Seminar on Combinatorial Aspects of Commutative Algebra and Algebraic Geometry (in IITB)

  • Update from 7 July, 2018:
  • Last semester, we explored various aspects of Green’s conjecture. This semester we plan to continue some parts of it that were not covered last time. We plan to have a mix of topics around combinatorial algebraic geometry such as graph complexes, Tutte polynomial and so on…

  • Update from the 17 January, 2018 meeting:
  • This semester we’ll have talks on two themes i. Tropical curves and their divisor theory, ii. Green’s conjecture on syzygies of canonical curves. We’ll start with the basics of tropical part (about 2-3 weeks) and then move to basics of Green’s conjecture part (2-3 weeks). We’ll then move on to more advanced aspects of both these topics.

  • Update from 8-th September meeting:
  • This semester (probably the next) we plan to focus on aspects of syzygies: combinatorial, algebraic, geometric. We’ll devote the first few lectures to the basics of syzygies.

    In the future, we plan to cover various aspects of the interplay between Combinatorics and Commutative Algebra-Algebraic Geometry. Some possible topics are

    i. Syzygies: connections to Combinatorics, Syzygies of curves particularly Green’s conjecture and the related conjecture for Graph Curves, Boij-Soederberg theory, Syzygies in Tropical geometry.

    ii. Vector Bundles with potential connections to Tropical Geometry.

    iii. Dessins D’enfants:

    iv. Brill-Noether theory on Algebraic curves and Tropical curves, Combinatorial Aspects of Moduli Spaces.

    v. Non-archimedean Geometry in connection with Tropical Geometry, particular Berkovich Spaces,

    vi. Combinatorics and Hodge theory, Vector Partition Functions.

    vii. Ideals of Powers of Linear Forms.

    viii. Tutte Polynomials of Graphs, Zeta Functions associated to Graphs.

    ix. Syzygies in Real Algebraic Geometry.

    x. Tropical Geometry in Dimensions Two and Higher.

    We plan to pick a theme every semester followed by talks by local faculty and students. These talks will be introductory and will assume very little background. We will then invite speakers from outside for more specialized talks on this theme.

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    Madhusudan Manjunath

    Welcome to my homepage! I am an Assistant Professor in the Department of Mathematics, IIT Bombay.

    I was visiting the IHES, October 20-December 2, 2017.
    Earlier this year I spent four months at the Erwin Schroedinger Institute for Mathematics and Physics, Vienna as a Junior Research Fellow and three months at the Mathematics Institute at Oberwolfach (MFO) as a Leibniz Fellow .
    Until recently, I have been a postdoctoral researcher at the School of Mathematical Sciences at Queen Mary University of London and a Visiting Assistant Professor at the Mathematics Department at University of California Berkeley. My position in Berkeley was partially funded by Feoder-Lynen Fellowship of the Humboldt Foundation. My mentor was Prof. Bernd Sturmfels. Previously, I was a visiting assistant professor at the School of Mathematics, Georgia Institute of Technology where my mentor was Prof. Matt Baker.

    News

    June 25, 2018, The ATM school on Combinatorial Commutative Algebra last week was great fun! Sincere thanks to all the participants. Really enjoyed lecturing in it and listening to the other lectures. We’re currently preparing lecture notes that we plan to put up in the arxiv.

    February 28, 2018, I’ll visit ICTP, Trieste, Italy for one month (May 15- June 15, 2018) under the Mathematics Visiting Fellowship.

    December 20, 2017, Back to IITB after IHES visit, Pune conference on Commutative Algebra and Algebraic Geometry.

    August 4, 2017, Currently in Bangalore, will join IITB mid-August. Visiting IHES, Paris October 20-December 20, 2017.

    July 6, 2017, We are planning to a start a seminar on “Combinatorial Aspects of Algebraic Geometry and Commutative Algebra” in IIT Bombay starting August, 2017. More details will follow soon.

    May 23, 2017, I’ll be starting as an Assistant Professor in the Department of Mathematics, IIT Bombay from August 2017.

    April 1, 2017, Commutative Algebra of Generalised Frobenius Numbers is on the arxiv.

    January 8, 2017: I will spend time at MFO, Schroedinger Institute in Vienna and IHES Paris this year.

    November 4, 2016: Back to London and expect to be here till the end of December.

    October 24, 2016: Visiting Rice University, 26, 27 October and then Minneapolis for the AMS Special Session on “Chip Firing, Divisors on Graph and Simplicial Complex”, October 28-30. My talk is on Saturday, 29 October.

    October 6, 2016: We are holding a Tropical Geometry Seminar at Imperial this term. Our first meeting will be at Huxley 109, 4pm tomorrow (October 7). Please drop by or write to me if you want to know more about it.

    September 13, 2016: Back to London and expect to be in this area throughout September.

    August 1, 2016: Visiting India August 5-September 7, 2016. I will visit TIFR, Mumbai, IIT Bombay and IIT Madras.

    July 25, 2016: I am attending Latin American Algebra at Buones Aires, Argentina 26-29 July. I’ll speak at the special session on Computational Algebra and Applications of Algebra on Friday 29 July.

    July 20, 2016: I will visit MFO, Oberwolfach on a Leibniz Fellowship from January-March 2017.

    June 24, 2016: Syzygies over the Polytope Semiring is up on the arxiv.

    Address

    School of Mathematical Sciences,
    Queen Mary University of London,
    Mile End Road,
    London, E1 4NS.

    Email

    Research


    My research interests are in the combinatorial aspects of algebraic geometry and commutative algebra. Specifically,

    • Tropical and Non-archimedean Geometry
    • Combinatorial Commutative Algebra
    • Algebraic Combinatorics
    • My google scholar page gives an impression of my papers. Here are some topics I have been working on.

    • Counting Special Divisors on Graphs:
    • I have been working on counting special divisors on graphs using methods from commutative algebra.

    • Syzygies over Polytope Semirings:
    • Tropical geometry and its applications indicate a theory of syzygies over polytope semirings. The broad goal of this project is to developed this notion and study applications, for instance to canonical embeddings of tropical curves (as developed by Haase, Musiker and Yu) along the lines of Green’s study of syzygies of canonical algebra curves. I recently completed a paper that takes first steps in this direction.

    • Tropical Graph Curves:
    • The broad goal in this project is to investigate the construction of “good” tropicalizations of canonical algebraic curves. More precisely, we mean tropicalizations that provide valuable information about the Berkovich analytification of the algebraic curve. This led to the concept of tropical graph curves. Using this we construct tropicalizations that capture the topology of the Berkovich analytification for algebraic curves whose graph underlying the Berkovich analytification is a three-connected planar graph. See below for preprints.

    • Combinatorial Brill-Noether for Graphs via Commutative Algebra:
    • We are investigating the Brill-Noether theory for Graphs a la Baker and Norine via commutative algebra. It turns out that the Brill-Noether theory on graphs is closely related to Boij-Seoderberg style problems on quotient rings of certain binomial ideals associated to the graph. We are currently investigating the free resolution of the residue field over these quotient rings. For instance, classifying graphs for which the ring is Golod. This is work in progress with A. Fink.

    • Commutative Algebra of Generalised Frobenius Numbers:
    • One goal of this project is to develop commutative algebraic interpretations of generalised Frobenius numbers. The combinatorial aspects have been developed in the work of Beck and Robins and the recent work of Aliev, De Loera and Louveaux. I have recently competed a paper with B. Smith in this direction.

       

      Research Articles

      2018

      July

      June

      2017

      October

      2016

      January

      2015 and Before: 

       

       

       

       

       

       

       

       

       

       

      Dissertation


      Teaching

      In January 2016, I taught an LTCC course on Tropical Geometry. I previously taught introduction to Abstract Algebra, Math 113 in Berkeley. Previously, I have taught courses at both undergraduate and graduate levels. At the undergraduate level I have taught courses on linear and discrete mathematics and calculus at Georgia Tech and the graduate level a course on combinatorial commutative algebra at Saarland University.

      Working with Students

      As a postdoc, I have particularly enjoyed working with graduate
      students and undergraduate students: mentoring, sharing my experiences with them and learning from them has been great fun! I look forward to working more with undergraduate and graduate students in the future.

      Travelling

      Upcoming

    • AMS Fall Central Sectional Meeting, Special Session on Chip-Firing and Divisors on Graphs and Complexes, October 2016.

      Recent Past

    • Latin American Algebra, July 2016
    • Tropical Day at Ecole Polytechnique, November 2015.
    • Sandpile Groups at Banff, Oaxaca, Mexico, July 2016.