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

The CACAAG seminar at IIT Bombay is now taking a new avatar as Facets of CACAAG.

Facets of CACAAG will highlight various aspects of combinatorial commutative algebra-algebraic geometry, broadly interpreted.  This includes closely related topics such as algebraic combinatorics and combinatorial representation theory.  The talks will start at a level accessible to graduate students and  will eventually touch upon some of the modern developments and future directions. 

Update on 26 November, 2020.

Upcoming talks:

December 03, Ezra Miller, Duke University.

December 10, Raman Sanyal, Goethe-Universität Frankfurt.

December 15, Satoshi Murai, Waseda University.

Update on 18 July, 2020.

The schedule:

July 9, Nathan Ilten, Simon Fraser University. Here is the recording of Nathan’s talk.


July 23, Mateusz Michalek, Max Planck Institute for Mathematics in the Sciences. Here is the recording of Mateusz’s talk.

July 30, R Venkatesh, Indian Institute of Science, Bangalore.

August 7, Gunnar Fløystad, University of Bergen.

August 13, Arvind Ayyer, Indian Institute of Science, Bangalore.

September 3, Amritansu Prasad, Institute of Mathematical Sciences, Chennai.

October 8, Alex Fink, Queen Mary University of London.

November 26: Alexey Garber, The University of Texas Rio Grande Valley.

We plan to start with a talk by Nathan Ilten (https://www.sfu.ca/~nilten/) next week, July 9 and R Venkatesh (http://math.iisc.ac.in/~rvenkat/) will speak on July 30. More updates will follow soon.

We will create a repository of the recordings of the talks and upload the lecture material (with the permission of the speaker).

The following is a record of the CACAAG seminar.


  • Update from 5 June, 2020.

Greg Blekherman spoke yesterday “Do Sums of Squares Dream of Free Resolutions”. The next CACAAG seminar in on June 11 and the speaker is Timo de Wolff, Technische Universität Braunschweig (had to be postponed from May 28). He will speak on Nonnegativity, Discriminants and Tropical Geometry.

  • Update from 22 May, 2020

The next CACAAG seminar in on May 28 and the speaker is Timo de Wolff, Technische Universität Braunschweig. He will speak on Nonnegativity, Discriminants and Tropical Geometry.

After that we have Greg Blekherman, Georgia Institute of Technology speaking on June 4.

  • Update from 30 April, 2020:

The next CACAAG seminar is on May 7 and the speaker is Justin Chen, Georgia Institute of Technology on Free resolutions of function classes via order complexes.

  • Update from 18 April, 2020:

We are currently hosting the CACAAG seminar via video conferencing (so far via Zoom). We had a series of three Lectures by Jugal Verma on toric ideals and integer programming. Our next seminar is by Ayush Kumar Tewari, TU Berlin and is scheduled for Thursday 23 April 2020 (6pm IST=2:30 PM CET=1:30 PM GMT).

  • Update from 2 March, 2020:

This semester we have had introductory lectures on tropical algebraic geometry. In the next few weeks, we will have lectures on uniformization of elliptic curves (complex and at some point non-archimedean) (Speaker: Poornima) and Bruhat-Tits trees (Speaker: Dipendra Prasad).

  • Update from 7 August 2019:

This semester we plan to have following list of topics:

  • An Introduction to the Geometry of Numbers: Madhusudan Manjunath.
  • Gubeladze’s proof of Anderson’s Conjecture (on projective modules over monoid rings), Manoj Keshari and Maria Matthew.
  • The Polynomial Method in Discrete geometry, Mrinal Kumar.
  • Aspects of Toric Ideals: Integer Programming and Invariant Theory: Jugal Verma.
  • Tightness of Surfaces and Betti numbers of Ideals (as in the work of Murai): Sameer Shukla and Madhusudan Manjunath.
  • Brill-Noether theory (and/ or Toric Local Cohomology): Madhusudan Manjunath.

Update from January 2019:

This semester we plan to have talks on three themes:

i. Zeta Functions Associated to Graphs: We will study the Ihara Zeta Function, the two variable Zeta Function due to Lorenzini.

ii. Combinatorial Brill-Noether Theory: I will talk about some ongoing work on this topic.

iii. Hilbert Series and Free Resolutions in Complexity Theory.

  • 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.

Madhusudan Manjunath

Welcome to my homepage! I am an Assistant Professor in the Department of Mathematics, IIT Bombay.
I am a part of the Seminar committee of the IITB, Mathematics department.
So, if you are visiting this part of the world, please get in touch.

Graduate Students

Rahul Singh Karki is interested in commutative algebra and its interface with combinatorics.


Amit Roy is currently at Bhaskaracharya Pratisthana, Pune.

Postdoc Positions

Early career researchers with research interests in the combinatorial aspects of commutative algebra and algebraic geometry, broadly interpreted are encouraged to get in touch with me. We have several postdoctoral positions, please see the department webpage and the institute webpage . The duration of the postdoc can vary from 1-3 years.

As a postdoc at IITB, you are encouraged to take advantage of the large research groups in commutative algebra and algebraic geometry in Mumbai and actively participate in the seminars.

More about me

I was visiting the IHES, October 20-December 2, 2017.
Earlier in 2017,  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.


28 January 2021: I’ll participate in the semester program in combinatorial algebraic geometry at ICERM.

28 January 2021: Our (with Arvind Ayyer and Pranav Pandit) proposal for the two week workshop titled “Combinatorial Algebraic Geometry – Tropical and Real” has been accepted. The workshop is scheduled  for June 27 – July 8, 2022.

3 December 2020: Here is video of a talk on Poincare Series of Divisors on Graphs.

25 November, 2020: Just posted a preprint “Poincare Series of Divisors on Graphs and Chains of Loops” to the arXiv.

March 30, 2020, Our (with Arvind Ayyer and Pranav Pandit) proposal for the two week workshop titled “Combinatorial Algebraic Geometry – Tropical and RealJuly 12 – 23, 2021 at ICTS has been accepted. Details will be out in a bit. This workshop is scheduled right after FPSAC 2021 (July 5-9, 2021).

February 18, 2020, Visit to Japan from mid-April to mid-May, 2020 and will be based in Waseda University, Tokyo (host: Satoshi Murai). I also plan to visit Osaka and Kyoto. (March 30, 2020: has to be postponed because of the COVID 19 situation)

January 27, 2020, Obtained a MATRICS grant from SERB for the next three years (http://www.serb.gov.in/matrix.php).

January 27, 2020, I will be at ICTS 8-14 February for the workshop (https://www.icts.res.in/discussion-meeting/mbrs2020)

November 11, 2019, I am in the organising committee of FPSAC (http://fpsac.org/confs/fpsac-2021/) to be held in IISc, Bangalore, July 5-21, 2021.

May 26, 2019, I will be in CIRM, Marseille June 17-28, 2019 and then visit Ecole Polytechnique, Paris July 1-3, MPI-MIS, Leipzig July 3-7.

March 13, 2019, Spencer Backman, Ye Luo and me will visit CIRM, Marseille on a research in pairs from June 17-28, 2019.

November 30, 2018, Attended the 84th annual meeting of the Indian Mathematical Society in Jammu and Kashmir. Spoke at the session “Commutative Algebra and Algebraic Geometry”.

November 4, 2018, Attended the 84th annual meeting of the Indian Academy of Sciences in Varanasi.

July 4, 2018, Have been selected as an associate of the Indian Academy of Sciences .

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.


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



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.

  • Commutative Algebra of Brill-Noether Theory on Graphs:

We study Brill-Noether theory on graphs from a commutative algebra point of view. We define modules on affine semigroup rings associated to graphs whose Hilbert series captures information related to special divisors on graphs.

  • Tutte Short Exact Sequences Associated to Graphs:

We construct short exact sequences corresponding to Tutte deletion-contraction recurrences.

Other recent projects:

  • 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.

  • 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













2015 and Before:



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.


Some past travel