L6

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.

Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown, and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X. 

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally has the structure of an algebraic variety, which we denote by X(B). In ongoing work with Tijs Buggenhout and Floris Vermeulen, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components  of X(B) of largest possible dimension​ analogously to dimension growth bounds. We do this by developing a geometric determinant method, and by using results on rational points on curves over function fields. 

Joint with Tijs Buggenhout and Floris Vermeulen.

A classical approach to studying the topology of a manifold is through the analysis of its submanifolds. The realm of 3-manifolds is particularly rich and diverse, and we aim to explore the complexity of surfaces within a given 3-manifold. After reviewing the fundamental definitions of the Thurston norm, we will present a constructive method for computing it on Seifert fibered manifolds and extend this approach to graph manifolds. Finally, we will outline which norms can be realized as the Thurston norm of some graph manifold and examine their key properties.

The hyperbolic plane and its higher-dimensional analogues are well-known
objects. They belong to a larger class of spaces, called rank-one
symmetric spaces, which include not only the hyperbolic spaces but also
their complex and quaternionic counterparts, and the octonionic
hyperbolic plane. By a result of Pansu, two of these families exhibit
strong rigidity properties with respect to their self-quasiisometries:
any self-quasiisometry of a quaternionic hyperbolic space or the
octonionic hyperbolic plane is at uniformly bounded distance from an
isometry. The goal of this talk is to give an overview of the rank-one
symmetric spaces and the tools used to prove Pansu's rigidity theorem,
such as the subRiemannian structure of their visual boundaries and the
analysis of quasiconformal maps.

Originally posed in the 1950s, the Hadwiger-Nelson problem interrogates the ‘chromatic number of the plane’ via an infinite unit-distance graph. This question remains open today, known only to be 5,6, or 7. We may ask the same question of the hyperbolic plane; there the lack of homogeneous dilations leads to unique behaviour for each length scale d. This variance leads to other questions: is the d-chromatic number finite for all d>0? How does the d-chromatic number behave as d increases/decreases? In this talk, I will provide a summary of existing methods and results, before discussing improved bounds through the consideration of semi-regular tilings of the hyperbolic plane.

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.