Seminários de até

This talk consists of an expanded reading of the paper A Note on the Sobolev Inequality by G. Bianchi and H. Egnell, which addresses stability for the Sobolev inequality in the case p=2. This detailed review will include some remarks on the techniques used, including the concentration-compactness principle and spectral theory for the corresponding linearized problem. This study is supervised by Delia Schiera and Giuseppe Negro.

Consider a probability measure $\nu$ with compact support in $GL(d)$ and a linear cocycle $\mathcal{F}:GL^{\mathbb{N}}(d)\times {\mathbb{R}}^d\rightarrow GL^{\mathbb{N}}(d)\times {\mathbb{R}}^d$ defined as $\mathcal{F}((g_j)_j,v)=((g_{j+1})_j,g_0 v)$. The cocycle associated with a random product of $2\times 2$ invertible matrices (i.e., $d=2$) under a probability distribution $\nu$ have two (possibly equal) Lyapunov exponents $\lambda_1(\nu)\ge\lambda_2(\nu)$. When $\lambda_1>\lambda_2$ we can prove that those two exponents are pointwise Hölder continuous with respect to the probability measure $\nu$. A natural question arises: does this result generalize to higher dimensions? In this talk, we'll explore the concept of a stochastic dynamical system, define a random $GL(d)$-cocycle and investigate the above-mentioned generalization. This is a work in progress with Adriana Sánchez, El Hadji Yaya Tall and Marcelo Viana.

For measuring possible concentrations of the eigenfunctions of the Laplace operator on a manifold, Burq-Gerard-Tzvetkov studied $L^p$ norm of the restrictions of the eigenfunctions to submanifolds. They proved sharp $L^p$ estimates restricted to the geodesic or a curve having nonvanishing geodesic curvature. I will talk about $L^p$ estimates restricted to a curve which is not geodesic and has vanishing geodesic curvature. The proof involves semiclassical analysis.

We define the atomic Hardy space $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$, $0\lt p≤ 1$, for the twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy space defined using the maximal function corresponding to the heat semigroup $e^{-\mathcal{L}t}$, $t\gt 0$. We also prove sharp $L^p$, $0\lt p≤ 1$, estimates for $\mathcal{L}^{β/2}e^{i\sqrt{\mathcal{L}}}$. More precisely, we prove that it is a bounded operator on $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$ when $β≥(2n-1)(1/p-1/2)$.

We consider a torus embedded in the 3-dimensional Euclidean space. It has a natural two-parameter scaling structure. Under this structure, we can consider a two parameter maximal average over the tori. We study the sharp boundedness of this maximal function, its Sobolev regularity, and local smoothing properties. We compare this result with the one parameter maximal function.

Recently, an approach to constructing topological invariants of gapped ground-states of lattice systems has been developed in our joint work with N. Sopenko. It applies to arbitrary gapped states of infinite-volume lattice spin systems with rapidly decaying interactions and employs C*-algebraic techniques. In this talk, I will explain an interpretation of these invariants as obstructions to gauging, i.e. to promoting a symmetry to a local symmetry. The key observation is that locality on a lattice is an asymptotic notion sensitive only to the large-scale geometry of the support set. Following Kashiwara and Schapira, one can encode locality using a natural Grothendieck topology on a category of semilinear subsets of Eucludean space. Infinitesimal symmetries of a gapped state form a cosheaf over the corresponding site, and the topological invariants are encoded in its Cech complex.

We will discuss a weak universality phenomenon in the context of two-dimensional fractional nonlinear wave equations. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_2^4$, we will present a sufficient and almost necessary criteria for the convergence of invariant measures to the fractional $Φ_2^4$. Then we will discuss the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. This extends a result of Gubinelli-Koch-Oh to a situation where we do not have any local Cauchy theory with highly supercritical nonlinearities. This is a joint work with Chenmin Sun and Weijun Xu.

I will begin by reviewing geometric and deformation quantization of a symplectic vector space. The goal will be to explain an analogy between these objects and Rozansky–Witten theory (along with a certain four-dimensional TQFT). This analogy will factor through an analogy concerning three-dimensional TQFTs generated by pointed fusion categories. Throughout, there will be an emphasis on equivariance and anomalies.

The “alpha” version of factorization homology pairs framed n-manifolds with $E_n$-algebras. This construction generalizes the classical homology of a manifold, yields novel results concerning configuration spaces of points in a manifold, and supplies a sort of state-sum model for sigma-models (i.e., mapping spaces) to (n-1)-connected targets. This “alpha” version of factorization homology novelly extends Poincaré duality, shedding light on deformation theory and dualities among field theories. Being defined using homotopical mathematical foundations, “alpha” factorization homology is manifestly functorial and continuous in all arguments, notably in moduli of manifolds and embeddings between them, and it satisfies a local-to-global expression that is inherently homotopical in nature.

Now, $E_n$-algebras can be characterized as $(\infty,n)$-categories equipped with an (n-1)-connected functor from a point. The (full) “beta” version of factorization homology pairs framed n-manifolds with pointed $(\infty,n)$-categories with adjoints. Applying 0th homology, or $\pi_0$, recovers a version of the string net construction on surfaces, as well as skein modules of 3-manifolds. In some sense, the inherently homotopical nature of (full) “beta” factorization homology affords otherwise unforeseen continuity in all arguments, and local-to-global expressions.

In this talk, I will outline a definition of “beta” factorization homology, focusing on low-dimensions and on suitably reduced $(\infty,n)$-categories (specifically, braided monoidal categories). I will outline some examples, and demonstrate some features of factorization homology. Some of this material is established in the literature, some a work in progress, and some conjectural — the status of each assertion will be made clear. I will be especially interested in targeting this talk to those present, and so will welcome comments and questions.

A what?!

We will give a gentle introduction to gerbes and other assorted "higher structures" from topology and mathematical physics. Gerbes are a generalization of line bundles. For a line bundle, the space of sections forms a vector space, and a little extra geometric structure can make it into a Hilbert space, beloved by quantum physicists everywhere. After introducing gerbes, we will ponder the analogous construction: how do we define a "Hilbert space of sections" for a gerbe?

This expository talk is based on the work of Bunk and Szabo.