Research Summary

My research interests lie in areas of category theory and topos theory, higher category theory, derived algebraic geometry, categorical logic and homotopy type theory. I am also interested in application of topology, topos theory, and algberaic topology to problems of computation; either concerning theoretical underpinning of the notion of computation such as foundational problems "what is computation?", "what is a proof?",etc, or application of algebro topological invariants such as homology and homotopy to study patterns in big data.

My current research work is pretty much focused on the study of 2-categorical (really bicategorical!) aspects of toposes; bi-limits, bi-colimits, fibrations and opfibrations, factorization systems, etc. Topoi can be approached from at least three distinct angles; a topos as a generalized topological space (or even better a generalized locale), as a presentation-independent geometric theory, and as a set theory ("when a topos looks into itself!"). My research aim is to make these connections more clear in the bicategory of toposes (a commune where all toposes over different bases live together.), particularly the connection of the first and second viewpoints. In this direction, there is an essential theme from constructive mathematics: developing approaches to reason about various constructions in topoi predicatively, that is systematically avoiding the use of power-set strucuture provided by subobject classifer of elementary topoi in the base.



Talks And Presentations

Toposes In Como

June 29, 2018

Conference talk, Toposes in Como, Chiostro di Sant’Abbondio, Università degli Studi dell’Insubria, Como, Italy

Fibrations Of Bicategories And Fibrations Of Toposes

March 21, 2018

Conference talk, YaMCATS, School of Mathematics, University of Sheffield, Sheffield, UK

Fibrations Of Toposes

September 16, 2017

Conference talk, PSSL 101, School of Mathematics, University of Leeds, Leeds, UK



Publications

First Year Progress Report

Published in School of Computer Science, University of Birmingham, 2015

A general report of the first year including background studies submitted to my thesis committee

Download here

Reading Group on Higher Topos Theory

With Alexander Oldenziel, Joost van Dijk and a few others we have an informal weekly reading sessions on Categorical Homotopy Theory and Higher Topos Theory in Utrecht and Amsterdam. If you are in the Randstad area, and interested in this sort of things, send me an email.



CARGO

I am one of the organizers of category theory reading group (CARGO). The meetings takes place at the school of computer science on a weekly basis. The particiants are mostly PhD students and research fellows in theoretical computer science.

Read more about it here .

  • From 2015-2017, CARGO meetings used to takes place at the school of computer science on every Tuesday at 2:00 p.m. in room 245. For the academic year of 2017-2018 we have our meetings on Thursdays at 11:00 a.m. in the same room.
  • Here is the archive of talks: CARGO archive. Feel free to subscribe to our mailing list. Let me also mention the archive of our Theory Seminars talks on Fridays. I have given four talks in theory seminars and numerous talks in CARGO. Sometimes I find enough time and I write the notes of the talks which can be found in below.
  • Here is a list of topics considered to be discussed in Autumn 2017 in Category Reading Group seminars as well as informal archive for suggested topics since Summer 2015: Potential topics and Informal archive.



Expository Papers And Notes

Writing notes helps me to grasp a mathematical concept or construction and they become less elusive in my mind. Here are the notes from some of my talks for which I had time to type them up. Any reports of mistakes and errors from the reader are always welcome. Also, suggestions and recommendations on how to improve a presentation of a concept or how to communicate the essential ideas of a construction are greatly appreciated.

A topos without points

Published:

An example of a topos (from logic) that does not have any points.

A learning wishlist

Published:

A list of tricks/theorems/constructions/concepts/theories I wished to know but never had time/opportunity to learn them!

Principal Bundles

Incomplete as they stand (April, 2018)

A review of theory of principal bundles and higher principal bundles

Principal Bundles

Incomplete as they stand (April, 2018)

A review of theory of principal bundles and higher principal bundles

Three Sites Of Relative Schemes

Incomplete as they stand (April, 2018)

Warning: in construction! We introduce three Grothendieck topologies on the category of based schemes; the Zariski, the étale and the fpqc topology



Academic Records

Here is my Curriculum Vitae