I was a PhD student at the Department of Electrical Engineering in Indian Institute of Technology Delhi.
I had been investigating the fundamental problem of understanding the abstract cause behind the observed phenomena in nature through the application of powerful relative approach and intuition of category theory by combining it with generative intuition of the work of Prof. Micheal Leyton.
My PhD thesis "Functorial Signal Representation & Base Structured Categories" was a step in this direction at the fundamental level of signal representation, signal redundancy and systems analysis from an engineering perspective.
The thesis contributed some novel results and perspectives such as:
1. The concept and precise mathematical formulation of base structured categories, associating the concept of a functor with the generative cause underlying real world observed effect particularly for signals.
2. The concept of trivial categorification, collapsing the dual argument between set-theory versus category-theory and demonstrating that both can be simultaneously utilized in signals and systems applications. The set-theory can be utilized at local object level while category-theory is lurking at global level especially relating objects to each other.
3. The concept of a new functorial framework carrying the potential for SYNTHESIZING unifying existing models of structured signal representations and explaining the underpinning of redundancy and compression.
My resume and the PhD thesis can be downloaded by clicking on the highlighted words.
The publications so far from this work are as below:
1. S. Samant., “Fibred signal representation", Abstract presented at the International Category Theory Conference CT, 2016.
2. S. Samant.,S. D. Joshi., “Functorial Signal Representation: Foundations and Redundancy", Selected in Proceedings of IEEE Twenty Fourth National Conference on Communications (NCC) 2018. Preprint.
3. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation I: From Grothendieck fibration to Base structured categories.", arXiv:1610.05926
4. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation II: Category action, Base Hierarchy, Geometries as Base structured categories.", arXiv:1611.02437 (major revision to appear with set-theoretic manifold theory).
5. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation III: Foundations, Redundancy, L0 and L2 functors.", arXiv:1710.10227 (revised version to appear shortly).
An old photograph with ethereally white and pristine snow-clad Himalayan peaks in the background.
Some words on thesis and the ongoing journey -
One can use the thesis to derive synthesizing sampling theorems in signal processing and data sciences more generally. When Grothendieck said - "Arrows are more important than objects" intuitvely it meant - Synthesis is much more significant than pure separative analysis of set theory or Whole is more than the sum of parts. Even if objects appear to be separated they are interconnected to form a larger whole. Unbalanced tilt towards analysis and fragmentation has halted sciences today."
Category theory is the synthesis of all mathematics. Synthesis (psychologically leading to right human relations) is the keyword for the incoming Aquarian age which Grothendieck sensed intuitively !