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College of Arts & Sciences
Advanced Solutions Group

About ASG

Advanced Solutions Group (ASG)    -     Joseph E. Johnson, PhD


ASG is the R&D group led by Professor Johnson who has been PI for over $14M in multidisciplinary grants and contracts since 1993.  His primary interests are in problems involving Lie algebras and groups, Markov processes, network theory, cluster analysis, artificial intelligence, mathematical physics, foundations of quantum theory, and most recently general relativity with programing in Python, Sage, SymPy, and other Python modules. ASG has just completed three new grants, partnering with 18 USC faculty and 22 students from various disciplines. Six new proposals are underway with research and development currently continuing in the following domains:

1.     An Optimal Numerical Data Standardization Algorithm: Johnson has developed an algorithm for attaching the units of measurement, level of uncertainty, and exactly defining metadata to numerical values with automatic dimensional analysis and error processing with full tracking of all metadata & meaning. This constitutes a new data standard that can support automated data exchange, IoT, and artificial intelligence on new levels. It totally eliminates the problems of adapting the metric (SI) standard as all units are equal (with metric default). In this standard, called MetaNumber (MN), every single numerical value has a unique name defined by the internet path to retrieve and process it. Machines can process such information without any human intervention allowing a new level of automation and machine communication.  (

2.     Network Analytics Based upon Lie algebras and Markov transformations: He has developed a method of mathematically analyzing any network by proving that every network is isomorphic to a continuous Markov transformation that models network flows. The eigenvalues and eigenvectors of that transformation provide an innovative cluster analysis for the network, which, along with its Renyi’ and Shannon entropies, allow for extensive classification, comparison, and analytics. The importance of this rests on the fact that networks are pervasive in living systems. This work rests on his novel decomposition of the general linear group GL(n,R).

3.     Cluster Analysis of Tabular Numerical Data: Imagine a numerical data table with various entities in rows and with their properties in columns (such as the chemical elements or personal health data).  He devised a method of converting such data into two networks: an entity network (rows) and a property network (columns) thus allowing the cluster and entropy analysis above to be invoked on the networks created by each. This cluster analysis can outperform existing cluster analysis. This process is now automated and can classify clusters in all tabular data tables with rank order of the degree of “tightness”. This is important since cluster analysis is fundamental for both human & artificial intelligence. The algorithm can be deployed on standardized data to intelligently find relationships and patterns and thus to “learn”. We are now working on constructing an advanced user interface.

4.     An Integration of General Relativity (GR) with Quantum Theory (QT) and the Standard Model (SM):  QT and the SM are built upon a Lie algebra foundation with representations on a Hilbert space. But GR is built with nonlinear differential equations for the metric of space-time as determined by the Einstein equation in terms of the energy momentum tensor.  Thus QT and the SM are fundamentally incompatible with GR and their integration is one of the most challenging problems in physics. Johnson previously proposed an extension of the 10 parameter Poincare Algebra to include four-vector position operators giving a 15 parameter Lie algebra that includes a covariant Heisenberg Lie algebra (HA)). He now proposes to generalize the HA structure constants from the flat Minkowsky metric to be the Riemann metric as determined in GR. The representations of this algebra are shown to convert the Christoffel symbols, and the Ricci and Riemann tensors from differential equations into commutator equations for operators thus giving GR and Einstein’s equations a solid foundation in Lie algebras of operators. The resulting representations also very naturally allow for the SM strong and electroweak vector boson terms to appear in parallel with the Riemann metric (for either a classical or quantized gravitational force as a graviton) and optionally with a scalar field. The constants “c”, ћ, and “G” now all reside in the structure constants supporting Plank units. More recently he has rebuilt the basis of Riemannian geometry in terms of Lie algebras and groups in order to apply that mathematics here. Available at


Joseph E. Johnson, PhD, is a Distinguished Professor Emeritus in the Department of Physics and Astronomy, University of South Carolina, Columbia, SC, 29208 and an Affiliate in the USC Department of Mechanical Engineering. Email:, Office: 405 Physical Science Center USC, Office Phone: 803-777-6431 and Cell: 803-920-1229. Web:


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