Abstract: We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

Abstract: We will first discuss the notion of stochastic integration with respect to infinite-dimensional semimartingales. Then we put it in the context of mathematical finance and consider the problem of pricing in large markets in a framework where the large market limits the small ones with finitely many traded assets. We show that this framework allows accommodating indifference pricing in stochastic utility settings and arbitrage-free pricing. Adopting a stochastic integration theory with respect to a sequence of semimartingales, we introduce the notion of indifference prices for the large post-limit market and establish their existence, uniqueness, and relation to arbitrage-free prices. These results rely on a theorem of independent interest on utility maximization with a random endowment in a large market that we state and prove first. Further, we provide approximation results for the indifference and arbitrage-free prices in the large market by those in small markets. In particular, our framework allows for pricing the asymptotically replicable claims, where we also show a consistency of the pricing methodologies and provide positive examples. This talk is based on the joint work with Pietro Siorpaes (Imperial College London).

Abstract: In the late '90s J. Cheeger introduced and started to study the notion of Lipschitz differentiability space (LDS).


A LDS is a metric measure space with an atlas of charts, whose values are in Euclidean spaces with uniformly bounded dimension, such that every real-valued Lipschitz function is almost everywhere differentiable in every chart.


As of today, after the contributions of several authors, being a LDS is known to be equivalent to both having enough independent curves on the space (Alberti representations) and, under a mild dimensionality condition, to Euclidean rectifiability.


In this talk I will introduce the analogue of the notion of Lipschitz differentiability space, but using Carnot-valued charts and Pansu derivatives as models. I will call such spaces Pansu differentiability spaces (PDS). A PDS is in particular a LDS, but the notion of differentiability considered is stronger when the models are non-Abelian.


I will discuss how to extend classical results on LDS to the setting of PDS, with a particular focus on the link with rectifiability.


This is based on a recent work with E. Le Donne and A. Merlo.

Abstract: The local central limit theorem describes the asymptotic behavior of
convolution powers of a probability distribution on the integer lattice, Z^d. Specifically, under certain mild assumptions on the distribution, the theorem guarantees that its convolution powers are well approximated by a single appropriately scaled Gaussian density, e.g., the heat kernel on R^d. When such distributions are allowed to take on complex values, their convolution powers exhibit rich and striking behavior not seen in the probabilistic setting. In this talk, I will first discuss some background on this topic dating back to its initial investigation by E. L. De Forest and subsequent study by T. N. E. Greville, I. J. Shoenberg, V. Thomee, P. Diaconis, and L. Saloff-Coste (all in the context of d = 1). I will then discuss a number of recent results on this topic including generalized local limit theorems. Under certain mild assumptions, these theorems show that the convolution powers of a complex-valued function on Z^d are well approximated by a sum of (appropriately scaled) "heat" kernels associated to a class of higher order partial differential operators on R^d.

Time permitting, I will also discuss these heat kernels and their properties.

To find the most efficient shape of a noise-absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with damping on the boundary modeled by a complex-valued Robin Boundary condition. We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall ensures the infimum of the acoustic energy. Then we also introduce a larger compact class of (ε, ∞) - or uniform domains with possibly non-Lipschitz (for example, fractal) boundaries in which an optimal shape exists, giving the minimum of the energy. The boundaries are described as the supports of Radon measures ensuring their Hausdorff dimension in the segment [n−1, n) . A byproduct of our proof is that the class of bounded (ε, ∞) -domains with fixed ε is stable under Hausdorff convergence. Another related result is the Mosco convergence of Robin-type energy functionals on converging domains.

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