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Tuesday

Geometria em Lisboa


, Stanford University.

Abstract

A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussen’s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo).


Wednesday

Probability and Stochastic Analysis


, Universität Ausburg.

Abstract

Dynamic large deviations for additive path functionals of stochastic processes have attracted recent research interest, in particular in the context of stochastic particle systems and statistical physics. Efficient numerical 'cloning' algorithms have been developed to estimate the scaled cumulant generating function, based on importance sampling via cloning of rare event trajectories. Adapting previous results from the literature of particle filters and sequential Monte Carlo methods, we use Feynman-Kac models to establish fully rigorous bounds on systematic and random errors of cloning algorithms in continuous time. To this end we develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization and related to the propagation of chaos for mean-field models. Our results apply to a large class of jump processes on locally compact state space, and provide a framework that can also be used to evaluate and improve the efficiency of algorithms. This is joint work with Letizia Angeli, Adam Johansen and Andrea Pizzoferrato.


Thursday

Mathematics, Physics & Machine Learning


, Department of Electrical and Electronic Engineering, Imperial College, London.

Abstract

The revolution in sensing, with the emergence of many new imaging techniques, offers the possibility of gaining unprecedented access tothe physical world, but this revolution can only bear fruit through the skilful interplay between the physical and computational worlds. This is the domain of computational imaging which advocates that, to develop effective imaging systems, it will be necessary to go beyond the traditional decoupled imaging pipeline where device physics, image processing and the end-user application are considered separately. Instead, we need to rethink imaging as an integrated sensing and inference model. In this talk we cover two research areas where computational imaging is likely to have an impact.

We first focus on the heritage sector which is experiencing a digital revolution driven in part by the increasing use of non-invasive, non-destructive imaging techniques. These new imaging methods provide a way to capture information about an entire painting and can give us information about features at or below the surface of the painting. We focus on Macro X-Ray Fluorescence (XRF) scanning which is a technique for the mapping of chemical elements in paintings. After describing in broad terms the working of this device, a method that can process XRF scanning data from paintings is introduced. The method is based on connecting the problem of extracting elemental maps in XRF data to Prony's method, a technique broadly used in engineering to estimate frequencies of a sum of sinusoids. The results presented show the ability of our method to detect and separate weak signals related to hidden chemical elements in the paintings. We then discuss results on the Leonardo’s The Virgin of the Rocks and show that our algorithm is able to reveal, more clearly than ever before, the hidden drawings of a previous composition that Leonardo then abandoned for the painting that we can now see.

In the second part of the talk, we focus on two-photon microscopy and neuroscience. To understand how networks of neurons process information, it is essential to monitor their activity in living tissue. Multi-photon microscopy is unparalleled in its ability to image cellular activity and neural circuits, deep in living tissue, at single-cell resolution. However, in order to achieve step changes in our understanding of brain function, large-scale imaging studies of neural populations are needed and this can be achieved only by developing computational tools that can enhance the quality of the data acquired and can scan 3-D volumes quickly. In this talk we introduce light-field microscopy and present a method to localize neurons in 3-D. The method is based on the use of proper sparsity priors, novel optimization strategies and machine learning.

This is joint work with A. Foust, P. Song, C. Howe, H. Verinaz, J. Huang and Y.Su from Imperial College London, and C. Higgitt and N. Daly from The National Gallery in London


Monday

String Theory


Sourav Roychowdhury, Chennai Mathematical Institute.

Abstract

In this talk I will discuss Klebanov-Tseytlin background and its non-Abelian T-dual geometry. In particular I will show that the T-dual background admits pp-wave geometry in the neighbourhood of appropriate null geodesic. I will make comments on possible dual gauge theory for our pp-wave background.

Monday

QM3 Quantum Matter meets Maths


, Caltech.

Abstract

M. Berry showed how to attach a line bundle and a connection on it to a family of quantum Hamiltonians with a non-degenerate ground state, under the assumption that the Hilbert space is finite-dimensional. The first Chern class of this line bundle is a topological invariant of the family. It is far from obvious if this construction can be generalized to quantum many-body Hamiltonians. Indeed, naive generalizations fail because ground states of different Hamiltonians typically correspond to inequivalent representations of the algebra of observables. Nevertheless, it is possible to construct such invariants by making use of a certain differential graded Lie algebra (DGLA) attached to a quantum lattice system. For example, it turns out that to any family of gapped Hamiltonians on a 1d lattice one can attach a “quantized” degree-3 cohomology class on the parameter space. In this talk I will outline a construction of this DGLA as well as the construction of higher Berry classes. The talk is based on a work in progress with Nikita Sopenko.


Wednesday

Probability and Statistics


Jennifer Israelsson, Helga Kristín Ólafsdóttir and Jessica Silva Lomba, University of Reading, Gothenburg University/Chalmers and UL Centre of Statistics and its Applications.

Abstract

In this theme session, three speakers will share their work regarding the extreme aspects of rainfall phenomena, in smaller 25-min presentations.

TALK 1: Jennifer Israelsson (University of Reading)

Estimating the dependence structure for extreme tropical rainfall; many issues and some success.
Abstract:
A great deal of research has been done on rainfall extremes over Europe and the US both in a univariate and bivariate setting thanks to the availability of high-quality data. There has however been very limited amount of work done over Africa, and close to none in a multivariate setting, due to the general lack of rain gauge observations and the poor performance of weather models. In this talk, I will present some of my PhD work on estimating the dependence structure in extreme daily rainfall over west Africa and how this connects with the monsoon cycle. I will also talk about some of the many issues and limitations we faced and how some of these might be addressed.
Bio:
Jennifer Israelsson is a postdoctoral researcher at the University of Reading where she currently works on creating risk scenarios for local hospitals by translating regional climate projections to admissions at a hospital level. Her PhD was in the intersection of Statistics and Meteorology and focused on developing new methods to estimate dependence structures in daily tropical rainfall, and the application of those to better understand differences between rainfall intensities.

TALK 2: Helga Kristín Ólafsdóttir (Gothenburg University/Chalmers)

Frequency changes in extreme rainfall in the Northeastern USA
Abstract:
Extreme daily rainfall can increase with the individual extreme rainfalls becoming more frequent, more intense, or both more intense and more frequent. Based on the Generalized Extreme Value (GEV) distribution for annual maxima series and the General Pareto (GP) distribution for exceedances over threshold for the partial duration series, we develop a new statistical extreme value model, the PGEV model, allowing the use of high quality annual maximum series data instead of less well-checked daily data to estimate trends in intensity and frequency separately. The method is applied to annual maxima data from the NOAA Atlas 14, Volume 10. With increasing mean temperature, the frequency of extreme rainfall events increases as mean temperature increases while the distribution of the intensities of individual extreme rainfall events remains constant in the Northeastern US. We also study three other large areas in the contiguous US, the Midwest, the Southeast, and Texas, where similar but weaker trends than those in the Northeast are found.
Bio:
Helga is a PhD student at Gothenburg University/Chalmers in Applied Mathematics and Statistics with focus on modelling and model evaluation of extremes with applications on extreme rainfall under climate change.

TALK 3: Jessica Silva Lomba (UL Centre of Statistics and its Applications)

Mixed Moment estimator for pooling spatio-temporal extreme rainfall data in a heteroscedastic context
Abstract:
Extreme Value Theory provides the ideal framework for forecasting the frequency of extreme and hazardous events that are unlikely to occur and hard to predict. In this context, the estimation of the extreme value index is key. Due to accelerating climate change, extreme meteorological phenomena such as heavy precipitation seem to be growing more severe and frequent, but estimation of this evolution remains subject to large uncertainty. Thus, inferential methods for the underlying non-stationary spatio-temporal processes are currently object of widespread interest. A recent development is the concept of scedasis, through which a trend in extremes of space-time indexed observations can be captured and tactfully modelled within a semi-parametric framework. In this talk, we will look at how one can use series of data collected at several isolated locations to model extremes of the whole space-time process, enabling the mixed moment estimator of the extreme value index to seamlessly incorporate space-time non-stationarity and dependence. Application of the extended mixed moment estimator is illustrated with daily rainfall data from a homogeneous region in the UK.
Bio:
Jessica is a PhD student for Statistics and Operations Research at FCUL, University of Lisbon, on the topic of Extreme Values Theory and Statistics. Currently, her research is focused in estimation for spatio-temporal extreme data under heteroscedasticity.

Wednesday

UL Extremes Webinar


Jennifer Israelsson, Helga Kristín Ólafsdóttir and Jessica Silva Lomba, University of Reading, Gothenburg University/Chalmers and UL Centre of Statistics and its Applications.

Abstract

In this theme session, three speakers will share their work regarding the extreme aspects of rainfall phenomena, in smaller 25-min presentations.

TALK 1: Jennifer Israelsson (University of Reading)

Estimating the dependence structure for extreme tropical rainfall; many issues and some success.
Abstract:

A great deal of research has been done on rainfall extremes over Europe and the US both in a univariate and bivariate setting thanks to the availability of high-quality data. There has however been very limited amount of work done over Africa, and close to none in a multivariate setting, due to the general lack of rain gauge observations and the poor performance of weather models. In this talk, I will present some of my PhD work on estimating the dependence structure in extreme daily rainfall over west Africa and how this connects with the monsoon cycle. I will also talk about some of the many issues and limitations we faced and how some of these might be addressed.
Bio:
Jennifer Israelsson is a postdoctoral researcher at the University of Reading where she currently works on creating risk scenarios for local hospitals by translating regional climate projections to admissions at a hospital level. Her PhD was in the intersection of Statistics and Meteorology and focused on developing new methods to estimate dependence structures in daily tropical rainfall, and the application of those to better understand differences between rainfall intensities.

TALK 2: Helga Kristín Ólafsdóttir (Gothenburg University/Chalmers)

Frequency changes in extreme rainfall in the Northeastern USA
Abstract:

Extreme daily rainfall can increase with the individual extreme rainfalls becoming more frequent, more intense, or both more intense and more frequent. Based on the Generalized Extreme Value (GEV) distribution for annual maxima series and the General Pareto (GP) distribution for exceedances over threshold for the partial duration series, we develop a new statistical extreme value model, the PGEV model, allowing the use of high quality annual maximum series data instead of less well-checked daily data to estimate trends in intensity and frequency separately. The method is applied to annual maxima data from the NOAA Atlas 14, Volume 10. With increasing mean temperature, the frequency of extreme rainfall events increases as mean temperature increases while the distribution of the intensities of individual extreme rainfall events remains constant in the Northeastern US. We also study three other large areas in the contiguous US, the Midwest, the Southeast, and Texas, where similar but weaker trends than those in the Northeast are found.
Bio:
Helga is a PhD student at Gothenburg University/Chalmers in Applied Mathematics and Statistics with focus on modelling and model evaluation of extremes with applications on extreme rainfall under climate change.

TALK 3: Jessica Silva Lomba (UL Centre of Statistics and its Applications)

Mixed Moment estimator for pooling spatio-temporal extreme rainfall data in a heteroscedastic context
Abstract:

Extreme Value Theory provides the ideal framework for forecasting the frequency of extreme and hazardous events that are unlikely to occur and hard to predict. In this context, the estimation of the extreme value index is key. Due to accelerating climate change, extreme meteorological phenomena such as heavy precipitation seem to be growing more severe and frequent, but estimation of this evolution remains subject to large uncertainty. Thus, inferential methods for the underlying non-stationary spatio-temporal processes are currently object of widespread interest. A recent development is the concept of scedasis, through which a trend in extremes of space-time indexed observations can be captured and tactfully modelled within a semi-parametric framework. In this talk, we will look at how one can use series of data collected at several isolated locations to model extremes of the whole space-time process, enabling the mixed moment estimator of the extreme value index to seamlessly incorporate space-time non-stationarity and dependence. Application of the extended mixed moment estimator is illustrated with daily rainfall data from a homogeneous region in the UK.
Bio:
Jessica is a PhD student for Statistics and Operations Research at FCUL, University of Lisbon, on the topic of Extreme Values Theory and Statistics. Currently, her research is focused in estimation for spatio-temporal extreme data under heteroscedasticity.

Wednesday

Topological Quantum Field Theory


Vladimir M. Stojanovic, TU Darmstadt.

Abstract

In this talk I will try to demonstrate the use of Lie-algebraic concepts in the quantum control of interacting qubit arrays, with examples from both operator (gate)- and state control. I will start from the basics of quantum control and briefly review the Lie-algebraic underpinnings of the concept of complete controllability. I will then specialize to qubit arrays with Heisenberg-type interactions, summarizing the conditions for their complete controllability and showing a few examples of gate realization. The second part of the talk will be devoted to a rather unconventional use of Lie-algebraic concepts within a dynamical-symmetry-based approach to the deterministic conversion between W- and Greenberger-Horne-Zeilinger (three-qubit) states. The underlying physical system consists of three neutral atoms subject to several external laser pulses, where the atomic ground- and a highly-excited Rydberg state play the role of the two relevant logical qubit states.



Tuesday

Geometria em Lisboa


, Universitat Politècnica de Catalunya.

Abstract

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [6, 7, 8] launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, in the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension [1].

A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist in [4] more than two decades ago. We end this talk addressing a question raised by Moore in [5] : “Is hydrodynamics capable of performing computations?”. The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. Can this result be improved? In [2] we construct a Turing complete steady Euler flow in dimension 3. Time permitting, we discuss this and other generalizations for t-dependent Euler flows contained in [3].

In all the constructions above, the metric is seen as an additional "variable" and thus the method of proof does not work if the metric is prescribed.

Is it still possible to construct a Turing complete Euler flow on a 3-dimensional space with the standard metric? Yes, see our recent preprint https://arxiv.org/abs/2111.03559 (joint with Cardona and Peralta).

This talk is based on several joint works with Cardona, Peralta-Salas and Presas.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb embeddings, arXiv:1911.01963.
[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in dimension 3. PNAS May 11, 2021 118 (19) e2026818118; https://doi.org/10.1073/pnas.2026818118.
[3] R. Cardona, E. Miranda and D. Peralta-Salas, Turing universality of the incompressible Euler equations and a conjecture of Moore, International Mathematics Research Notices, rnab233, https://doi.org/10.1093/imrn/rnab233
[4] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture. Nonlinearity 13 (2000) 441–458.
[5] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991) 199–230.
[6] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.
[7] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete Cont. Dyn. Sys. A 38 (2018) 1553–1565.
[8] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.



Monday

String Theory


Kiril Hristov, Sofia University.

Abstract

We reinterpret the OSV formula for the on-shell action/entropy function of asymptotically flat BPS black holes as a fixed point formula that is formally equivalent to a recent gluing proposal for asymptotically AdS$_4$ black holes. This prompts a conjecture that the complete perturbative answer for the most general gravitational building block of 4d $N = 2$ supergravity at a single fixed point takes the form of a Nekrasov-like partition function with equivariant parameters related to the higher-derivative expansion of the prepotential. In turn this leads to a simple localization-like proposal for a set of supersymmetric partition functions in (UV completed) 4d $N = 2$ supergravity theories. The conjecture is shown to be in agreement with a number of available results for different BPS backgrounds with both Minkowski and AdS asymptotics. In particular, it follows that the OSV formula comes from the unrefined limit of the general expression including only the so-called $\mathbb{W}$ tower of higher derivatives, while the on-shell action of pure (Euclidean) AdS$_4$ with round $S^3$ boundary comes from the NS limit that includes only the $\mathbb{T}$ tower.



Thursday

Mathematics, Physics & Machine Learning


, MIT, Center for Theoretical Physics.

Abstract

Deep learning is an exciting approach to modern artificial intelligence based on artificial neural networks. The goal of this talk is to provide a blueprint — using tools from physics — for theoretically analyzing deep neural networks of practical relevance. This task will encompass both understanding the statistics of initialized deep networks and determining the training dynamics of such an ensemble when learning from data.

This talk is based on a book, The Principles of Deep Learning Theory, co-authored with Sho Yaida and based on research also in collaboration with Boris Hanin. It will be published next year by Cambridge University Press.


Wednesday

Topological Quantum Field Theory


, Instituto Superior Técnico, Lisbon.

Abstract

The classical Dold–Kan correspondence for simplicial objects in an abelian category is one of the cornerstones of homological algebra. When the abelian category is that of vector spaces, it gives a full identification between simplicial vector spaces and chain complexes of vector spaces vanishing in negative degrees. The Grothendieck construction for fibered categories, on the other hand, is a cornerstone of category theory. It relates the fibered category point of view with the pseudo-functor point of view and lies at the heart of the theory of stacks. Our main result can be understood as a far-reaching simultaneous generalization of both ideas within the contexts of linear algebra and differential geometry. In our result, simplicial vector spaces and chain complexes of vector spaces are replaced respectively by vector fibrations over a given (higher) Lie groupoid G and by representations up to homotopy of G. (Joint work with Matias del Hoyo.)





Tuesday

Geometria em Lisboa


, University of Lancaster.

Abstract

(Joint with Y. Lekili) If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its “link”, that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp $y^2 = x^3$ is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). We use symplectic cohomology to distinguish the contact structures on many of these links.


Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT