The Critical 2d Stochastic Heat Flow arises as a non-trivial solution
of the Stochastic Heat Equation (SHE) at the critical dimension 2 and at a phase transition point.
It is a log-correlated field which is neither Gaussian nor a Gaussian Multiplicative Chaos.
We will review the phase transition of the 2d SHE, describe the main points of the construction of the Critical 2d SHF
and outline some of its features and related questions. Based on joint works with Francesco Caravenna and Rongfeng Sun.

In this talk I will consider properties of the disordered elastic manifold, describing an N-dimensional field u(x) defined for sites x of a d-dimensional lattice of linear size L. This prototypical model is used to describe interfaces in a wide range of physical systems [1]. I will consider properties of the ground-state energy for this model whose optimal configuration u_0(x) results from a compromise between the disorder which tend to favour sharp variations of the field and elastic interactions that smoothen them. I will study in particular the limit of large N>>1 and finite d which has been studied extensively in the physics literature (notably using the replica approach) [1,2] and has recently been considered in a series of paper by Ben Arous and Kivimae [3,4]. For this model, we compute exactly the large deviation function of the ground-state energy E_0, showing that it displays replica-symmetry breaking transitions. As an interesting outcome of this study, we show analytically the validity of the scaling law conjectured by Mezard and Parisi [2] for the variance of the ground-state energy. The latter relates the exponent of the variance Var(E_0)\sim L^{2\theta} such that \theta=2\zeta+d-2 with \zeta the exponent characterising the transverse fluctuations of the optimal configuration u_0(x), i.e.  (u_0(x)-u_0(x+y))^2\sim |y|^{2\zeta}. This work is done in collaboration with Y.V. Fyodorov (KCL) and P. Le Doussal (LPENS, CNRS).

[1] Giamarchi, T., & Le Doussal, P. (1998). Statics and dynamics of disordered elastic systems. In Spin glasses and random fields (pp. 321-356).

[2] Mézard, M., & Parisi, G. (1991). Replica field theory for random manifolds. Journal de Physique I, 1(6), 809-836.

[3] Ben Arous, G., & Kivimae, P. (2024). The Free Energy of the Elastic Manifold. arXiv preprint arXiv:2410.19094.

[4] Ben Arous, G., & Kivimae, P. (2024). The larkin mass and replica symmetry breaking in the elastic manifold. arXiv preprint arXiv:2410.22601.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

The Ciarlet definition of a finite element has been core to our understanding of the finite element method since its inception. It has proved particularly useful in structuring the implementation of finite element software. However, the definition does not encapsulate all the details required to uniquely implement an element, meaning each user of the definition (whether a researcher or software package) must make further mathematical assumptions to produce a working system. 

The talk presents a new definition built on Ciarlet’s that addresses these concerns. The novel definition forms the core of a new piece of software in development, FUSE, which allows the users to consider the choice of finite element as part of the data they are working with. This is a new implementation strategy among finite element software packages, and we will discuss some potential benefits of the development.

A number of algorithms are now available---including Halko-Martinsson-Tropp, interpolative decomposition, CUR, generalized Nystrom, and QR with column pivoting---for computing a low-rank approximation of matrices. Some methods come with extremely strong guarantees, while others may fail with nonnegligible probability. We present methods for efficiently estimating the error of the approximation for a specific instantiation of the methods. Such certificate allows us to execute "responsibly reckless" algorithms, wherein one tries a fast, but potentially unstable, algorithm, to obtain a potential solution; the quality of the solution is then assessed in a reliable fashion, and remedied if necessary. This is joint work with Gunnar Martinsson. 

Time permitting, I will ramble about other topics in Randomised NLA.