In this talk I will give an overview of the theory of directional averages and singular integrals, firstly in dimension 2. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for directional singular integrals in codimension one, and general ambient dimension.
In this talk we will see a brief history of sharp Fourier restriction theory and some recent developments related to Fourier restriction estimates on spheres. We will discuss the problem of finding sharp constants for such inequalities, as well as the questions of existence and classification of extremizers of these estimates.
We consider large systems of jump processes that interact locally with respect to an underlying (possibly random) graph. Such processes model diverse phenomena including the spread of diseases, opinion dynamics and gas dynamics. Under a broad set of assumptions, we show that the empirical measure satisfies a large deviation principle in the sparse regime, that is, when the sequence of graphs converges locally to a limit graph. As a corollary we establish (quenched) hydrodynamic limits for the sequence of interacting jump processes. In addition, for a sub-class of processes that include the SIR process, we obtain a fairly explicit characterization of this limit and provide numerical evidence to show that it serves as a good approximation for finite systems of moderate size.
This is based on various joint works with I-Hsun Chen, Juniper Cocomello and Sarath Yasodharan.
We develop a general theory of 1-, 2-, and 3-dimensional "orbifold completion", to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. This can be viewed as the "oriented version" of condensation completion. We give a basic introduction to TQFTs and their orbifolds, and discuss applications which include defect TQFTs for state sum models, Reshethikin-Turaev and Crane-Yetter theory. This is joint work with Lukas Müller.
In this talk, we propose a new approach to solving the Buckley-Leverett System, which is to consider a compressible approximation model characterized by a stiff pressure law. Passing to the incompressible limit, the compressible model gives rise to a Hele-Shaw type free boundary limit of Buckley-Leverett System, and it is shown the existence of a weak solution of it.
We obtain the hydrodynamic limit of symmetric long-jumps exclusion in $\mathbb{Z}^{d}$ (for d≥1), where the jump rate is inversely proportional to a power of the jump's length with exponent $\gamma+1$, where $\gamma≥2$. Moreover, movements between $\mathbb{Z}^{d-1} \times \mathbb{Z}_{-}^{*}$ and $\mathbb{Z}^{d-1} \times \mathbb{N}$ are slowed down by a factor $\alpha n^{-\beta}$ (with $α>0$ and $β≥0$). In the hydrodynamic limit we obtain the heat equation in $\mathbb{R}^{d}$ without boundary conditions or with Neumann boundary conditions, depending on the values of $β$ and $γ$. The (rather restrictive) condition in previous works (for $d=1$) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.
Agências nacionais de estatística do mundo inteiro têm experimentado uma necessidade crescente de fornecer estimativas confiáveis de índices económicos e sociais, como proporções ou taxas, a nível de pequenas áreas ou pequenos domínios a partir de dados de pesquisas amostrais. No entanto, devido ao pequeno tamanho da amostra nessas áreas, não é viável obter estimativas com um nível de precisão aceitável sem usar abordagens baseadas em modelos. Este trabalho propõe modelar conjuntamente o estimador direto de índices no intervalo (0,1) e suas respectivas precisões utilizando-se as distribuições Beta e Beta prime. A novidade é modelar também o estimador de precisão amostral como uma distribuição Beta prime. Um estudo de avaliação com dados reais mostra que há ganho extra na modelagem conjunta do estimador direto e seu estimador de precisão com relação ao modelo Beta que não utiliza informação amostral sobre a precisão das estimativas. Uma aplicação para estimar o índice de insegurança alimentar em pequenas áreas do Estado de Minas Gerais, usando dados da Pesquisa Nacional de Orçamentos Familiares (POF) para o ano de 2018 é também apresentada.
Trabalho conjunto com Soraia Pereira (CEAUL/FCUL) e Giovani Silva (CEAUL/IST).
Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a "barcode" to any any finite binary tree embedded in ${\mathbb R}^3$. Using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.
In this talk, I will introduce the TMD, then focus on a very recent application to comparing mouse and human cortical neurons and characterizing the differences between them. I'll also briefly discuss the role of machine learning in our work.
This talk is based on collaborations led by Lida Kanari of the Blue Brain Project.
Do you know what is an Interacting Particle System? Do you know what is Stochastic Duality? If yes, then this talk might interest you for a different perspective on the field and recent applications of this tool to different classes of models. If not, this talk is for you to get to know a bit about it. I will only be assuming knowledge on the level of a basic course on measure theory.
The local information of a 2d rational conformal field theory (RCFT) is encoded in a vertex operator algebra, whose modules constitute a modular fusion category C. The collection of global observables of the theory is given by conformal blocks and carries actions of mapping class groups, which is described mathematically by a modular functor that assigns the Drinfeld center Z(C) to a circle. The string-net construction, which first appeared in the study of topological phases of matter, not only provides such a modular functor but also supplies a graphical construction of correlators. A generalization of the string-net construction takes a pivotal bicategory as input. When such a bicategory is taken to be C (considered as a bicategory with one object), it recovers the modular functor of conformal blocks. On the other hand, the modular functor associated with the Morita bicategory of separable symmetric Frobenius algebras internal to C classifies stratified worldsheets up to "categorical symmetries". In this talk we explain, using the framework of double categories, that RCFT correlators exhibit an equivalence between these two modular functors. This is in fact a consequence of the functoriality of the string-net construction: the lax biadjunction between a pivotal bicategory and its orbifold completion induces an equivalence between their string-net modular functors.
We study transitions from chaotic to integrable Hamiltonians in the double scaled SYK and $p$-spin systems. The dynamics of our models is described by chord diagrams with two species. We begin by developing a path integral formalism of coarse graining chord diagrams with a single species of chords, which has the same equations of motion as the bi-local Liouville action, yet appears otherwise to be different and in particular well defined. We then develop a similar formalism for two types of chords, allowing us to study different types of deformations of double scaled SYK and in particular a deformation by an integrable Hamiltonian. The system has two distinct thermodynamic phases: one is continuously connected to the chaotic SYK Hamiltonian, the other is continuously connected to the integrable Hamiltonian, separated at low temperature by a first order phase transition.
A longstanding conjecture of Carleson stated that the tangent points of the boundaries of certain planar domains can be characterized by the behavior of the Carleson ε-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that having some Dini type control of the Carleson ε-function implied the existence of tangents. A natural question is whether quantitative control on this function implies better regularity results. In this talk, we will present results that give a positive answer to this question. This is ongoing work.
We use 3d defect TQFTs and state sum models with defects to give a gauge theoretical formulation of Kitaev's quantum double model (for a finite group) and the (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedded quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFT is then formulated in terms of spans of groupoids and their representations.
This is work in progress with João Faría Martins (University of Leeds).
The main aim of this talk is to discuss the existence of nontrivial (and non semi-trivial) least energy solutions for a Neumann elliptic system with a critical nonlinearity, characterized by a cooperative-competitive behaviour, namely \[ \begin{cases} -\Delta u+\lambda_1 u=u^3+\beta uv^2 & \text{ in } \Omega\\ -\Delta v+\lambda_2 v=v^3+\beta u^2v & \text{ in } \Omega\\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial \nu}=0 &\text{ on } \partial\Omega,\\ \end{cases} \label{eq:pbeta} \tag{$\mathcal P_\beta$} \] where $\Omega\subset \mathbb{R}^4$ is a $C^2$ bounded domain, and $\lambda_1,\lambda_2>0$ and the parameter $\beta\in\mathbb{R}$ captures the essence of cooperation-competition, assuming positive or negative values respectively.
The approach is variational and the idea is to minimize the energy functional on a suitable manifold of the Nehari type. In addition, to deal with the critical power, we estimate the energy level, using the solutions of $-\Delta w=w^3$ in $ \mathbb{R} ^4$ and the solution for the scalar equations $-\Delta u_i+\lambda_iu_i=u_i^3$ in $\Omega$, to establish a compactness condition based on the classical Cherrier's inequality: if $\partial\Omega\in C^1$ then for each $\varepsilon\gt 0$ there exists $M_{\varepsilon}>0$ such that $$\|u\|_{2^*}\le\left(\frac{2^{2/N}}{S}+\varepsilon\right)^{1/2}\|\nabla u\|_2+M_{\varepsilon}\|u\|_2,\ \ \forall\ u\in H^1(\Omega),$$ where $S$ is the best Sobolev constant. Additionally, I will discuss the more difficult cases in which $\lambda_1,\lambda_2\le0$, that I have started to study recently.
Nonstandard analysis (NSA), founded by Abraham Robinson in the 1960s, was to a great extent inspired by Leibniz’s ideas and intuitions towards the use of infinitesimal and infinitely large quantities. One of the greatest features of NSA is that, by allowing a correct formulation of infinitesimals, one is now able to reason using orders of magnitude. This means that one can give precise meaning, and reason formally, about otherwise vague terms such as "small" or "large". Recently, accounts of vagueness relying on NSA were introduced [2, 8, 4]. In particular, and unlike other accounts of vagueness, the so-called nonstandard primitivist account [4, 5] embraces transitivity for marginal differences (i.e. "small" differences), but not for large differences in a soritical series. Nonstandard primitivism also seems to be particularly adequate to deal with the ship of Theseus paradox [3, 6] and may also shed some light in doxastic reasoning by considering infinitesimal probabilities and associating them to infinitesimal credences [1, 7]. We aim at assessing the relative merits of nonstandard primitivism and to show some lines of future research regarding the connections between NSA and philosophy.
In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.
Severals SPDEs arise from SDE dynamics under partial conditioning of the noise. My talk will circulate on three concrete examples, the Zakai equation from non-linear filtering, the pathwise control problem suggested by Lions-Sougandis, and last not least a rough PDE approach to pricing in non-Markovian stochastic volatility models. Underlying all these examples is the notion of rough stochastic differential equations, recently introduced, recently introduced by K. Lê, A. Hocquet and the speaker.
Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory. In joint work with Alex McDonald and Krystal Taylor, we study a variant of Davies's efficient covering theorem in which we replace lines with curves. This has a dual formulation in terms of nonlinear projections.
I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.
Topological quantum field theories (TQFTs) have attracted much attention from the physics and mathematical communities over the last thirty years, and for good reason: in low dimensions they let simple topology inform less-understood algebraic constructions. In the first half of this talk we will introduce factorization homology, a powerful procedure for constructing TQFTs out of homotopical gadgets, called $\mathsf{E}_n$-algebras. We explore this in dimension $n=1$, by using string-nets. In the second half of the talk, we introduce TQFTs with defects and factorization homology for stratified spaces and for an appropriate notion of stratified $\mathsf{E}_n$-algebra. Once again, we focus on dimension $n=1$, and will end by offering a conjectural connection between string-nets on stratified cylinders, Drinfel'd centres for bimodule categories and quasiparticles in a topological quantum computer.
Sharp restriction theory and the finite field extension problem have both received much attention in the last two decades, but so far they have not intersected. In this talk, we discuss our first results on sharp restriction theory on finite fields. Even though our methods for dealing with paraboloids and cones borrow some inspiration from their euclidean counterparts, new phenomena arise which are related to the underlying arithmetic and discrete structures. The talk is based on recent joint work with Cristian González-Riquelme.
