Ashay Dharwadker
Distinguished Professor of Mathematics & Natural Sciences
Endowed Chair
Institute of Mathematics
H501 Palam Vihar
District Gurgaon
Haryana 122017
India
ashay@dharwadker.org
Honours:
Research:
Fundamental research in mathematics and its applications. Algebra, topology, graph theory, computer science and the foundations of physics.
:: Mathematical Genealogy
:: Google Scholar Citations
:: Amazon Book Reviews
The Theory of Everything
Endowment Lecture at the Institute of Mathematics, Gurgaon.
The Cosmological Constant
Proceedings of the Institute of Mathematics, Amazon Books, 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 sevenyear Wilkinson Microwave Anisotropy Probe (WMAP) observations. Thus dark matter, dark energy and the cosmological constant are intrinsic properties of the gauge in the Grand Unified Theory.
Space, Time and Matter
Proceedings of the Institute of Mathematics, Amazon Books, 2010 :: ISBN 1466403926
Baltic Horizons No. 14 (111), Special Issue on Fundamental Problems in Mathematics, 2010
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. Electric and gauge charges of all the particles of the standard model can now be interpreted as elements of the Poincaré group. We define the space and time chiralities of all spin 1/2 fermions in agreement with Dirac's relativistic wave equation. All the particles of the standard model now correspond to irreducible representations of the Poincaré group according to Wigner's classification. Finally, we construct the Steiner system of fermions and show 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
:: arXiv:0912.5189
Proceedings of the Institute of Mathematics, Amazon Books, 2009 :: 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 M_{H}^{0} = 125.992 ~ 126 GeV, as a direct consequence of our theory.
The Graph Isomorphism Algorithm
Proceedings of the Institute of Mathematics, Amazon Books, 2009 :: ISBN 1466394374
The Structure Semiotics Research Group, S.E.R.R., Euroacademy, Tallinn, 2009
We present a new polynomialtime 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 polynomialtime, thus showing that the Graph Isomorphism Problem is in P. The semiotic theory for the recognition of graph structure is used to define a canonical form of the sign matrix of a graph. We prove that the canonical form of the sign matrix is uniquely identifiable in polynomialtime for isomorphic graphs. The algorithm is demonstrated by solving the Graph Isomorphism Problem for many of the hardest known examples. We implement the algorithm in C++ and provide a demonstration program.
Grand Unification of the Standard Model with Quantum Gravity
Proceedings of the Institute of Mathematics, Amazon Books, 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. Our first goal is to construct all the particles constituting the classic standard model, in exact agreement with 't Hooft's table. We are able to predict the exact mass of the Higgs particle and the CP violation and mixing angle of weak interactions. Our second goal is to construct the gauge groups and explicitly calculate the gauge coupling constants of the force fields. We show how the gauge groups are embedded in a sequence along the cosmological timeline in the grand unification. Finally, we calculate the mass ratios of the particles of the standard model. Thus, the mathematical proof of the four colour theorem shows that the grand unification of the standard model with quantum gravity is complete, and rules out the possibility of finding any other kinds of particles.
Applications of Graph Theory
Proceedings of the Institute of Mathematics, Amazon Books, 2007 :: ISBN 1466397098
Journal of The Korean Society for Industrial and Applied Mathematics (KSIAM), Vol. 11, No. 4, 2007
New graph theoretical proofs of Fermat’s Little Theorem and the NielsonSchreier Theorem. Applications of minimum vertex covers in graphs to DNA sequencing (the SNP assembly problem) and computer network security (worm propagation). We show how to apply edge coloring and matching in graphs for scheduling (the timetabling problem) and vertex coloring in graphs for map coloring and the assignment of frequencies in GSM mobile phone networks. We revisit the classical problem of finding reentrant knight’s tours on a chessboard using Hamiltonian circuits in graphs.
The Vertex Coloring Algorithm
Proceedings of the Institute of Mathematics, Amazon Books, 2006 :: ISBN 1466391324
A new polynomialtime algorithm for finding proper mcolorings
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 mcoloring of the
vertices of G with m less than or equal to Δ+1.
Furthermore, we prove that this condition is the best possible in terms
of n and Δ by explicitly constructing
graphs for which the chromatic number is exactly
Δ+1.
In the special case when G is a connected simple graph and is neither
an odd cycle nor a complete graph, we show that the algorithm will always
find a proper mcoloring of the vertices of G with m
less than or equal to Δ. In the process, we
obtain a new constructive proof of Brooks' famous theorem of 1941. For
all known examples of graphs, the algorithm finds a proper mcoloring
of the vertices of the graph G for m equal to the chromatic
number χ(G). In view of the importance
of the P versus
NP question, we ask: does there exist
a graph G for which this algorithm cannot find a proper mcoloring of the
vertices of G with m equal to the chromatic number χ(G)?
The algorithm is demonstrated with several examples of famous graphs, including
a proper fourcoloring of the map of India and two large Mycielski benchmark
graphs with hidden minimum vertex colorings. We implement the algorithm
in C++ and provide a demonstration program.
:: The
Math Forum Review
The Clique Algorithm
Proceedings of the Institute of Mathematics, Amazon Books, 2006 :: ISBN 1466391219
Baltic Horizons, No. 8 (107), 270 Years of Graph Theory Conference, Euroacademy, 2007
A new polynomialtime 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 finds a maximum clique in all known examples of graphs. In
view of the importance of the
P versus NP question, we ask
if there exists a graph for which the algorithm cannot find a maximum clique.
The algorithm is demonstrated by finding maximum cliques for several famous
graphs, including two large benchmark graphs with hidden maximum cliques.
We implement the algorithm in C++ and provide a demonstration program.
:: The
Math Forum Review
The Independent Set Algorithm
Proceedings of the Institute of Mathematics, Amazon Books, 2006 :: ISBN 1466387696
A new polynomialtime 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 finds a maximum independent set in all known examples of
graphs. In view of the importance of the
P versus NP question,
we ask if there exists a graph for which the algorithm cannot find a maximum
independent set. The algorithm is demonstrated by finding maximum independent
sets for several famous graphs, including two large benchmark graphs with
hidden maximum independent sets. We implement the algorithm in C++ and
provide a demonstration program.
:: The
Math Forum Review
The Vertex Cover Algorithm
Proceedings of the Institute of Mathematics, Amazon Books, 2006 :: ISBN 1466384476
A new polynomialtime 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 finds a minimum vertex cover in all known examples of graphs.
In view of the importance of the
P versus NP question, we
ask if there exists a graph for which the algorithm cannot find a minimum
vertex cover. The algorithm is demonstrated by finding minimum vertex covers
for several famous graphs, including two large benchmark graphs with hidden
minimum vertex covers. We implement the algorithm in C++ and provide a
demonstration program.
:: The
Math Forum Review
Common Systems of Coset Representatives
Proceedings of the Institute of Mathematics, Amazon Books, 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.
A New Algorithm for finding Hamiltonian Circuits
Proceedings of the Institute of Mathematics, Amazon Books, 2004 :: ISBN 146638137X
A new polynomialtime 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 algorithm finds a Hamiltonian circuit (respectively, tour)
in all known examples of graphs that have a Hamiltonian circuit (respectively,
tour). In view of the importance of the P versus NP question,
we ask: does there exist a graph that has a Hamiltonian circuit (respectively,
tour) but for which this algorithm cannot find a Hamiltonian circuit (respectively,
tour)? The algorithm is implemented in C++ and the program is demonstrated
with several examples.
:: The Math Forum Review
:: University of Rome  Computing Large Square Loops
Heptahedron and Roman Surface
Electronic Geometry Models, Model 2003.05.001, 2004
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
Electronic Geometry Models, Model 2002.05.001, 2003
Riemann surfaces were first studied by Bernhard Riemann in his Inauguraldissertation
at Göttingen in 1851. This paper shows the construction of the surfaces
w = z^{n}.
The Witt Design
Proceedings of the Institute of Mathematics, Amazon Books, 2002 :: 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 group M(24) was the first sporadic simple group discovered in 1860 amongst exactly 26 that were
eventually found, many of them arising from the exceptional geometry of the Witt design. This led to the complete classification of all
finite simple groups accomplished during the 1980's. The combinatorial properties of the Witt design also played a major role in the proof
of the Four Colour Theorem in 2000 and the Grand Unification of the Standard Model with Quantum Gravity
in 2008. This paper presents a straightforward and explicit construction of the Witt design, listing the blocks in dictionary order.
:: Design Resources at Queen Mary, University of London
:: The Math Forum Review
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. Recent research in physics shows that this 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 with precision. Thus, nature itself demonstrates the logical completeness and consistency of the proof.
:: Canadian Mathematical Society Announcement October 5, 2000
:: The
Math Forum Review
:: Tölvunot
Fréttahorn
:: Higgs Boson Mass predicted by the Four Color Theorem
Split Extensions and Representations of Moufang Loops
Communications in Algebra 23(11), 42454255, 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 uptodate 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.
:: The
Math Forum  Single Variable Calculus
My Students Database
2003 
A prototype online relational database management system in BoyceCodd
normal form using MySQL, PHP and Apache web server.
Teaching:
Today's
Lecture
Lectures at the Institute of Mathematics, 2000 
