Leonid Ryvkin

Mathematician · Maître de conférences in Université Claude Bernard Lyon 1

I am a mathematician working in differential geometry and mathematical physics. My research interests include multisymplectic geometry, singular foliations, simplicial and graded manifolds. I am a Maître de conférences (Assistant/ Associate professor) at the Institut Camille Jordan of the University Claude Bernard Lyon 1.

In Fall 2023, I will be one of the lecturers of the International masters program in mathematical Physics here in Lyon. It is an M2 program, i.e. intended for students who have already completed their first master year.

Publications and Preprints

Scientific publications

1. On Dirac structures admitting a variational approach, accepted in Mathematics and Mechanics of Complex Systems, 2022 with Oscar Cosserat, Camille Laurent-Gengoux, Alexei Kotov and Vladimir Salnikov, arXiv:2109.00313
2. The holonomy of a singular leaf, with Camille Laurent-Gengoux, 2021, Selecta Mathematica, also available on the arXiv:1912.05286.
3. The L-infinity-algebra of a symplectic manifold, with Bas Janssens and Cornelia Vizman, 2021, the Pacific Journal of Mathematics, available on the arXiv:2012.03836.
4. The neighbourhood of a singular leaf, with Camille Laurent-Gengoux, 2021, Journal de l'École polytechnique.
5. On the extension problem for weak moment maps, with Leyli Mammadova, 2021, Homology, Homotopy and Applications, also available on the arXiv:2001.00264.
6. Multisymplectic actions of compact Lie groups on spheres, with Antonio Michele Miti, 2020, Journal of Symplectic Geometry.
7. An invitation to multisymplectic geometry, with Tilmann Wurzbacher, Journal of Geometry and Physics, 2019. Also on the arXiv.
8. Conserved quantities on multisymplectic manifolds, with Tilmann Wurzbacher and Marco Zambon, Journal of the Australian Mathematical Society, 2019. Also on the arXiv.
9. Observables and Symmetries of n-Plectic Manifolds , BestMasters. Springer Spektrum Verlag, Wiesbaden, 2016.
10. Existence and unicity of co-moments in multisymplectic geometry. with Tilmann Wurzbacher, Journal of Differential Geometry and Applications, 2015. Also on the arXiv.

Preprints

• New paper: Reduction of L-infinity-algebras of observables on multisymplectic manifolds, 2022, with Casey Blacker and Antonio Miti, arXiv:2206.03137

Events and Activities

2022

• From 07. to 09. December 2022, I organized the Atelier on higher structures in differential geometry in Lyon.
• On the 12. October 2022, I gave a short talk at the meeting of the PSPM group at the Institut Camille Jordan in Lyon.
• On the 10. October 2022, I spoke in the virtual SFARS seminar.
• On the 30. September 2022, I gave a talk at the mathematical physics seminar at the Institut Camille Jordan in Lyon.
• In August 2022, I gave a talk at the thematic programme on Higher Structures and Field Theory at the ESI in Vienna. A youtube-recording of the talk can be accessed here.
• On June 09. 2022, I gave a talk at the conference on Noncommutative Geometry and Higher Structures in Scalea, Italy.
• On March 15. 2022, I gave a talk at the Göttingen-Würzburg Geometry day.
• From Februaray 28. 2022 to March 04. 2022, Jérémy Mougel, Thomas Schick and I are organizing a winter school on Foliations, Pseudodifferential operators and groupoids.
• In February 2022, I gave an introductory lecture on Foliations, Lie groupoids and Lie algebroids in Göttingen.

Puzzles and hunts

Scavenger Hunt

Alfonso Garmendia and I have created a scavenger hunt for the participants of the SFARS seminar, that we organize. It is still available here, the access code to start the hunt is "Request Mission".

A few elementary riddles and excercises

I know some of the following from Benjamin Böhme, who in turn credits Wim Martens. If you know a source I should refer to for some of them, please let me know.

• Subsets: Let $$n\in \mathbb N$$ be a natural number. Consider an $$n+1$$-element subset $$A$$ of $$\{1,...,2n\}$$. Show that $$A$$ contains two distinct numbers $$a,b$$, such that $$a$$ divides $$b$$.

• Checkerboard: Consider a checkerboard of $$n\times n$$ fields. Some fields contain pieces, others don't. Now put pieces on every field that touches (through its edges) at least two fields already containing pieces. Repeat the process until no further pieces are added. What is the minimal number of pieces on the field at the beginning of the procedure, if every field contains a piece at the end? (And why?)

• Hexagonal checkerboard How is the situation for a hexagonal board with sides of 6 fields, and hexagonal fields, if we need 3 neighbors of a field to already carry pieces for it to also get a piece? (communicated by Anne Marlen Prepeneit)

• Coins: $$2n$$ coins lie in a line on the table. Alice and Bob take turns in taking a coin away from the bounday of the row. Each coin is labelled with a value. In the end, after all coins are picked, the person who has taken coins with the larger value wins. In case of a draw Alice, who also takes the first turn, wins. Does someone have a winning strategy, if yes how does the strategy look?

• Burning cords: You have two cords (or fuses), that each take exactly 60 seconds to burn. You don't know whether they burn down at a constant speed and whether they burn down in the same way. How can you use them to measure exactly 45 seconds?

• Poisoned chocolate: ALice and Bob have an $$n\times m$$ chocolate bar. The top left corner piece is poisoned. Starting with Alice, they take turns in picking a point in the chocolate bar and eating all chococolate pieces, that are right of this point and all that are below this point. They have to choose in a way that makes them eat at least one piece of chocolate. The person eating the poisoned piece loses. Does one of them have a winning strategy? If yes, who is it? Click for an example game in which Bob loses.

• Poisoned chocolate (addendum): Does the answer change if the chocolate is infinitely big? (communicated by: Anne Marlen Prepeneit)

More riddles and excercises

• Pirates: A group of a thousand pirates earns 1000 pieces of gold. They have a very complicated gold distribution procedure, your task will be to find out how the gold will be distributed. First of all, all gold pieces are identically large, individual gold pieces can not be split and the pirates are totally hierarchically ordered. Pirate 1 is more important than pirate 2, pirate 2 is more important than pirate 3 etc. Their distribution procedure works as follows:

1. The most important alive pirate proposes a distribution.
2. The alive pirates vote whether they accept this distribution, if more than half of them are against it, the most important alive pirate is killed and they restart the procedure. Otherwise, the proposed distribution is accepted and the gold is distributed.

All pirates behave perfectly rationally (other than choosing to be a pirate)m know that all other pirates also behave rationally and have the following preferences:

1. From two options a pirate would always choose the one, where she survives.
2. If she survives in both cases, she prefers the options where she gets more gold.
3. If there are two cases with survival and equal amount of gold, she chooses the one where more other pirates die.

So, how will the gold be distributed among the pirates and how many of them will survive?

• An unknown square: We are interested in a square with (unknown) corners $$A,B,C,D$$ and unknown side length $$a$$. Assume, that for some point $$P$$, we know the distance between $$A$$ and $$P$$, the distance between $$B$$ and $$P$$, and the distance between $$C$$ and $$P$$. Can we determine $$a$$ from this data? (There is a beautiful solution using straightedge and compass).

Even More riddles and excercises

• Prisoners' hats: A group of 100 prisoners discovers that it will be executed (or freed) on the next day. They will have to stand in a row on a staircase, such that each prisoner sees all the prisoners in front of them but none behind them. Each prisoner will wear a hat, which is either black or white. From back to front, the henchman will ask each prisoner "What color is your hat?" If the answer is correct, then the prisoner is free, otherwise he/she is heheaded. All the remaining prisoners hear the question and answer and also hear whether the person leaves or is beheaded. Before the procedure, the prisoners have time to discuss and agree on a strategy. Assuming a reasonable level of solidarity among them, what should their strategy be and how many would be sure to survive? (communicated by: Anna Marklová)

• Toasting coordination: A group of people are standing in a circle and want to toast (let their glasses of wine touch). It is considered impolite for the glasses to cross. Assuming that only 2 glasses are allowed to touch at a time (during one "round") and that the people don't change places and don't toast behind each others backs, how many rounds of toasting are necessary so everyone has toasted with everyone? (communicated by: Oscar Cosserat)

• Prime Sudoku: Can you fill a Sudoku field with prime numbers, so all rows, columns and the 9 3x3-subfields contain a number at most once and sum up to prime numbers? (Additional question to which I do not know the answer: Can you fill it completely with distinct numbers). (proposed by Anna Marklová)

• Semi-integer rectangles Take a rectangle which is decomposed into smaller rectangles, which all have at least one integer side-length. Must the original rectangle also have an integer side length? (communicated by: Anne Marlen Prepeneit)

• Monotony In whatever order one puts the numbers $$1,...,n^2+1$$, there will always be an $$n$$-element subsequence which is monotonous.

Contact

Leonid Ryvkin
Leonid@ryvkin.eu

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