Seminars from until

A classical result of Plotkin (IRE Transactions on Information Theory, 1960) establishes that over binary alphabets, positive rate codes cannot have minimum distance greater than $1/2$, and thus can uniquely correct at most a $1/4$ fraction of bit-flip errors. It is additionally known that if one insists on constructing a code correcting a $1/4+ε$ fraction of errors (for small $ε>0)$, then this code can have size at most $O(1/ε)$, and that this is tight.

If one moves to list-decoding binary codes with list-size $L$ — that is, the decoder may output up to $L$ guesses for the transmitted message, as long as one of the guesses is correct — Blinovsky (Problems of Information Transmission, 1986) computed a similar threshold $p_L$. Additionally, his argument establishes that any $(p_L+ε, L)$-list-decodable code must have size at most $O_{ε,L}(1)$ — a constant, but with a (massive) dependence on $ε$. Later, Alon, Bukh and Polyanskiy (IEEE Transactions on Information Theory, 2018) showed that for odd $L$, such codes have size $O_L(1/ε)$ (as with Plotkin’s bound), but already with $L=2$ such codes of size $O(1/ε^{3/2})$ exist.

In this talk, we will generalize all of these results to any (constant) alphabet size $q > 2$. A crucial tool in the proof is the concept of Schur convexity, which in certain cases allows one to show that the optimizing value for a function on a space of distributions is the uniform distribution.

We derive a scaling limit in law as N tends to infinity for the cover time by a simple random walk of the subgraph of the square lattice obtained by discretizing an N-scale blow up of a planar domain and adding a wired boundary. The limiting distribution is that of a Gumbel random variable shifted by an independent (random) quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the rescaled location of the last visited vertex by the walk. Here the limit turns out to be precisely the (expected) critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well-known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field and, in particular, with that of the discrete Gaussian Free Field conditioned to have zero average. Joint work with Oren Louidor (Technion).

Na terceira sessão serão abordados os seguintes tópicos de análise $p$-ádica:

• Breve sumário sobre a topologia $p$-ádica.
• Breve resumo sobre séries $p$-ádicas.
• Funções continuas e diferenciabilidade.

We consider the one-dimensional stirring process on the segment $\{−N , . . . , N \}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of fixed and finite size. In this talk, I will present the non-equilibrium fluctuations of the system when the initial configuration is given by a product measure associated with a smooth macroscopic profile. In this regime, the fluctuations are described by an Ornstein–Uhlenbeck process driven by the Laplacian and gradient operators, with boundary conditions determined by the hydrodynamic profile. A central step in the analysis is the derivation of sharp bounds for space and space–time v-functions of arbitrary degree associated with the centered occupation variables. In particular, we prove that the v-functions of degree 2 and 3 are of order $N^{−1}$, while those of degree at least 4 are of order $N^{ −1−\zeta}$ for some $\zeta> 0$.

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenaria. In particular, alternating patterns between synchronization and desynchronization behaviors are given by studying the asymptotics of the Markov perturbed stationary distributions. This talk is based on joint works with Arno Berger, Wen Huang, Hong Qian, Felix X.-F. Ye, and Yingfei Yi.

The development of the theory of martingale Hardy spaces has been strongly influenced by the classical theory. Because of this it is inevitable to compare results of this theory to those on classical Hardy spaces and during the decades this new direction was developing in this way and many similarities between these theories have already been found. But nowadays a lot of new results were obtained in the theory of martingale Hardy spaces which are new in classical case.

This lecture is devoted to review theory of martingale Hardy spaces. In particular, we give atomic decomposition of these spaces and show how this theorem simplify the proofs of boundedness of any sub-linear operators on these spaces. For the illustration of bounded operators on the martingale Hardy spaces we define Vilenkin groups and their characters which are called Vilenkin functions. We also define Fourier series with respect to Vilenkin system and maximal operators related to these operators. Moreover, we define modulus of continuity in martingale Hardy spaces and derive necessary and sufficient conditions in the terms of modulus of continuity such that partial sums with respect to one Vilenkin-Fourier series converge in $H^p$ norm. We also investigate classical Hardy spaces, modulus of continuity in these spaces and present one to one analogues of these results for partial sums in this classical case.

At the end of this presentation we state some open problem how to generate these two similar results for Hardy spaces on some locally compact Abelian groups and partial sums with respect to character functions of these groups. Moreover, we also state similar open problems for variable martingale Hardy spaces. Finally, we will choose opposite approach and will try to state and investigate results for partial sums of the two-dimensional Fourier series in martingale Hardy spaces for classical case of Hardy spaces.

Diffusion probabilistic models have become the state-of-the-art tool in generative methods, used to generate high-resolution samples from very high-dimension distributions (e.g. images). Although very effective, they suffer some drawbacks:

1. as opposed to variational encoders, the dimension of the problem remains high during the generation process and
2. they can be prone to memorization of the training dataset.

In this talk, we first provide an introduction to generative modeling, with a focus on diffusion models from the point of view of stochastic PDEs. Then, we introduce a kernel-smoothed empirical score and study the bias-variance of this estimator. We find improved bounds on the KL-divergence between a true measure and an approximate measure generated by using the smoothed empirical score. This score estimator leads to less memorization and better generalization. We demonstrate these findings on synthetic and real datasets, combining diffusion models with variational encoders to reduce the dimensionality of the problem.

A spectral gap bound is an essential ingredient in the rigorous derivation of the macroscopic equations from microscopic stochastic dynamics. This work establishes a spectral gap of order $\ell^{-2}$ for a special class of the simple symmetric mass migration misanthrope process on a finite box $\Lambda_\ell$ with reflecting boundaries. The model allows for cooperative jumps of blocks of any size up to the origin site occupancy, with misanthrope rates governed by the occupation numbers of the departure and arrival sites. By imposing and subsequently exploiting a linear structure in the expected total particle jump rates, using results from the theory of Ollivier-Ricci curvature and classical results on symmetric random walks, we extend the framework of Gobron and Saada [GS10] to accommodate these large-block dynamics.

[GS10] T. Gobron and E. Saada, Couplings, attractiveness and hydrodynamics for conservative particle systems, Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1132–1177.

In this talk we examine some aspects of the classical-quantum correspondence induced by symplectic maps and metaplectic operators. We first recall the notion of the metaplectic group, the double covering of the symplectic group.

We then extend this construction to the complex setting and define the metaplectic semigroup associated with the semigroup of positive complex symplectic linear maps. In this context, we review the various definitions appearing in the literature, notably those due to M. Brunet and P. Kramer, L. Hörmander, and R. Howe.

We finally establish several properties of the metaplectic semigroup, with particular emphasis on applications to time-frequency analysis and to evolution equations with complex quadratic Hamiltonians.

This talk is based on joint work with G. Giacchi, M. Malagutti, A. Parmeggiani and L. Rodino.

Let us consider two notions of concentration for homogeneous polynomials in d complex variables on the unit sphere: a local notion measuring the fraction of the $L^2$-norm supported on a measurable subset and a global notion given by the generalized Wehrl entropy. Lieb and Solovej proved that the extremizers in both cases are monomials up to a unitary rotation. Their result generalizes the one by Lieb in 1978 on the Wehrl entropy conjecture for coherent states in representations of the Heisenberg group to symmetric representations of the groups $\operatorname{SU}(d)$.

In this talk, we will focus on the stability of the previous inequalities. Namely, if the concentration is close to the optimal one, we will quantify how close the polynomial is to the extremizers. This is obtained in full generality in the case $d=2$, while in the case of higher dimensions restrictions on the size of the subset or on the degree of the polynomials arise. We will finally recover analogous stability results in the Bargmann–Fock space.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).