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Friday

Mathematics, Physics & Machine Learning


, Technische Universitaet Darmstadt.

Abstract

Autonomous robots that can assist humans in situations of daily life have been a long standing vision of robotics, artificial intelligence, and cognitive sciences. A first step towards this goal is to create robots that can learn tasks triggered by environmental context or higher level instruction. However, learning techniques have yet to live up to this promise as only few methods manage to scale to high-dimensional manipulator or humanoid robots. In this talk, we investigate a general framework suitable for learning motor skills in robotics which is based on the principles behind many analytical robotics approaches. It involves generating a representation of motor skills by parameterized motor primitive policies acting as building blocks of movement generation, and a learned task module that transforms these movements into motor commands. We discuss learning on three different levels of abstraction, i.e., learning for accurate control is needed to execute, learning of motor primitives is needed to acquire simple movements, and learning of the task-dependent „hyperparameters“ of these motor primitives allows learning complex tasks. We discuss task-appropriate learning approaches for imitation learning, model learning and reinforcement learning for robots with many degrees of freedom. Empirical evaluations on a several robot systems illustrate the effectiveness and applicability to learning control on an anthropomorphic robot arm. These robot motor skills range from toy examples (e.g., paddling a ball, ball-in-a-cup, juggling) to playing robot table tennis against a human being and manipulation of various objects.


Monday

LisMath


João Rijo, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

Abstract

We consider a nonautonomous dynamical system given by a sequence of bounded linear operators acting on a Banach space. We introduce the notion of an exponential dichotomy which is central in the stability theory of dynamical systems. Our results give a characterization of the existence of an exponential dichotomy in terms of the invertibility of a certain linear operator between so-called admissible spaces. Using this characterization, we show that the notion of an exponential dichotomy is robust for sufficiently small linear perturbations.

We also introduce the notion of an exponential dichotomy for difference equations with infinite delay. This requires considering an appropriate class of phase spaces that are Banach spaces of sequences satisfying a certain axiom motivated by the work of Hale and Kato for continuous time. We present a result that establishes the existence of stable manifolds for any sufficiently small perturbation of a difference equation having an exponential dichotomy.

Finally, we briefly describe the formulation of the previous results for the more general case of a tempered exponential dichotomy. This is a nonuniform version of an exponential dichotomy that is ubiquitous in the context of ergodic theory.

References

[1] L. Barreira, J. Rijo, C. Valls. Characterization of tempered exponential dichotomies. J. Korean Math. Soc., 57(1):171–194, 2020.

[2] L. Barreira, J. Rijo, C. Valls. Stable manifolds for difference equations with infinite delay. J. Difference Equ. Appl., 26(9-10):1266–1287, 2020.

Monday

LisMath


Carlos Sotillo Rodríguez, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Abstract

We consider the space of polygons with edges in a complex projective space, generalizing the classical space of polygons with edges in $\mathbb{R}^3$.

Using the Gelfand-McPherson correspondence and wall crossing techniques we will prove a recursive formula for the Poincaré polynomial of these spaces and compute their symplectic volume.

Monday

LisMath


Pedro Aniceto, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

Abstract

This talk is divided into two parts. In the first part, we consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics. We prove that the canonical Wiener-Hopf factorization of a matrix obtained from a general monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the gravitational field equations. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields distinct solutions to the field equations.

In the second part, we approach the problem of constructing the holographic dictionary for the AdS$_2$/CFT$_1$ correspondence for higher derivative gravitational actions in AdS$_2$ spacetimes obtained by an $S^2$ reduction of 4-dimensional $\mathcal{N}=2$ Wilsonian effective actions with Weyl squared interactions. We focus on BPS black hole solutions, for which we show how the Wald entropy of these black holes is holographically encoded in the dual CFT. Additionally, using a 2D/3D lift we show that the dual CFT$_1$ is naturally embedded in the chiral half of the CFT$_2$ dual to the AdS$_3$ spacetime.

Monday

LisMath


Inês Rodrigues, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Abstract

Gillespie, Levinson and Purbhoo recently introduced a crystal-like structure for shifted tableaux, called the shifted tableau crystal. Following a similar approach as Halacheva, for crystals of finite Cartan type, we exhibit a natural internal action of the cactus group on this structure, realized by the restrictions of the shifted Schützenberger involution to all primed intervals of the underlying crystal alphabet. This includes the shifted crystal reflection operators, which agree with the restrictions of the shifted Schützenberger involution to single-coloured connected components, but unlike the case for type A crystals, these do not need to satisfy the braid relations of the symmetric group.

In addition, we define a shifted version of the Berenstein-Kirillov group, by considering shifted Bender-Knuth involutions. Paralleling the works of Halacheva and Chmutov, Glick and Pylyavskyy for type A semistandard tableaux of straight shape, we exhibit another occurrence of the cactus group action on shifted tableau crystals of straight shape via the action of the shifted Berenstein-Kirillov group. We also conclude that the shifted Berenstein-Kirillov group is isomorphic to a quotient of the cactus group. Not all known relations that hold in the classic Berenstein-Kirillov group need to be satisfied by the shifted Bender-Knuth involutions, namely the one equivalent to the braid relations of the type A crystal reflection operators, but the ones implying the relations of the cactus group are verified, thus we have another presentation for the cactus group in terms of shifted Bender-Knuth involutions.

Monday

LisMath


Vicente Lorenzo García, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

Abstract

The main numerical invariants of a complex projective algebraic surface $X$ are the self-intersection of its canonical class $K^2_X$ and its holomorphic Euler characteristic $\chi(\mathcal{O}_X)$. If we assume $X$ to be minimal and of general type then $K^2_X\geq 2\chi(\mathcal{O}_X)-6$ by Noether's inequality.

Minimal algebraic surfaces of general type $X$ such that $K^2_X=2\chi(\mathcal{O}_X)-6$ or $K^2_X=2\chi(\mathcal{O}_X)-5$ are called Horikawa surfaces and they admit a canonical $\mathbb{Z}_2$-action. In this talk we will discuss other possible group actions on Horikawa surfaces. In particular, $\mathbb{Z}_2^2$-actions and $\mathbb{Z}_3$-actions on Horikawa surfaces will be studied.

In the case of $\mathbb{Z}_2^2$-actions we will not settle for Horikawa surfaces and results regarding the geography of minimal surfaces of general type admitting a $\mathbb{Z}_2^2$-action will be discussed. They will yield some consequences on the moduli spaces of stable surfaces $\overline{\mathfrak{M}}_{K^2,\chi}$.

Monday

LisMath


Matthias Hofmann, LisMath, Faculdade de Ciências, Universidade de Lisboa.

Abstract

We present our thesis work dealing with several topics in PDE theory on metric graphs. Firstly, we present our framework and present existence results for nonlinear Schroedinger (NLS) type energy functionals as generalizations and unification of various results obtained by several authors, most notably from [1] and [2], among others. Secondly, we consider spectral minimal partitions of compact metric graphs recently introduced in [3]. We show sharp lower and upper estimates for various spectral minimal energies, estimates between these energies and eigenvalues of the Laplacian and discuss their asymptotical behaviour. Thirdly, we present Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators of Schroedinger type on compact metric graphs. Among other things, these results characterize the accumulation points of the sequence $(\frac{\nu_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. Finally, we introduce a numerical method for calculating the eigenvalues for a special operator in the beforementioned class, the standard Laplacian, based on a discrete graph approximation.

References

[1] Adami, Serra and Tilli, Journal of Functional Analysis 271 (2016), 201-223

[2] Cacciapuoti, Finco and Noja, Nonlinearity 30 (2017), 3271-3303

[3] Kennedy et al, Calculus of Variations and Partial Differential Equations 60 (2021), 61

Monday

LisMath


Grace Mwakyoma, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

Abstract

In this talk I want to present my thesis work towards the classification of Hamiltonian circle actions on compact orbifolds of dimension 4, with isolated cyclic singularties.

I will start by explaining what orbifolds are and how circle actions are defined on them. Then I will explain how to associate a graph to any such space, and show that two 4 dimensional orbifolds with Hamiltonian circle actions are isomorphic, if and only if their graphs are isomorphic. I will then give a list of minimal models from which many orbifolds with a Hamiltonian circle action can be constructed.This work is inspired and based on the work of Yael Karshon, who classified Hamiltonian circle actions on compact manifolds of dimension four.

Monday

LisMath


Renato Ricardo de Paula, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

Abstract

In recent years, there has been intensive research activity around the derivation of partial differential equations with boundary conditions from interacting particle systems. This derivation is known as hydrodynamic limit. In this seminar we will discuss how to derive the porous medium equation with Dirichlet, Neumann, and Robin boundary conditions, from a random microscopic dynamics with slow reservoirs.

Monday

String Theory


Michal P. Heller, Albert Einstein Institute.

Abstract

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.

Monday

QM3 Quantum Matter meets Maths


, Massachusetts Institute of Technology.

Abstract

Recently, the notion of symmetry has been extended from 0-symmetry described by group to higher symmetry described by higher group. In this talk, we show that the notion of symmetry can be generalized even further to "algebraic higher symmetry". Then we will describe an even more general point of view of symmetry, which puts the (generalized) symmetry charges and topological excitation at equal footing: symmetry can be viewed as gravitational anomaly, or symmetry can be viewed as shadow topological order in one higher dimension. This picture allows us to see many duality relations between seeming unrelated symmetries.


Tuesday

Geometria em Lisboa


, University of Illinois at Chicago.

Abstract

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.


Wednesday

Probability and Statistics


, CMA, FCT NOVA e NOVA LINCS, FCT NOVA.

Abstract

Tumor heterogeneity plays a critical role in cancer progression and therapy resistance. Not only intertumoral heterogeneity leads to the definition of distinct tumor subtypes, but also intratumoral heterogeneity shows at distinct cell clones with different selective advantages. Emerging biomedical technologies, in particular, those generating omics data (e.g., genomics, transcriptomics, proteomics) now make it possible to ask which molecular entities govern tumor heterogeneity and can be candidates for disease biomarkers and therapeutical targets. Omics data are high-dimensional, with the number of features greatly outnumbering the number of observations. This calls for the need to develop statistical and machine learning methods able to translate vast amounts of data into meaningful biological solutions. Learning high-dimensional ‘omic data poses many challenges, in particular for parameter estimation and generation of interpretable solutions. In this talk, I will cover strategies for unveiling relevant information from high-dimensional omic data, including model regularization for feature selection and network-based modeling, with examples of application in the cancer research domain.

Wednesday

Applied Mathematics and Numerical Analysis

Online


, Departamento de Matemática FCUL and CEMAT, Lisboa, Portugal.

Abstract

Mathematical analysis is a useful tool to give insights in very different mathematical biology problems.

In this talk we will consider predator-prey and competition population dynamics models. We will give an overview of recent results in the case of seasonally forced models not entering in technical details.

First of all we consider predator-prey models with or without Allee effect and prove results on extinction or persistence. We will give some examples such as models including competition among predators, prey-mesopredator-superpredator models and Leslie-Gower systems. When Allee effect is considered, we deal with the cases of strong and weak Allee effect.

Then we consider competition models of two species giving conditions for the extinction of one or both species and for coexistence.

This talk is based in joint works with I. Coelho, M. Garrione, C. Soresina and E. Sovrano.

References:

[1] I. Coelho and C. Rebelo, Extinction or coexistence in periodic Kolmogorov systems of competitive type, submitted.

[2] M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Analysis: Real World Applications, 30, (2016) 73-98.

[3] C. Rebelo and C. Soresina, Coexistence in seasonally varying predator-prey systems with Allee effect, Nonlinear Anal. Real World Appl. 55 (2020), 103140, 21 pp.

Wednesday

Mathematics, Physics & Machine Learning


, Halicioğlu Data Science Institute, University of California San Diego.

Abstract

Recent empirical successes of deep learning have exposed significant gaps in our fundamental understanding of learning and optimization mechanisms. Modern best practices for model selection are in direct contradiction to the methodologies suggested by classical analyses. Similarly, the efficiency of SGD-based local methods used in training modern models, appeared at odds with the standard intuitions on optimization.

First, I will present evidence, empirical and mathematical, that necessitates revisiting classical notions, such as over-fitting. I will continue to discuss the emerging understanding of generalization, and, in particular, the "double descent" risk curve, which extends the classical U-shaped generalization curve beyond the point of interpolation.

Second, I will discuss why the landscapes of over-parameterized neural networks are generically never convex, even locally. Instead, as I will argue, they satisfy the Polyak-Lojasiewicz condition across most of the parameter space instead, which allows SGD-type methods to converge to a global minimum.

A key piece of the puzzle remains - how does optimization align with statistics to form the complete mathematical picture of modern ML?


Thursday

Lisbon WADE — Webinar in Analysis and Differential Equations


, IST Austria.

Abstract

Many stochastic systems can be viewed as gradient flow ('steepest descent') in the space of probability measures, where the driving functional is a relative entropy and the relevant geometry is described by a dynamical optimal transport problem. In this talk we focus on these optimal transport problems and describe recent work on the limit passage from discrete to continuous.

Surprisingly, it turns out that discrete transport metrics may fail to converge to the expected limit, even when the associated gradient flows converge. We will illustrate this phenomenon in examples and present a recent homogenisation result.

This talk is based on joint work with Peter Gladbach, Eva Kopfer, and Lorenzo Portinale.

Thursday

Lisbon WADE — Webinar in Analysis and Differential Equations


, Université de Paris.

Abstract

The Caffarelli contraction theorem states that optimal transport maps (for the quadratic cost) from a Gaussian measure onto measures that satisfy certain convexity properties are globally Lipschitz, with a dimension-free estimate. It has found many applications in probability, such as concentration and functional inequalities. In this talk, I will present an alternative proof, using entropic interpolation and variational arguments. Joint work with Nathael Gozlan and Maxime Prod'homme.


Monday

QM3 Quantum Matter meets Maths


, Centre de Physique Theorique, École Polytechnique, CNRS.

Abstract

Topological states of matter are characterized by a gap in the bulk of the system referring to an insulator or a superconductor and topological edge modes as well which find various applications in transport and spintronics. The bulk-edge correspondence is associated to a topological number. The table of topological states include the quantum Hall effect and the quantum anomalous Hall effect, topological insulators and topological superconductors in various dimensions and lattice geometries. Here, we discuss classes of states which can be understood from mapping onto a spin-1/2 particle in the reciprocal space of wave-vectors. We develop a geometrical approach on the associated Poincare-Bloch sphere, developing smooth fields, which shows that the topology can be encoded from the poles only. We show applications for the light-matter coupling when coupling to circular polarizations and develop a relation with quantum transport and the quantum Hall conductivity. The formalism allows to include interaction effects. We show our recent developments on a stochastic approach to englobe these interaction effects and discuss applications for the Mott transition of the Haldane and Kane-Mele models. Then, we develop a model of coupled spheres and show the possibility of fractional topological numbers as a result of interactions between spheres and entanglement allowing a superposition of two geometries, one encircling a topological charge and one revealing a Bell or EPR pair. Then, we show applications of the fractional topological numbers C=1/2 in bilayer honeycomb models describing topological semi-metals characterized by a quantized Berry phase at one Dirac point.

- Joel Hutchinson and Karyn Le Hur, arXiv:2002.11823 (under review)
- Philipp Klein, Adolfo Grushin, Karyn Le Hur, Phys. Rev. B 103, 035114 (2021)


Tuesday

Geometria em Lisboa


, Boston University.

Abstract

Tropical geometry is a useful tool to study the Gromov-Witten type invariants, which count the number of holomorphic curves with incidence conditions. On the other hand, holomorphic discs with boundaries on the Lagrangian fibration of a Calabi-Yau manifold play an important role in the quantum correction of the mirror complex structure. In this talk, I will introduce a version of open Gromov-Witten invariants counting such discs and the corresponding tropical geometry on (log) Calabi-Yau surfaces. Using Lagrangian Floer theory, we will establish the equivalence between the open Gromov-Witten invariants with weighted count of tropical discs. In particular, the correspondence theorem implies the folklore conjecture that certain open Gromov-Witten invariants coincide with the log Gromov-Witten invariants with maximal tangency for the projective plane.


Thursday

Lisbon WADE — Webinar in Analysis and Differential Equations

Online


, University of Bergen.

Abstract

The Dysthe equation is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. After reviewing the derivation of the Dysthe and related equations, we will focus on the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$.

Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S $ and its dual $V^2_S $ of square bounded variation functions.

We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s\gt 0$.

Our results extend to the finite depth version of the Dysthe equation.
This talk is based on a joint work with Razvan Mosincat (University of Bergen) and Jean-Claude Saut (Université Paris-Saclay).


Friday

Mathematics, Physics & Machine Learning


, University of Chicago.

Abstract

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional approaches by large margins. In this talk, I will describe the central prevailing themes of this emerging area and a taxonomy that can be used to categorize different problems and reconstruction methods. We will also explore mechanisms for model adaptation; that is, given a network trained to solve an initial inverse problem with a known forward model, we propose novel procedures that adapt the network to a perturbed forward model, even without full knowledge of the perturbation. Finally, I will describe a new class of approaches based on "infinite-depth networks" that can yield up to a 4dB PSNR improvement in reconstruction accuracy above state-of-the-art alternatives and where the computational budget can be selected at test time to optimize context-dependent trade-offs between accuracy and computation.


Monday

String Theory


Tarek Anous, University of Amsterdam.

Abstract

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.

Monday

QM3 Quantum Matter meets Maths


, Fribourg University.

Abstract

The eight-vertex model is an useful description that generalizes several spin systems, as well as the more common six-vertex model, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. However, no general theory is known for dimers in the non-bipartite case, contrary to the extensive rigorous description of Gibbs measures by Kenyon, Okounkov, Sheffield for bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, on generic planar graphs. I will mention a few consequences: computation of long-range correlations, criticality and critical exponents, and their "exact" application to Z-invariant regimes on isoradial graphs.




Thursday

Mathematical Relativity


, Sorbonne Université.

Abstract

We will discuss the large time behavior of solutions to Einstein's equations with a positive cosmological constant. This is a subject first addressed by Friedrich ('86) in vacuum, showing in particular the global stability of de Sitter space. Since then numerous works have investigated the asymptotic behavior of solutions with an accelerated expansion at infinity, in different contexts, and the effect of the expansion to various matter fields. We will give an account of known results and present a recent work that classifies the future dynamics of the FLRW solution for the massless-scalar field system.


Friday

Topological Quantum Field Theory


, Tsinghua University.

Abstract

We describe a general framework to study the quantum geometry of $\sigma$-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have exact descriptions at all loops in terms of target geometry and can be rigorously formulated. We illustrate this idea by the example of topological quantum mechanics which will lead to an explicit construction of the universal trace map on periodic cyclic chains of matrix Weyl algebras. As an application, we explain how to implement the idea of exact semi-classical approximation into a proof of the algebraic index theorem using Gauss Manin connection.

This is joint work with Zhengping Gui and Kai Xu.

Zhengping Gui, Si Li, Kai Xu,
Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index
https://arxiv.org/pdf/1911.11173




Friday

Mathematics, Physics & Machine Learning


, Georgia Institute of Technology.

Abstract

Cyber-physical systems (CPS) comprise interacting digital, analog, physical, and human components engineered for function through integrated physics and logic. Incorporating intelligence in CPS, however, makes their physical components more exposed to adversaries that can potentially cause failure or malfunction through actuation attacks. As a result, augmenting CPS with resilient control and design methods is of grave significance, especially if an actuation attack is stealthy. Towards this end, in the first part of the talk, I will present a receding horizon controller, which can deal with undetectable actuation attacks by solving a game in a moving horizon fashion. In fact, this controller can guarantee stability of the equilibrium point of the CPS, even if the attackers have an information advantage. The case where the attackers are not aware of the decision-making mechanism of one another is also considered, by exploiting the theory of bounded rationality. In the second part of the talk, and for CPS that have partially unknown dynamics, I will present an online actuator placement algorithm, which chooses the actuators of the CPS that maximize an attack security metric. It can be proved that the maximizing set of actuators is found in finite time, despite the CPS having uncertain dynamics.


Monday

String Theory


Daniel Waldram, Imperial College London.

Abstract

“G-algebroids” are natural extension of Lie and Courant algebroids that give a unified picture of the symmetries that underlie generalised and exceptional geometry as well as new “non-exact” versions. We analyse their structure in the exceptional case, and translate the problem of finding maximally supersymmetric consistent truncations to a relatively simple algebraic condition. We then show how Poisson-Lie U-duality is encoded in this framework and prove, in particular, that it is compatible with the supergravity equations of motion.


Friday

Mathematics, Physics & Machine Learning


, Universitat de València.

Abstract

Most problems in Earth sciences aim to do inferences about the system, where accurate predictions are just a tiny part of the whole problem. Inferences mean understanding variables relations, deriving models that are physically interpretable, that are simple parsimonious, and mathematically tractable. Machine learning models alone are excellent approximators, but very often do not respect the most elementary laws of physics, like mass or energy conservation, so consistency and confidence are compromised. I will review the main challenges ahead in the field, and introduce several ways to live in the Physics and machine learning interplay that allows us (1) to encode differential equations from data, (2) constrain data-driven models with physics-priors and dependence constraints, (3) improve parameterizations, (4) emulate physical models, and (5) blend data-driven and process-based models. This is a collective long-term AI agenda towards developing and applying algorithms capable of discovering knowledge in the Earth system.



Monday

String Theory


Syo Kamata, National Centre for Nuclear Research, Warsaw.

Abstract

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with $N $ minima on $S^{1}$. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of $N$-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with 't Hooft anomaly for even $N$ and global inconsistency for odd $N$. By using Delabaere Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

This talk is based on:

  1. On exact-WKB analysis, resurgent structure, and quantization conditions, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv:2008.00379.
  2. Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$, N.Sueishi, S.K, T.Misumi, and M.Ünsal, arXiv.2103.06586



Monday

String Theory


Sven Krippendorf, LMU Munich.

Abstract

Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of relevant structures, the Lax pair and Lax connection respectively. By formulating this search as an optimization problem, we are able to identify appropriate structures via machine learning techniques. We test our method on standard systems of classical integrability and find that we can distinguish between integrable and non-integrable deformations of a system. Due to the ambiguity in defining a Lax pair our algorithm identifies novel Lax pairs which can be easily verified analytically.


Tuesday

Geometria em Lisboa


, University of Toronto.

Abstract

Together with Jihyeon Jessie Yang, we are resurrecting an old idea of Raoul Bott for using large torus actions to construct canonical bases for unitary representations of compact Lie groups. Our methods are complex analytic; we apply them to families of Bott-Samelson manifolds parametrized by C^n. Our construction requires the vanishing of higher cohomology of sheaves of holomorphic sections of certain line bundles over the total spaces of such families; this vanishing is conjectural, hence the question mark in the title.


Friday

Topological Quantum Field Theory


, Mathematical Institute, University of Oxford.

Abstract

This is a report on joint work in progress with L. Katzarkov, M. Kontsevich, and P. Pandit. The Homological Mirror Symmetry conjecture is stated as an equivalence of triangulated categories, one coming from algebraic geometry and the other from symplectic topology. An enhancement of the conjecture also identifies stability conditions (in the sense of Bridgeland) on these categories. We adopt the point of view that triangulated (DG/A-infinity) categories are non-commutative spaces of an algebraic nature. A stability condition can then be thought of as the analog of a Kähler class or polarization. Many, often still conjectural, constructions of stability conditions hint at a richer structure which we think of as analogous to a Kähler metric. For instance, a type of Donaldson and Uhlenbeck-Yau theorem is expected to hold. I will discuss these examples and common features among them, leading to a tentative definition.


Monday

String Theory


Ingmar Saberi, University of Heidelberg.

Abstract

We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a $p$-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one $p$-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated).




Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT