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Quinta feira

Lisbon WADE — Webinar em Análise e Equações Diferenciais

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients.
Boyan Sirakov, PUC - Rio.

Resumo

In this joint work with P. Souplet we develop a new, unified approach to the following two classical questions on elliptic PDE:

(i) the strong maximum principle for equations with non-Lipschitz nonlinearities; and

(ii) the at most exponential decay of solutions in the whole space or exterior domains.

Our results apply to divergence and nondivergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

Quinta feira

Probabilidades e Estatística

Bayesian state-space models for understanding and managing infectious disease challenges.
Mafalda Viana, Institute of Biodiversity, Animal Health and Comparative Medicine, University of Glasgow, Scotland, UK.

Resumo

Understanding the ecological and epidemiological processes that govern the transmission of complex multi-host, multi-pathogens systems remains challenging. One of the key reasons is that these are difficult to observe directly, which makes it necessary to rely on less direct, and often ‘weak’, sources of inference. In this talk I will show the power of Bayesian state-space models to overcome some of these difficulties and reveal hidden patterns and relationships from field and experimental data from wildlife and human diseases. Specifically, I will show examples from the Canine Distemper Virus and Canine Parvovirus in lions and dogs in the Serengeti, and mosquito vectors of human Malaria, for which these approaches enabled us to identify the disease dynamics, quantify the impacts of intervention on those dynamics and ultimately identify optimal control strategies for these infectious diseases.

Quinta feira

Escola de Inverno de Matemática

Online

Applied algebraic topology.
Gustavo Granja, Instituto Superior Técnico.

Resumo

Algebraic topology is the area of Mathematics which studies shapes or "large scale geometry” by algebraic means. During the 20th century it was recognised that the basic ideas of algebraic topology play an organizing role in most areas of pure mathematics. More recently, these ideas have also been successfully applied to other areas of science, from computer vision and data analysis, to neuroscience. This talk will introduce some of the most fundamental ideas in algebraic topology, which go back to Poincaré at the beginning of the 20th century, and briefly describe some ot their recent applications.

Quinta feira

Escola de Inverno de Matemática

Online

The hidden geometry.
José Natário, Instituto Superior Técnico.

Resumo

Geometry is all around us, but it is many times hidden from our eyes.

In this talk we will see several examples of this, ranging from everyday situations to astrophysics and elementary particles.

Quinta feira

Escola de Inverno de Matemática


Paulo Mourão, Mathematics PhD Student at Université de Genève.

Resumo

Even though we can easily model the behaviour of a charged particle under the influence of a magnetic field, the problem quickly grows very complicated as soon as we start considering many particle systems, due to the electromagnetic interaction between them. A breakthrough was achieved in 1983 when Robert Laughlin proposed an ansatz for the ground state of simple two dimensional systems of charged particles under the influence of a uniform magnetic field. This was later proven to be an excellent approximation to the exact wavefunctions. In my Master's thesis, we used techniques of Kähler geometry, geometric quantization and imaginary time Hamiltonian flows to deform these systems and thus obtain possible Laughlin states on deformed geometries. With this talk, I hope to provide some insight into modern applications of abstract mathematics to theoretical physics.


Sexta feira

Escola de Inverno de Matemática

Online


, Instituto Superior Técnico.

Resumo

If we are given a linear system of equations on finite dimensional spaces, we know from Linear Algebra how many solutions there are and how to find them. On the other hand, most mathematical modelling involves the analysis of some system of differential equations describing our model; here the solutions are functions, living in an infinite dimensional world. Finding the actual solutions may be very hard in practice, but if we are lucky, we have a powerful tool: there is an integer, called the index, that gives us information about the number of solutions without computing them. In the right setting, this number is very stable and behaves well under many natural transformations.

In this talk, we place this invariant in the setting of Operator theory, that is, the study of linear maps between normed vector spaces, let it be Euclidean space, spaces of sequences or function spaces. We make a tour through some of the main concepts and examples, and consider the class of Fredholm operators, the ones that lead to a well-defined index. We see what stability means and explain how the strong properties of the index lead in some cases to remarkable index formulas, depending on the shape of the space our model takes place, thus establishing a strong link between analysis and topology.

Sexta feira

Escola de Inverno de Matemática

Online


, Instituto Superior Técnico.

Resumo

We present basic concepts of mathematical epidemiology, that date from Kermack–McKendrick, 1927. The theory was later on developed to study the spread of other infeccious deseases with latency time or in more complex situations. We analyse equilibrium points, and limit cycles in different situations. We make a brief overview of the role of discrete dynamical systems in these theories. The final point of this talk is the discussion of the use of compartimental models with ordinary differential equations, both autonomous and nonautonomous, to forecast and analyse the situation of COVID-19 in Portugal.

Sexta feira

Teoria Quântica do Campo Topológica


, University of Oregon.

Resumo

The group algebra of the symmetric group has a large commutative subalgebra generated by Young-Jucys-Murphy elements, which acts diagonalizably on any irreducible representation. The goal of this talk is to give an accessible introduction to the categorification of this story. The main players are: Soergel bimodules, which categorify the Hecke algebra of the symmetric group; Rouquier complexes, which categorify the braid group where Young-Jucys-Murphy elements live; and the Elias-Hogancamp theory of categorical diagonalization, which allows one to construct projections to "eigencategories."

Sexta feira

Escola de Inverno de Matemática


Renata Amado, Software Component Engineer at Edisoft.

Resumo

A system of reaction-diffusion partial differential equations was recently proposed for modelling the interaction of predators competing for the same prey. The most counterintuitive conclusion drawn from the experiments is that strong competition, besides causing emergence of territories, leads also to an increase in both predators and prey populations, alongside with the emergence of a refuge area for the prey population. The ecological implications of varying the different parameters present in the model were also tested regarding the emergence of territories and if and how this variation influences the size of the territories.


Segunda feira

Teoria de Cordas


Jan Manschot, School of Mathematics, Trinity College, Dublin.

Resumo

Supersymmetric D-branes supported on the complex two-dimensional base $S$ of the local Calabi-Yau threefold $K_S$ are described by semi-stable coherent sheaves on $S$. Under suitable conditions, the BPS indices counting these objects (known as generalized Donaldson-Thomas invariants) coincide with the Vafa-Witten invariants of $S$ (which encode the Betti numbers of the moduli space of semi-stable sheaves). For surfaces which admit a strong collection of exceptional sheaves, we develop a general method for computing these invariants by exploiting the isomorphism between the derived category of coherent sheaves and the derived category of representations of a suitable quiver with potential $(Q,W)$ constructed from the exceptional collection. We spell out the dictionary between the Chern class $\gamma$ and polarization $J$ on $S$ vs. the dimension vector $\vec N$ and stability parameters $\vec\zeta$ on the quiver side. For all examples that we consider, which include all del Pezzo and Hirzebruch surfaces, we find that the BPS indices $\Omega_\star(\gamma)$ at the attractor point (or self-stability condition) vanish, except for dimension vectors corresponding to simple representations and pure D0-branes. This opens up the possibility to compute the BPS indices in any chamber using either the flow tree or the Coulomb branch formula. In all cases we find precise agreement with independent computations of Vafa-Witten invariants based on wall-crossing and blow-up formulae. This agreement suggests that i) generating functions of DT invariants for a large class of quivers coming from strong exceptional collections are mock modular functions of higher depth and ii) non-trivial single-centered black holes and scaling solutions do not exist quantum mechanically in such local Calabi-Yau geometries.

Segunda feira

QM3 Matéria Quântica & Matemática


, University of Cambridge.

Resumo

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. Please do not be scared by the physics though: I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a recent PR Letter by Tarnopolsky–Kruchkov–Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hoermander’s bracket condition in a very simple setting). The results will be illustrated by colourful numerics which suggest some open problems. This is joint work with M. Embree, J. Wittsten, and M. Zworski.



Quarta feira

Matemática, Física & Aprendizagem Automática


, Faculdade de Engenharia, Universidade do Porto.

Resumo

The past few decades have witnessed a significant research effort in the field of Lyapunov model based control design. In parallel, optimal control and optimization model based design have also expanded their range of applications, and nowadays, receding horizon approaches can be considered a mature field for particular classes of control systems.

In this talk, I will argue that Lyapunov based techniques play an important role for analysis of model based optimization methodologies and moreover, both approaches can be combined for control design resulting in powerful frameworks with formal guarantees of robustness, stability, performance, and safety. Illustrative examples in the area of motion control of autonomous robotic vehicles will be presented for Autonomous Underwater Vehicles (AUVs), Autonomous Surface Vehicles (ASVs) and Unmanned Aerial Vehicles (UAVs).


Quinta feira

Lisbon WADE — Webinar em Análise e Equações Diferenciais


, Università di Pisa.

Resumo

The vectorial Bernoulli problem is a variational free boundary problem involving the Dirichlet energy of a vector-valued function and the measure of its support. It is the vectorial counterpart of the classical one-phase Bernoulli problem, which was first studied by Alt and Caffarelli in 1981.

In this talk, we will discuss some results on the regularity of the vectorial free boundaries obtained in the last years by Caffarelli-Shahgholian-Yeressian, Kriventsov-Lin, Mazzoleni-Terracini-V., and Spolaor-V.. Finally, we will present some new results on the rectifiability of the singular set obtained in collaboration with Guido De Philippis, Max Engelstein and Luca Spolaor.

Quinta feira

Lisbon WADE — Webinar em Análise e Equações Diferenciais


, Università di Pavia.

Resumo

First of all, we recall the basic notions and results concerning shape optimization problems for the eigenvalues of the Dirichlet Laplacian.
Then we focus on the study of the regularity of the optimal shapes and on the link with the regularity of related free boundary problems.

The main topic of the talk is the regularity of the optimal sets for a "degenerate'" functional, namely the second Dirichlet eigenvalue in a box. Given $D\subset \mathbb{R}^d$ an open and bounded set of class $C^{1,1}$, we consider the following shape optimization problem, for $\Lambda>0$,\begin{equation}\label{eq:main}\min{\Big\{\lambda_2(A)+\Lambda |A| : A\subset D,\text{ open}\Big\}},\end{equation}where $\lambda_2(A)$ denotes the second eigenvalue of the Dirichlet Laplacian on $A$.

In this talk we show that any optimal set $\Omega$ for \eqref{eq:main} is equivalent to the union of two disjoint open sets, $\Omega^\pm$, which are $C^{1,\alpha}$ regular up to a (possibly empty) closed singular set of Hausdorff dimension at most $d-5$, which is contained in the one-phase free boundaries.

In particular, we are able to prove that the set of two-phase points, that is, $\partial \Omega^+\cap \partial \Omega^-\cap D$, is contained in the regular set.


This is a joint work with Baptiste Trey and Bozhidar Velichkov.


Sexta feira

Teoria Quântica do Campo Topológica


, University of Massachusetts, Amherst.

Resumo
Factorization algebras provide a flexible language for describing the observables of a perturbative QFT, as shown in joint work with Kevin Costello. Those constructions extend to a manifold with boundary for a special class of theories. I will discuss work with Eugene Rabinovich and Brian Williams that includes, as an example, a perturbative version of the correspondence between chiral ${\rm U}(1)$ currents on a Riemann surface and abelian Chern-Simons theory on a bulk 3-manifold, but also includes a systematic higher dimensional version for higher abelian CS theory on an oriented smooth manifold of dimension $4n+3$ with boundary a complex manifold of complex dimension $2n+1$. Given time, I will discuss how this framework leads to a concrete construction of the center of higher enveloping algebras of Lie algebras, in work with Greg Ginot and Brian Williams.

Segunda feira

QM3 Matéria Quântica & Matemática


, University of Birmingham.

Resumo

Spatial dimensionality plays a key role in our understanding of topological physics, with different topological invariants needed to characterise systems with different numbers of spatial dimensions. In a 2D quantum Hall system, for example, a robust quantisation of the Hall response is related to the first Chern number: a 2D topological invariant of the electronic energy bands. Generalising to more spatial dimensions, it was shown that a new type of quantum Hall effect could emerge in four dimensions, but where the quantised response was related to a four-dimensional topological invariant, namely the second Chern number. While systems with four spatial dimensions may seem abstract, recent developments in ultracold atoms and photonics have opened the door to exploring such higher-dimensional topological physics experimentally. In this talk, I will introduce the theory of 4D topological phases of matter, before discussing recent experiments in cold atoms, photonics and electrical circuits that have begun to probe aspects of this physics in the laboratory.






Segunda feira

Teoria de Cordas


David Berman, Queen Mary University of London.

Resumo

We examine various properties of double field theory and the doubled string sigma model in the context of geometric quantisation. In particular we look at T-duality as the symplectic transformation related to an alternative choice of polarisation in the construction of the quantum bundle for the string. Following this perspective we adopt a variety of techniques from geometric quantisation to study the doubled space. One application is the construction of the double coherent state that provides the shortest distance in any duality frame and a stringy deformed Fourier transform.



Quarta feira

Matemática, Física & Aprendizagem Automática


, Caltech.

Resumo

We compare the complexity of training classical and quantum machine learning (ML) models for predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve the execution of a (possibly unknown) quantum process $E$. Our figure of merit is the number of runs of $E$ needed during training, disregarding other measures of complexity. A classical ML performs a measurement and records the classical outcome after each run of $E$, while a quantum ML can access $E$ coherently to acquire quantum data; the classical or quantum data is then used to predict outcomes of future experiments. We prove that, for any input distribution $D(x)$, a classical ML can provide accurate predictions on average by accessing $E$ a number of times comparable to the optimal quantum ML. In contrast, for achieving accurate prediction on all inputs, we show that exponential quantum advantage exists in certain tasks. For example, to predict expectation values of all Pauli observables in an $n-$qubit system, we present a quantum ML using only $O(n)$ data and prove that a classical ML requires $2^{\Omega(n)}$ data.



Segunda feira

QM3 Matéria Quântica & Matemática


, Max-Planck Institute for the Physics of Complex Systems, Dresden.

Resumo

In the last two decades the field of nonequilibrium quantum many-body physics has seen a rapid development driven, in particular, by the remarkable progress in quantum simulators, which today provide access to dynamics in quantum matter with an unprecedented control. However, the efficient numerical simulation of nonequilibrium real-time evolution in isolated quantum matter still remains a key challenge for current computational methods especially beyond one spatial dimension. In this talk I will present a versatile and efficient machine learning inspired approach. I will first introduce the general idea of encoding quantum many-body wave functions into artificial neural networks. I will then identify and resolve key challenges for the simulation of real-time evolution, which previously imposed significant limitations on the accurate description of large systems and long-time dynamics. As a concrete example, I will consider the dynamics of the paradigmatic two-dimensional transverse field Ising model, where we observe collapse and revival oscillations of ferromagnetic order and demonstrate that the reached time scales are comparable to or exceed the capabilities of state-of-the-art tensor network methods.




Segunda feira

Teoria de Cordas


Dmitry Melnikov, ITEP Moscow.

Resumo

One of the attractive ideas of building a quantum computer is based on the topological properties of matter. In such a realization, the Topological Quantum Field Theories (TQFT) become the main language to describe the functioning of the quantum computer. In my talk I will discuss some basic elements of the topological quantum computing. I will start from a description of TQFTs as instances of quantum mechanics in terms of category theory. Then I will review the notion of quantum entanglement in this context. As a further preparation to quantum computations I will discuss the question of complexity of quantum algorithms and quantum states. I will introduce a complexity measure for a simple class of the "torus knot states" and review some alternative recent measures and approaches from the literature.


Quarta feira

Matemática, Física & Aprendizagem Automática


, University of Washington.

Resumo

Many tasks in fluid mechanics, such as design optimization and control, are challenging because fluids are nonlinear and exhibit a large range of scales in both space and time. This range of scales necessitates exceedingly high-dimensional measurements and computational discretization to resolve all relevant features, resulting in vast data sets and time-intensive computations. Indeed, fluid dynamics is one of the original big data fields, and many high-performance computing architectures, experimental measurement techniques, and advanced data processing and visualization algorithms were driven by decades of research in fluid mechanics. Machine learning constitutes a growing set of powerful techniques to extract patterns and build models from this data, complementing the existing theoretical, numerical, and experimental efforts in fluid mechanics. In this talk, we will explore current goals and opportunities for machine learning in fluid mechanics, and we will highlight a number of recent technical advances. Because fluid dynamics is central to transportation, health, and defense systems, we will emphasize the importance of machine learning solutions that are interpretable, explainable, generalizable, and that respect known physics.


Quinta feira

Lisbon WADE — Webinar em Análise e Equações Diferenciais

Online


, CMAP, École Polytechnique.

Resumo

Additive manufacturing (or 3-d printing) is a new exciting way of building structures without any restriction on their topologies. However, it comes with its own difficulties or new issues. Therefore, it is a source of many interesting new problems for optimization. I shall discuss two of them and propose solutions to these problems, but there is still a lot of room for improvement!

First, additive manufacturing technologies are able to build finely graded microstructures (called lattice materials). Their optimization is therefore an important issue but also an opportunity for the resurrection of the homogenization method ! Indeed, homogenization is the right technique to deal with microstructured materials where anisotropy plays a key role, a feature which is absent from more popular methods, like SIMP. I will describe recent work on the topology optimization of these lattice materials, based on a combination of homogenization theory and geometrical methods for the overall deformation of the lattice grid.

Second, additive manufacturing, especially in its powder bed fusion technique, is a very slow process because a laser beam must travel along a trajectory, which covers the entire structure, to melt the powder. Therefore, the optimization of the laser path is an important issue. Not only do we propose an optimization strategy for the laser path, but we couple it with the usual shape and topology optimization of the structure. Numerical results show that these two optimizations are tightly coupled.

This is a joint work with many colleagues, including two former PhD students, P. Geoffroy-Donders and M. Boissier.




Segunda feira

Teoria de Cordas


Murad Alim, University of Hamburg.

Resumo

We study difference equations which are obtained from the asymptotic expansion of topological string theory on the deformed and the resolved conifold geometries as well as for topological string theory on arbitrary families of Calabi-Yau manifolds near generic singularities at finite distance in the moduli space. Analytic solutions in the topological string coupling to these equations are found. The solutions are given by known special functions and can be used to extract the strong coupling expansion as well as the non-perturbative content. The strong coupling expansions show the characteristics of D-brane and NS5-brane contributions, this is illustrated for the quintic Calabi-Yau threefold. For the resolved conifold, an expression involving both the Gopakumar-Vafa resummation as well as the refined topological string in the Nekrasov-Shatashvili limit is obtained and compared to expected results in the literature. Furthermore, a precise relation between the non-perturbative partition function of topological strings and the generating function of non-commutative Donaldson-Thomas invariants is given. Moreover, the expansion of the topological string on the resolved conifold near its singular small volume locus is studied. Exact expressions for the leading singular term as well as the regular terms in this expansion are provided and proved. The constant term of this expansion turns out to be the known Gromov-Witten constant map contribution.



Quinta feira

Lisbon WADE — Webinar em Análise e Equações Diferenciais


, Università degli Studi di Siena.

Resumo

We briefly discuss the definition of relaxation of the area functional. The relaxed area functional, denoted by $A$, extends the classical area functional, which, for any "regular" map $v:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^N$ evaluates the $n$-dimensional area of its graph over $U$. The problem of determining the domain and the expression of $A$ is open in codimension greater than 1. Specifically, this relaxed functional turns out to be nonlocal and cannot be expressed by an integral formula. We discuss how it is related to classical and nonclassical versions of the Plateau problem. As a main example, we try to understand what is the relaxed graph of the function $x/|x|$, a question that surprisingly remained open for decades.



Segunda feira

Teoria de Cordas


Michal P. Heller, Albert Einstein Institute.

Resumo

We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the "complexity equals volume" conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $T\bar{T}$, we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action. Based on arXiv:2101.01185.



Segunda feira

Teoria de Cordas


Tarek Anous, University of Amsterdam.

Resumo

Scalar unitary representations of the isometry group of d-dimensional de Sitter space $SO(1,d)$ are labeled by their conformal weights $\Delta$. A salient feature of de Sitter space is that scalar fields with sufficiently large mass compared to the de Sitter scale $1/l$ have complex conformal weights, and physical modes of these fields fall into the unitary continuous principal series representation of $SO(1,d)$. Our goal is to study these representations in $d=2$, where the relevant group is $SL(2,R)$. We show that the generators of the isometry group of $dS_2$ acting on a massive scalar field reproduce the quantum mechanical model introduced by de Alfaro, Fubini and Furlan (DFF) in the early/late time limit. Motivated by the ambient $dS_2$ construction, we review in detail how the DFF model must be altered in order to accommodate the principal series representation. We point out a difficulty in writing down a classical Lagrangian for this model, whereas the canonical Hamiltonian formulation avoids any problem. We speculate on the meaning of the various de Sitter invariant vacua from the point of view of this toy model and discuss some potential generalizations.







Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT