Welcome

Welcome to my personal website. Here, I mainly collect ideas related to my research and, more generally, interesting bits of mathematics and computer science and life.

Research interests

I am interested in understanding structures relevant to mathematical physics and, relatedly, to quantum algebra and higher algebra, using discrete (and computable) geometric foundations. Here, “discrete” foundations can be contrasted with “continuous” foundations, in which definitions of geometric structures are based on the continuum \(\mathbb{R}\). My interest stems from the observation that underlying many seemingly continuous phenomena we often find hidden discrete (or “algebraic”) laws that govern such phenomena. With respect to mathematical physics, the hope is that understanding these deeper foundational principles will eventually be helpful in providing the language for actual physical theories. This a view dates back many decades, and has lead to a highly fruitful interplay of higher category theory (the study of discrete “higher” algebraic structures) and fundamental physics. One of the originators of this relation writes:

My dream was that the cobordism hypothesis, tangle hypothesis and generalized tangle hypothesis would let us see how spacetime and strings/loops/membranes in spacetime arise from pure \(n\)-category theory, without having to put in \(\mathbb{R}^n\) “by hand”.

John Baez (2018)

The FCT research programme

The idea of combinatorial foundations for geometric structures that appear in the context of quantum algebra (TQFTs, knot theory, knotted surfaces, etc.) has led to the theory of framed combinatorial topology (FCT). The theory synthesizes known combinatorial ideas with a novel (similarly combinatorial) notion of framings, and this synthesis has surprising consequences.

Some research outcomes

FCT addresses several problems of previous approaches to describing the combinatorial interplay between higher algebra and the geometry of (stratified) manifolds.

  1. A resolution to the problem of undefinability of computadic cell shapes, formalized using the combinatorics of trusses.
  2. A formalization of the notion of manifold diagrams, both geometrically and combinatorially, which generalizes string diagrams to higher dimension and provides a Poincaré dual to pasting diagrams of computadic cells.
  3. A formalization of the paradigm of categorical homotopicity, providing a direct description of (higher) paths in spaces of composites.
  4. A combinatorial toolset for studying tangles, singularities and higher Morse theory, overcoming stumbling blocks of the classical differential singularity theory in high parameter ranges.
  5. More generally, but conjecturally, a faithful combinatorialization of smooth structure and diffeomorphism.

Many open problems

A list of current open problems is maintained here.

News

The Past

  • 2 Jun 2022: I gave a talk at the Geometric Structures Lab at the Fields Institute, University of Toronto. Here’s the video (password: #d8=Wm40vV where # is the lowest dimension in which exotic smooth spheres have been constructed)
  • 16 Mar 2022: I gave a talk at the TQFT Club at the University of Lisbon. Here’s the video
  • 29 Dec 2021: Chris Douglas and I uploaded the FCT book preprint
  • 22 Nov 2021: I gave a talk in the Advanced topology class on “Generalized differential cohomology theories”

The future

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