Scientific

The Grothendieck Lp problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit Lp ball. The p=2 case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for p=∞ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general p and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.

Divided power algebras were defined by H. Cartan in 1954 to study the homology of Eilenberg-MacLane spaces. They are commutative algebras endowed, for each integer n, with an additional monomial operation. Over a field of characteristic 0, this operation corresponds to taking each element to its n-th power divided by factorial n. This definition does not make sense if the base field is of prime characteristic, yet, Cartan's definition of divided power algebra applies in this situation as well. The notion of divided power algebra over a field of prime characteristics allows us to describe algebraic structures that appear in homology and homotopical algebra, and has found applications in a wide array or mathematical domains, for instance in crystalline cohomology, and deformation theory.In this talk we will review the motivations for the definition of divided power algebra. We will start by recalling some constructions of algebraic invariants from topological spaces, and we will show that divided power algebras arise naturally in this setting. We will give the generalised definition of a divided power algebra, given by B. Fresse in 2000, using the theory of operads. Finally, we will give a complete characterisation for generalised divided power algebras in terms of monomial operations and relations.

Speaker Biography

Sacha Ikonicoff was born and raised in the Paris region in France. He obtained his mathematics license degree in 2014, and his pure mathematics master's degree in 2016, both from Université Paris 6 - Pierre & Marie Curie (now "Sorbonne Université"). While studying for his master's degree, Sacha got more and more interested in the subject of algebraic topology. His master's thesis, written under the direction of Muriel Livernet, concerns the divided power algebra structures that appear on the homotopy of simplicial algebras. Muriel Livernet then became Sacha Ikonicoff's PhD advisor at Université de Paris. Throughout the course of his PhD, Sacha continued to work in the domain of algebraic operads and divided power algebras, and obtained a full characterisation of these structures in his article "Divided power algebras over an operad", published in the Glasgow Mathematical Journal in 2019. He also developed an operadic theory for unstable modules over the Steenrod algebra in the article "Unstable algebra over an operad", published in Homology, Homotopy and Applications in 2021.

Sacha obtained his PhD, entitled "Level algebras and applications to algebraic topology" in 2019. He is now a PIMSCNRS postdoctoral scholar at the University of Calgary in Alberta, Canada.

(Joint work with Perla Sousi.) A sequence of Markov chains is said to exhibit cutoff if it exhibits an abrupt convergence to equilibrium. Loosely speaking, this means that the epsilon mixing time (i.e. the first time the worst-case total-variation distance from equilibrium is at most epsilon) is asymptotically independent of epsilon. An emerging paradigm is that cutoff is related to entropic concentration. In the case of random walks on random graphs G=(V,E), the entropic time can be defined as the time at which the entropy of random walk on some auxiliary graph (often the Benjamini-Schramm limit) is log |V|. Previous works established cutoff at the entropic time in the case of the configuration model. We consider a random graph model in which a random graph G'=(V,E') is obtained from a given graph G=(V,E) with an even number of vertices by picking a random perfect matching of the vertices, and adding an edge between each pair of matched vertices. We prove cutoff at the entropic time, provided G is of bounded degree and its connected components are of size at least 3. Previous works were restricted to the case that the random graph is locally tree-like.

We study the critical Ising model with free boundary conditions on finite domains in Zd
with d ≥ 4. Under the assumption, so far only proved completely for high d, that the critical infinite volume two-point function is of order |x-y|-(d-2) for large |x−y|, we prove the same is valid on large finite cubes with free boundary conditions, as long as x, y are not too close to the boundary. This confirms a numerical prediction in the physics literature by showing that the critical susceptibility in a finite domain of linear size L with free boundary conditions is of order L2 as L → ∞. We also prove that the scaling limit of the near-critical (small external field) Ising magnetization field with free boundary conditions is Gaussian with the same covariance as the critical scaling limit, and thus the correlations do not decay exponentially. This is very different from the situation in low d or the expected behavior in high d with bulk boundary conditions.

The Loewner energy for simple closed curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle. This intriguing class of curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and string theory, and has been studied since the eighties. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of math it touches on. I will also highlight how ideas from probability theory inspire new results on Weil-Petersson quasicircles and discuss further directions.