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Sousbie, Thierry - ColDICE: a parallel Vlasov-Poisson solver using moving adaptative simplicial tessellation
http://library.cirm-math.fr/Record.htm?record=19283291124910014739
In this talk, I will present ColDICE[1, 2], a publicly available parallel numerical solver designed to solve the Vlasov-Poisson equations in the cold case limit. The method is based on the representation of the phase-space sheet as a conforming, self-adaptive simplicial tessellation whose vertices follow the Lagrangian equations of motion. In this presentation, I will mainly focus on describing the underlying algorithm and its practical implementation, as well as showing a few practical examples demonstrating its capabilities.Sousbie, Thierry Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283291124910014739Angel, Omer - Bootstrap percolation on Erdos-Renyi graphs
http://library.cirm-math.fr/Record.htm?record=19283219124910014919
We consider bootstrap percolation on the Erdos-Renyi graph: given an initial infected set, a vertex becomes infected if it has at least $r$ infected neighbours. The graph is susceptible if there exists an initial set of size $r$ that infects the whole graph. We identify the critical threshold for susceptibility. We also analyse Bollobas's related graph-bootstrap percolation model.
Joint with Brett Kolesnik.Angel, Omer Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283219124910014919Rinot, Assaf - Distributive Aronszajn trees
http://library.cirm-math.fr/Record.htm?record=19283195124910013779
It is well-known that the statement "all $\aleph_1$-Aronszajn trees are special'' is consistent with ZFC (Baumgartner, Malitz, and Reinhardt), and even with ZFC+GCH (Jensen). In contrast, Ben-David and Shelah proved that, assuming GCH, for every singular cardinal $\lambda$: if there exists a $\lambda^+$-Aronszajn tree, then there exists a non-special one. Furthermore:
Theorem (Ben-David and Shelah, 1986) Assume GCH and that $\lambda$ is singular cardinal. If there exists a special $\lambda^+$-Aronszajn tree, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
This suggests that following stronger statement:
Conjecture. Assume GCH and that $\lambda$ is singular cardinal.
If there exists a $\lambda^+$-Aronszajn tree,
then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
The assumption that there exists a $\lambda^+$-Aronszajn tree is a very mild square-like hypothesis (that is, $\square(\lambda^+,\lambda)$).
In order to bloom a $\lambda$-distributive tree from it, there is a need for a toolbox, each tool taking an abstract square-like sequence and producing a sequence which is slightly better than the original one.
For this, we introduce the monoid of postprocessing functions and study how it acts on the class of abstract square sequences.
We establish that, assuming GCH, the monoid contains some very powerful functions. We also prove that the monoid is closed under various mixing operations.
This allows us to prove a theorem which is just one step away from verifying the conjecture:
Theorem 1. Assume GCH and that $\lambda$ is a singular cardinal.
If $\square(\lambda^+,<\lambda)$ holds, then there exists a $\lambda$-distributive $\lambda^+$-Aronszajn tree.
Another proof, involving a 5-steps chain of applications of postprocessing functions, is of the following theorem.
Theorem 2. Assume GCH. If $\lambda$ is a singular cardinal and $\square(\lambda^+)$ holds, then there exists a $\lambda^+$-Souslin tree which is coherent mod finite.
This is joint work with Ari Brodsky. See: http://assafrinot.com/paper/29Rinot, Assaf Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283195124910013779Tao, Terence - An integration approach to the Toeplitz square peg problem
http://library.cirm-math.fr/Record.htm?record=19283176124910013589
The Toeplitz square peg problem asks if every simple closed curve in the plane inscribes a square. This is known for sufficiently regular curves (e.g. polygons), but is open in general. We show that the answer is affirmative if the curve consists of two Lipschitz graphs of constant less than 1 using an integration by parts technique, and give some related problems which look more tractable.Tao, Terence Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283176124910013589Métayer, François - Homotopy theory of strict $\omega$-categories and its connections with homology of monoids - Lecture 1
http://library.cirm-math.fr/Record.htm?record=19283162124910013449
In the first part, we describe the canonical model structure on the category of strict $\omega$-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as $\omega$-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular $\omega$-category, this polygraphic point of view will lead us to an alternative definition of monoid homology, which happens to coincide with the usual one.Métayer, François Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283162124910013449Welschinger, Jean-Yves - Expected topology of a random subcomplex in a simplicial complex
http://library.cirm-math.fr/Record.htm?record=19283142124910013249
I will explain how to bound from above and below the expected Betti numbers of a random subcomplex in a simplicial complex and get asymptotic results under infinitely many barycentric subdivisions. This is a joint work with Nermin Salepci. It complements previous joint works with Damien Gayet on random topology.Welschinger, Jean-Yves Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283142124910013249Schrijver, Alexander - The partially disjoint paths problem
http://library.cirm-math.fr/Record.htm?record=19283126124910013089
The partially disjoint paths problem asks for paths $P_1, \ldots,P_k$ between given pairs of terminals, while certain pairs of paths $P_i$,$P_j$ are required to be disjoint. With the help of combinatorial group theory, we show that, for fixed $k$, this problem can be solved in polynomial time for planar directed graphs. We also discuss related problems. No specific foreknowledge is required.Schrijver, Alexander Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283126124910013089Khanin, Konstantin - Interview at CIRM: Konstantin Khanin
http://library.cirm-math.fr/Record.htm?record=19283966124910011489
Chaire Jean-Morlet - First Semester 2017: 'KPZ Universality and Directed Polymers'
Konstantin "Kostya" Mikhailovich Khanin is a Russian mathematician and physicist. Khanin received his PhD from the Landau Institute of Theoretical Physics in Moscow and continued working there as a research associate until 1994. Afterwards, he taught at Princeton University, at the Isaac Newton Institute in Cambridge, and at Heriot-Watt University before joining the faculty at the University of Toronto. Khanin was an invited speaker at the European Congress of Mathematics in Barcelona in 2000. He was a 2013 Simons Foundation Fellow. He held the Jean-Morlet Chair at the Centre International de Rencontres Mathématiques in 2017, and he is an Invited Speaker at the International Congress of Mathematicians in 2018 in Rio de Janeiro.Khanin, Konstantin Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283966124910011489Achdou, Yves - Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes
http://library.cirm-math.fr/Record.htm?record=19283932124910011149
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:
- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments
- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit
- convergence to classical or weak solutions can be proved
Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows
1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.
2. Monotone finite difference schemes
3. Examples of applications
4. Variational MFG and related algorithms for solving the discrete system of nonlinear equationsAchdou, Yves Post-editedhttp://library.cirm-math.fr/Record.htm?record=19283932124910011149Benjamini, Yoav - A review of challenges in high dimensional multiple inferences
http://library.cirm-math.fr/Record.htm?record=19282844124910000269
I shall classify current approaches to multiple inferences according to goals, and discuss the basic approaches being used. I shall then highlight a few challenges that await our attention : some are simple inequalities, others arise in particular applications.Benjamini, Yoav Post-editedhttp://library.cirm-math.fr/Record.htm?record=19282844124910000269