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

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

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.

# 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
- Syzygies over Polytope Semirings:
- Tropical Graph Curves:
- Combinatorial Brill-Noether for Graphs via Commutative Algebra:
- Commutative Algebra of Generalised Frobenius Numbers:
- Commutative Algebra of Generalised Frobenius Numbers, Madhusudan Manjunath and Ben Smith, March 2017.
- Syzygies over the Polytope Semiring, Madhusudan Manjunath, accepted at the Journal of the London Mathematical Society, 2017.
- Tropical Graph Curves, Madhusudan Manjunath, March 2016.
- Embeddings and immersions of tropical curves, Dustin Cartwright, Andrew Dudzik, Madhusudan Manjunath, Yuan Yao, Collectanea Mathematica, Volume 67, Issue 1, pp 1-19, 2016.
- Smoothing of Limit Linear Series of Rank One on

Saturated Metrized Complexes of Algebraic Curves , Ye Luo and Madhusudan Manjunath, accepted at the Canadian Journal of Mathematics (CJM), 2017. - Explicit Deformation of Lattice Ideals via Chip Firing on Directed Graphs, Spencer Backman and Madhusudan Manjunath, Journal of Algebraic Combinatorics, Volume 42, Issue 4, pp 1097-1110, 2015.
- Minimal Free Resolutions of G-Parking Function Ideal and the Toppling Ideal, Madhusudan Manjunath, Frank-Olaf Schreyer and John Wilmes, Transactions of the American Mathematical Society, Volume 367, pp 2853-2874, 2015.
- Monomial, Binomials and Riemann-Roch,

Madhusudan Manjunath and Bernd Sturmfels, Journal of Algebraic Combinatorics, Volume 37, Issue 4, pp 737-756, 2013. - The Laplacian Lattice of a Graph under a Simplicial Distance Function,

Madhusudan Manjunath, European Journal of Combinatorics, Volume 34,

Issue 6, pp 1051–1070, 2013. - The Rank of a Divisor on a Finite graph: Geometry and Computation., Madhusudan Manjunath, 2011 available in the arxiv at arXiv:1111.7251v1.
- Riemann Roch for sublattices of the Root Lattice An, Omid Amini and Madhusudan Manjunath, Electronic Journal of Combinatorics 17 (2010), Research Paper 124.
- Applications of Dimensionality Reduction and Exponential Sums to Graph Automorphism, Madhusudan Manjunath and Vikram Sharma, Theoretical Computer Science, Volume 412, Issue 29, pp 3639-3649, 2011.
- Approximate Counting of Cycles in Streams, Madhusudan Manjunath, Kurt Mehlhorn, Konstantinos Panagiotou and He Sun, the proceedings of the European Symposium on Algorithms (ESA), 2011.
- Minimizing Absolute Gaussian Curvature over Meshes, Joachim Giesen and Madhusudan Manjunath, Computational Geometry, Dagstuhl Seminar Proceedings 2009.
- Riemann-Roch Theory for Sublattices of the Root Lattice An, Graph Automorphisms and Counting Cycles in Graphs, Madhusudan Manjunath, Dissertation, Saarland University
- 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.

My google scholar page gives an impression of my papers. Recently, I have been working the following topics:

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.

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.

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.

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. This joint work with B. Smith.

## (Pre-)Publications

2017

2016

June

March

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