Journal
I am happy to advertise a new open-access journal "Higher structures" where I serve on the editorial board. Its website is accessible via the URL https://journals.mq.edu.au/index.php/higher_structures/. If your research is related to the scope of this journal, please do consider publishing a paper there.
Book
In collaboration with Murray Bremner, I wrote a book "Algebraic Operads: An Algorithmic Companion".
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Table of Contents: 1. Normal forms for vectors and univariate polynomials 2. Noncommutative associative algebras 3. Nonsymmetric operads 4. Twisted associative algebras and shuffle algebras 5. Symmetric operads and shuffle operads 6. Operadic homological algebra and Gröbner bases 7. Commutative Gröbner bases 8. Linear algebra over polynomial rings 9. Case study of nonsymmetric binary cubic operads 10. Case study of nonsymmetric ternary quadratic operads Appendix. Maple code for Buchberger's algorithm |
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Papers and preprints
Various recent and forthcoming activities
- I will organise (together with Eduardo Marcos) the thematic section Homological methods" at XXIII Latin American Algebra Colloquium which will take place in Mexico City on August 5-9, 2019.
- I organised (together with Vincent Gélinas and Andrea Solotar) a workshop "Homotopy meets homology"; it took place in Dublin on May 27-31, 2019.
- I was one of the local organisers of "The 14th William Rowan Hamilton Geometry and Topology Workshop" "Group actions and ergodic theory in geometry and topology"; it took place in Dublin on July 30-August 3, 2018.
- I organised (together with Victoria Lebed) a workshop "Geometry and combinatorics of associativity"; it took place in Dublin on October 23-27, 2017.
- I was one of the organisers of the third edition of the conference "Non-associative algebras and Lie theory"; it took place in Mexico City on February 27 - March 3, 2017.
- I was one of the local organisers of "The 12th William Rowan Hamilton Geometry and Topology Workshop" "Low-dimensional topology, complex and symplectic geometry"; it took place in Dublin on August 22-26, 2016.
- I was on the program committee of the workshop Higher-Dimensional Rewriting and Applications; it took take place in Oporto, Portugal, on June 25-26, 2016.
- I was one of the local organisers of "The 11th William Rowan Hamilton Geometry and Topology Workshop" "Connections between three and four manifolds"; it took place in Dublin on August 25-29, 2015.
- I was on the program committee of the workshop Higher-Dimensional Rewriting and Applications, to take place in Warsaw, Poland, on June 28, 2015.
- I was one of the local organisers of "The 10th William Rowan Hamilton Geometry and Topology Workshop" "Homological invariants in low-dimensional topology and geometry"; it took place in Dublin on August 26-30, 2014.
- I organised (together with Uli Kraehmer) a mini-workshop Batalin-Vilkovisky algebras, operads, and Hopf algebroids, it took place in Mathematisches Forschungsinstitut Oberwolfach on February 24-28, 2014.
- I was one of the organisers of a conference dedicated to Borya Feigin on the occasion of his 60th anniversary, it took place in Moscow on December 13-19, 2013. Video recordings of talks at that conference are available here.
- I was one of the local organisers of "The 9th William Rowan Hamilton Geometry and Topology Workshop" "Geometry and Groups after Thurston"; it took place in Dublin on August 27-31, 2013.
- I organised (together with Norbert Poncin) a workshop "Higher algebras and Lie-infinity homotopy theory" at the University of Luxembourg, it took place on June 25-28, 2013.
- In February 2013 I gave a mini-course "Operads, shuffles, and Gröbner bases" at the meeting Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms (in honour of Jean-Louis Loday), at CIMAT, Guanajuato, México.
- In February 2012 I gave a mini-course "Rewriting methods for associative algebras and operads" at Algebra and computation, one of the workshops of the thematic programme "Logic and interactions 2012" held at CIRM Luminy, Marseille.
Upcoming travels
- July 20-24, 2020: Workshop "Quantum math, singularities and applications", OIST, Japan
- August 5-9, 2019: XXIII Latin American Algebra Colloquium, Mexico City
Some research talks
- "Graded algebras and the Lagrange inversion formula", a talk at the workshop "Graded Algebras, Geometry and related Topics" celebrating the 60th birthday of Adolfo Sánchez Valenzuela, Merida, November 2016:YouTube Video.
- "Noncommutative M0,n+1", a talk at Samuel Gitler Memorial Conference, CINVESTAV, September 2016: YouTube Video.
- "Shuffle operads", Introductory Workshop of the GDO programme: video of the talk on the Newton Institute website.
- "Homotopical meaning of the Givental group", Higher Structure 2013 Workshop of the GDO programme: video of the talk on the Newton Institute website.
- "Hierarchies of identities for differential operators and moduli spaces of curves", Algebra Seminar, University Lyon 1, February 9, 2012. Beamer presentation: PDF.
- "Filtered distributive laws", Conference "Operads and rewriting", Lyon, France, November 3, 2011. Beamer presentation: PDF.
- "Shuffle operads", Conference "New development in noncommutative algebra and its applications", Isle of Skye, Scotland, June 30, 2011. Beamer presentation: PDF.
- "Compatible associative products and trees", LMS--ARTIN workshop "Integrable systems: algebraic aspects", University of Glasgow, April 23, 2010. Beamer presentation: PDF.
- "Anick resolutions, shuffle algebras, and consecutive pattern avoidance", British Mathematics Colloquium, Edinburgh, April 6, 2010. Beamer presentation: PDF
Research supervision and mentorship
In 2017-20, I am supervising the PhD thesis of Pedro Tamaroff, who works on a number of topics in homological and homotopical algebra, his current research outputs are this paper he wrote himself, this paper we wrote together, and this paper he wrote himself.
In 2019/20, I was a mentor of Vincent Gélinas during his time as a postdoc at the Hamilton Mathematics Institute.
In 2017-19, I supervised the Masters thesis of Norah Alghamdi who worked on Gröbner bases for nonsymmetric operads and their applications; her work resulted in a thesis "The associative filtration of the dendriform operad".
In 2016-18, I was a mentor of Victoria Lebed during her time as a postdoc at the Hamilton Mathematics Institute; she is now a maître de conférences at Université de Caen - Normandie.
In 2016, I was a mentor of José Moreno during his exchange visit to Trinity College Dublin as a part of his PhD studies in Malaga. José then was a postdoc at MPIM Bonn, and in September 2019 is moving to Trinity College Dublin to start a competitive research fellowship awarded by the Irish Research Council.
In 2014/15, I supervised the Masters thesis of Soutrik Roy Chowdhury who studied an important computational method of homological algebra, the Anick's resolution.
In 2013/14, I supervised the Masters thesis of Lucas Mason-Brown who studied invariant (coordinate-independent) operations that map various differential-geometric constructions to one another.
For prospective postgraduate students
This webpage and various materials linked here (my papers, slides and videos of my talks etc.) should give you some ideas about the range of my research interests, as well as my style of doing maths. At any time, I have suggestions for PhD projects in a few areas of my immediate interest, however, if you have some ideas of your own for topics to work on, I would be happy to discuss these ideas, and possibly either serve as a supervisor for such a project or suggest a more suitable supervisor.
Doing research in maths is great fun, and I assume you share this opinion if you are reading this. However, it does not hurt anyone to bring a bit of structure in that fun: while typical problems you encounter in tutorials and exams during your undergraduate studies are designed in a way that one can finish them in 15-20 minutes, actual research questions can take days, weeks, months, and in some famous cases years of work. Approaching them without any system would not thus take one too far. There are some good examples of guidelines towards systematic mathematical reasoning (suggestions will depend a lot on who you ask for guidelines). I would recommend to have a glance at an excellent essay "Why is it plausible?" by Barry Mazur, and then maybe even follow to one of the sources he references, "Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics" by George Polya.
Ideally, you will have completed an undergraduate degree in maths that includes some research experience, such as a final year project, or a summer research internship, but in principle previous research experience, especially for a master student, is not mandatory. I strongly believe that apart from interest in maths, the only two qualities that someone aiming to do research in pure mathematics should develop at an early stage is, first, being prepared to work hard, and, second, not being discouraged too easily: unlike many other research areas, in pure maths we work with concepts, and this means that on many occasions working on a problem would continue without much tangible progress for quite a while (unlike doing experiments in a lab, let us say), and this can be seriously frustrating. However, over time your knowledge of a problem grows, and at times you will be rewarded by that unique feeling of having found something new and cool, and this kinda makes it all worth your while. Keep that in mind, and work hard, and we shall get along well.
On a more practical side: currently, the situation with funding for Masters and PhD in Ireland is tricky, but there still are quite a few ways for a motivated student to get funded. In any case, if you are considering me as a prospective supervisor for your PhD or Master thesis, please let me know at your earliest convenience; sometimes knowing that a good candidate for a postgraduate research position exists is crucial for securing funding that is suddenly made available.
My favourite instances of mathematical writing
There are three pieces of mathematical writing which I like a lot, both because of lovely style of writing and because of nearly infinite potential for discovering new amazing things about maths while reading them, regardless of how many times you read them before. These are Ian Macdonald's book called "Symmetric functions and Hall polynomials" (2nd edition) where every other fact mentioned in examples and exercises leads to a separate exciting mathematical story, Daniel Quillen's article "On the (co-)homology of commutative rings", featuring the most fantastic style of maths writing, and Victor Ufnarovskij's survey "Combinatorial and asymptotic methods in algebra" which is sort of similar to Macdonald's book but almost without theorems and proofs between examples and exercises.
School of maths summer internships
Recently, School of Maths has been offering several summer research internships to undergrads studying maths at Trinity College; schedules and some minimal info on projects can be seen here. Contact me for any further questions you may have.