics, 2009 :: arXiv:0912.5189 :: ISBN 1466403993
Based on the proof of the four color theorem and the grand unification of the standard model with quantum gravity, we show how to derive the values of the famous Cabibbo angle and CKM matrix, in excellent agreement with experimental observations. We make a precise prediction for the elusive Higgs boson mass MH0 = 125.992 ~ 126 GeV, as a direct consequence of our theory.

The Graph Isomorphism Algorithm
Proceedings of the Institute of Mathematics, 2009 :: ISBN 1466394374
We present a new polynomial-time algorithm for determining whether two given graphs are isomorphic or not. We prove that the algorithm is necessary and sufficient for solving the Graph Isomorphism Problem in polynomial-time, thus showing that the Graph Isomorphism Problem is in P.

Grand Unification of the Standard Model with Quantum Gravity
Proceedings of the Institute of Mathematics, 2008 :: ISBN 1466272317
We show that the mathematical proof of the four colour theorem directly implies the existence of the standard model together with quantum gravity in its physical interpretation. Conversely, the experimentally observable standard model and quantum gravity show that nature applies the mathematical proof of the four colour theorem at the most fundamental level. Thus, the grand unification of the standard model with quantum gravity is complete.

Applications of Graph Theory
Proceedings of the Institute of Mathematics, 2007 :: ISBN 1466397098
We present new proofs of Fermat's Little Theorem and the Nielson-Schreier Theorem using graph theory. The vertex cover algorithm is applied to DNA sequencing and computer network security problems. We apply edge coloring and matching in graphs for scheduling and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. The classical problem of finding re-entrant knight's tours on a chessboard is addressed using the Hamiltonian circuit algorithm.

The Vertex Coloring Algorithm
Proceedings of the Institute of Mathematics, 2006 :: ISBN 1466391324
A new polynomial-time algorithm for finding proper m-colorings of the vertices of a graph. We prove that every graph with n vertices and maximum vertex degree Δ must have chromatic number χ(G) less than or equal to Δ+1 and that the algorithm will always find a proper m-coloring of the vertices of G with m less than or equal to Δ+1. We obtain a new constructive proof of Brooks' famous theorem of 1941. The algorithm is demonstrated with several examples of famous graphs, including a proper four-coloring of the map of India and two large Mycielski benchmark graphs with hidden minimum vertex colorings.

The Clique Algorithm
Proceedings of the Institute of Mathematics, 2006 :: ISBN 1466391219
A new polynomial-time algorithm for finding maximal cliques in graphs. It is shown that every graph with n vertices and minimum vertex degree δ must have a maximum clique of size at least ⌈n/(n−δ)⌉ and that this condition is the best possible in terms of n and δ. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm is demonstrated by finding maximum cliques for several famous graphs, including two large benchmark graphs with hidden maximum cliques.

The Independent Set Algorithm
Proceedings of the Institute of Mathematics, 2006 :: ISBN 1466387696
A new polynomial-time algorithm for finding maximal independent sets in graphs. It is shown that every graph with n vertices and maximum vertex degree Δ must have a maximum independent set of size at least ⌈n/(Δ+1)⌉ and that this condition is the best possible in terms of n and Δ. As a corollary, we obtain new bounds on the famous Ramsey numbers in terms of the maximum and minimum vertex degrees of the corresponding Ramsey graphs. The algorithm is demonstrated by finding maximum independent sets for several famous graphs, including two large benchmark graphs with hidden maximum independent sets.

The Vertex Cover Algorithm
Proceedings of the Institute of Mathematics, 2006 :: ISBN 1466384476
A new polynomial-time algorithm for finding minimal vertex covers in graphs. It is shown that every graph with n vertices and maximum vertex degree Δ must have a minimum vertex cover of size at most n−⌈n/(Δ+1)⌉ and that this condition is the best possible in terms of n and Δ. The algorithm is demonstrated by finding minimum vertex covers for several famous graphs, including two large benchmark graphs with hidden minimum vertex covers.

Common Systems of Coset Representatives
Proceedings of the Institute of Mathematics, 2005 :: ISBN 1466265302
Using the axiom of choice, we prove that given any group G and a finite subgroup H, there always exists a common system of representatives for the left and right cosets of H in G. This result played a vital role in the proof of the four color theorem and the grand unification of the standard model with quantum gravity.

A New Algorithm for finding Hamiltonian Circuits
Proceedings of the Institute of Mathematics, 2004 :: ISBN 146638137X
A new polynomial-time algorithm for finding Hamiltonian circuits in graphs. It is shown that the algorithm always finds a Hamiltonian circuit in graphs that have at least three vertices and minimum degree at least half the total number of vertices. In the process, we also obtain a constructive proof of Dirac's famous theorem of 1952, for the first time. The program is demonstrated with several examples.
:: University of Rome - Computing Large Square Loops

Heptahedron and Roman Surface
Proceedings of the Institute of Mathematics, Electronic Geometry Models, 2003 :: Model 2003.05.001
Using Hilbert's definition of a heptahedron we show how to construct Steiner's Roman surface as a model of the projective plane.
:: MathWorld - Roman Surface
:: MathWorld - Heptahedron

Riemann Surfaces
Proceedings of the Institute of Mathematics, Electronic Geometry Models, 2002 :: Model 2002.05.001
Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation at Göttingen in 1851. This paper shows the construction of the surfaces w = zn.

The Witt Design
Proceedings of the Institute of Mathematics, 2001 :: ISBN 1466265302
The unique Steiner system S(5, 8, 24) was discovered by Ernst Witt in 1938 and is known as the Witt design. The blocks of the Witt design are the Golay codewords of weight 8. Witt's original construction of S(5, 8, 24) used the largest of the Mathieu groups, M(24), that turns out to be the automorphism group of the Witt design. The combinatorial properties of the Witt design played a major role in the proof of the four colour theorem and the grand unification of the standard model with quantum gravity.

A New Proof of The Four Colour Theorem
Proceedings of the Institute of Mathematics, Amazon Books, 2000 :: ISBN 1466265302
We present a new proof of the famous four colour theorem using algebraic and topological methods. The proof directly implies the Grand Unification of the Standard Model with Quantum Gravity in its physical interpretation and conversely the existence of the standard model of particle physics shows that nature applies this proof of the four colour theorem at the most fundamental level, giving us a grand unified theory. In particular, we have shown how to use this theory to predict the Higgs Boson Mass and Einstein's Cosmological Constant with precision.
:: Canadian Mathematical Society Announcement

Split Extensions and Representations of Moufang Loops
Communications in Algebra 23(11), 4245-4255, 1995
A representation theory of Moufang loops generalizing the traditional representation theory of groups.
:: European Mathematical Society Review

Textbook:

Graph Theory
Orient Longman and Universities Press of India, Amazon Books, 2008 :: ISBN 1466254998
This text offers the most comprehensive and up-to-date presentation available on the fundamental topics in graph theory. It develops a thorough understanding of the structure of graphs, the techniques used to analyze problems in graph theory and the uses of graph theoretical algorithms in mathematics, engineering and computer science. The climax of the book is a new proof of the famous four colour theorem.

Software:

Statistics 1.0
Software for Windows, 2007 -
Descriptive statistics, statistical inference, quality control, acceptance sampling, regression and correlation, time series and trends, analysis of variance (ANOVA), probability distributions with moment generating functions and random samples.

Calculus 1.0
Software for Windows, 2003 -
Compute and graph functions, derivatives, integrals, tangents, arc lengths, areas, roots, maxima/minima, points of inflection, Taylor series and Fourier series, areas and volumes of surfaces of revolution, estimate limits of functions, sequences and series.

My Students Database
2003 -
A prototype online relational database management system in Boyce-Codd normal form using MySQL, PHP and Apache web server.

Teaching:

Today's Lecture
Lectures at the Institute of Mathematics, 2000 -
 
 



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