priyavrat deshpande

Chennai Mathematical Institute
SIPCOT IT park,Siruseri, TN 603103
+91 (044)6748 0962;
pdeshpande AT cmi DOT ac DOT in

I am an assistant professor at the Chennai Mathematical Institute. I completed my PhD in 2011 in mathematics from University of Western Ontario, Canada under the supervision of Graham Denham and Ján Mináč. Then I spent a year at Northeastern in Boston followed by three years as a post-doc here at CMI. My mathematical interests broadly include Algebraic topology and Combinatorics. In particular I am interested in the theory of arrangements (hyperplanes, tori, submanifolds etc.), topological robotics and applications of algebraic topology.

I am one of the placement coordinators at CMI. If you are looking for hiring mathematicians to work in finance, machine learning, formal verification etc. feel free to write to me. Visit our placement page here.

I am also the local coordinator for the Madhava Mathematics Competition. If you are from around Chennai and are interested in participating in this competition then get in touch.

Research

My research involves combinatorial and topological understanding of hyperplane arrangements. In my PhD thesis I initiated a study of submanifold arrangements. In particular I am interested in spherical, projective and toric arrangements. I am also interested in discrete subgroups of diffeomorphisms that are generated by certain involutions, a generalization of Artin groups, enumerative aspects of collection of geodesics on compact surfaces and topology of certain algebraic varieties.

My current projects, in topological robotics, include computation of homotopy groups of planar polygon spaces, homology of arachnoid mechanisms and computation of navigational complexity of related spaces.

I was a recipient of SERB's MATRICS grant for the period of 3 years (June 2018 - March 2021).

Activities

  • I and Anurag Singh are organizing an NCM funded workshop titled "Cohen-Macaulay complexes in graph theory" in CMI from 10-15 July, 2023. Visit the link for more details.
  • I and Dheeraj Kulkarni organized a 3-day meeting (July 16-18, 2018) for graduate students and postdocs interested in various aspects of topology. The details can be found here.
  • I co-organized Computational Geometry Day at CMI on 25/01/2018. For more information click here.
  • I and Ronno Das ran a weekly seminar on configuration spaces with a focus on understanding their cohomology as representations of the symmetric group. The seminar page is here.

Publications

  1. The Borsuk-Ulam theorem for planar polygon spaces.
    (with N. Daundkar, S. Goyal and A. Singh), Topology Appl DOI. arXiv.
  2. A statistic for regions of braid deformations.
    (with K. Menon), Semin Lothar Combi 86B.23, proceedings of the 34th FPSAC, 2022. Link.
  3. On the topology of bi-cyclopermutohedra.
    (with N. Manikandan and A. Singh) Indian J. Pure Appl. Math. DOI and arXiv.
  4. Distance r-domination number and r-independence complexes of graphs.
    (with S. Shukla and A. Singh) Eur. J. Comb. 102, No. 103508 (2022), arXiv.
  5. The moment polytope of the abelian polygon space.
    (with N. Daundkar) Topology Appl. 302, Article ID 107834, 2021. arXiv.
  6. A combinatorial statistic for labeled threshold graphs.
    (with K. Menon and A. Singh) Enum. Comb. Appl. 1:3, Article S2R22, 2021. arXiv
  7. Counting regions of the boxed threshold arrangement.
    (with K. Menon and A. Singh) J. Integer Seq. 24, No. 5, Article 21.5.7, 2021. arXiv
  8. Higher Independence Complexes of graphs and their homotopy types.
    (with A. Singh) J.Ramanujan Math. Soc. 36, No. 1, 2021. arXiv
  9. Face counting formula for toric arrangements defined by root systems.
    (with K. Sutar) AKCE Int. J. Graphs Comb. 16, No. 1, 2019.
  10. Face enumeration for line arrangements in a 2-torus.
    (with K. Chandrashekhar) Indian J. Pure Appl. Math., 48(3), 2017.arXiv.
  11. On arrangements of pseudohyperplanes.
    Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 3, 2016, pp. 399–420.
  12. Coxeter transformation groups and reflection arrangements in smooth manifolds.
    (with R. Das), J. Homot. Rel. Struct.,11:571-597, 2016, arXiv.
  13. Arrangements of spheres and projective spaces.
    Rocky Mountain J. Math 46(5): 1447-1487, 2016, link, arXiv.
  14. Deletion-restriction in toric arrangements.
    (with K. Sutar) J. Ramanujan Math. Soc. 31(1):17-30, 2016. arXiv.
  15. On a generalization of Zaslavsky's theorem for hyperplane arrangements.
    Ann. Comb., 18 (1):35-55, 2014. arXiv
Accepted articles
  1. A branch statistic for trees: Interpreting coefficients of the characteristic polynomial of braid deformations. (with K. Menon) arXiv, to appear in Enumerative Combinatorics and Applications.
  2. Refinements of the braid arrangement and two-parameter Fuss-Catalan numbers. (with K. Menon and W. Sarkar) arXiv, to appear in Journal of Algebraic Combinatorics.
Preprints/ In preparation
  1. On the structure of finitely presented Bestvina-Brady groups. (with M. Roy) arXiv.
  2. Building planar polygon spaces from the projective braid arrangement. (with N. Daundkar) arXiv.
  3. The n-dimensional Klein bottle is a real Bott manifold. (with N. Daundkar) arXiv
  4. Chordal graphs, higher independence and vertex decomposable complexes. (with F. Abdelmalek, S. Goyal, A. Roy and A. Singh) arXiv
  5. The Goresky-MacPherson formula for toric arrangements arXiv.
  6. Arrangements of submanifolds and the tangent bundle complement arXiv.

teaching

This semester (Jan-May 2022) I am teaching Intro to manifolds and Enumerative combinatorics.

Past courses

  1. Mar-Apr 2022 Graduate topology 2
  2. Jan-Feb 2022 Topological data analysis
  3. Sep-Dec 2021 Intro to manifolds
  4. Apr-Jul 2021 Algebra 2
  5. Mar-Apr 2021 Topological data analysis
  6. Aug-Nov 2020 Differential topology
  7. Jan-Apr 2020 Graduate complex analysis
  8. Aug-Nov 2019 Topological data analysis
  9. Jan-Apr 2019 Graduate Topology 2
  10. Aug-Nov 2018 Topological Robotics.
  11. Aug-Nov 2018 Intro to Manifolds.
  12. Jan-Apr 2018 Topics in combinatorics .
  13. Aug-Nov 2017 Intro to Manifolds
  14. Jan-Apr 2017 Graduate Topology 2
  15. Aug-Nov 2016 Combinatorics 1
  16. Aug-Nov 2015 Homotopy theory
  17. Jan-Apr 2015 Differential topology
  18. Jan-Apr 2015 Topics in topology
  19. Aug-Nov 2014 Algebraic topology
  20. Jan-Apr 2014 Algebraic topology II
  21. Aug-Nov 2013 Algebraic topology
  22. Jan-Apr 2013 Intro to reflection groups.
  23. Aug-Nov 2012 Real analysis

supervision

Postdoc
  1. Mallika Roy (2021-2022): PhD (2020) from Universitat Politecnica de Catalunya.
  2. Shuchita Goyal (2019-2021): PhD (2019) from IIT Bombay. Currently a post-doc at IIT Delhi.
  3. Anurag Singh (2018-2021): PhD (2018) from IIT Kanpur. Now a faculty at IIT Bhilai.
PhD students
  1. K. Menon (2021-)
  2. N. Daundkar (2017-2022). Currently a post-doc at IIT Bombay.
MSc students
  1. Deepak M.S. (2021-22): Topological complexity.
  2. Shambhavi S. (2020-21, IISER Pune): Applications of topology to data analysis.
  3. Krihna Menon (2020-21): Counting regions of the deformations of the braid arrangement.
  4. Sahil Kulasreshtha (2019-20): Information cohomology.
  5. Naageswaran M. (2018-19): On the topology of cyclopermutohedron and its quotient.
  6. N. Adhikari (2016-17): Discrete Morse theory on moduli spaces of planar polygons (thesis).
  7. A. Khetan (2015-16): A cellular model for the configuration space of points on graphs (thesis).
  8. R. Das (2014-15): Salvetti complex construction for manifold reflection arrangements (thesis).
  9. Karthik C. (2013-14): Face numbers of toric arrangements.

Misc.