## Search

## Probability and Statistics

Quasi-analytical solution of an investment problem with decreasing investment cost due to technological innovations.

*Cláudia Nunes*, CEMAT-IST.

## Abstract

In this talk we address, in the context of real options, an investment problem with two sources of uncertainty: the price (reflected in the revenue of the firm) and the level of technology. The level of technology impacts in the investment cost, that decreases when there is a technology innovation. The price follows a geometric Brownian motion, whereas the technology innovations are driven by a Poisson process. As a consequence, the investment region may be attained in a continuous way (due to an increase of the price) or in a discontinuous way (due to a sudden decrease of the investment cost).

For this optimal stopping problem no analytical solution is known, and therefore we propose a quasi-analytical method to find an approximated solution that preserves the qualitative features of the exact solution. This method is based on a truncation procedure and we prove that the truncated solution converges to the solution of the original problem.

We provide results for the comparative statics for the investment thresholds. These results show interesting behaviors, particularly, the investment may be postponed or anticipated with the intensity of the technology innovations and with their impact on the investment cost.

(joint work with Carlos Oliveira and Rita Pimentel)

## Mathematics, Physics & Machine Learning

Path integral control theory.

*Hilbert Johan Kappen*, Donder Institute, Radboud University Nijmegen, the Netherlands.

## Abstract

Stochastic optimal control theory deals with the problem to compute an optimal set of actions to attain some future goal. Examples are found in many contexts such as motor control tasks for robotics, planning and scheduling tasks or managing a financial portfolio. The computation of the optimal control is typically very difficult due to the size of the state space and the stochastic nature of the problem. Special cases for which the computation is tractable are linear dynamical systems with quadratic cost and deterministic control problems. For a special class of non-linear stochastic control problems, the solution can be mapped onto a statistical inference problem. For these so-called path integral control problems the optimal cost-to-go solution of the Bellman equation is given by the minimum of a free energy. I will give a high level introduction to the underlying theory and illustrate with some examples from robotics and other areas.

## Topological Quantum Field Theory

Knot invariants from homotopy theory.

*Danica Kosanović*, Max-Planck Institut für Mathematik.

## Abstract

The embedding calculus of Goodwillie and Weiss is a certain homotopy theoretic technique for studying spaces of embeddings. When applied to the space of knots this method gives a sequence of knot invariants which are conjectured to be universal Vassiliev invariants. This is remarkable since such invariants have been constructed only rationally so far and many questions about possible torsion remain open. In this talk I will present a geometric viewpoint on the embedding calculus, which enables explicit computations. In particular, we prove that these knot invariants are surjective maps, confirming a part of the universality conjecture, and we also confirm the full conjecture rationally, using some recent results in the field. Hence, these invariants are at least as good as configuration space integrals.

## String Theory

Perturbative gravity via BRST Yang-Mills^{2}.

*Silvia Nagy*, University of Nottingham.

## Abstract

I will present a formulation of gravity as a double copy of gauge theories in the context of the Becchi-Rouet-Stora-Tyutin (BRST) formalism. I will show how this gives an algorithm for consistently mapping gauge choices from Yang-Mills to gravity. Moreover, it resolves the issue of the dilaton degree of freedom arising in the double copy, thus allowing for the consistent construction of solutions in General Relativity. I will describe the perturbative construction at higher orders. I will also give a formulation of the BRST double copy in a spherical background.

## Lisbon WADE — Webinar in Analysis and Differential Equations

Nonunique characteristic curves of Sobolev vector fields.

*Maria Colombo*, École Polytechnique Fédérale de Lausanne.

## Abstract

Given a vector field in $\mathbb{R}^d$, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow provided the vector field is sufficiently smooth; this, in turn, translates in existence and uniqueness results for the transport equation. In 1989, Di Perna and Lions proved that Sobolev regularity for vector fields, with bounded divergence and a growth assumption, is sufficient to establish existence, uniqueness and stability of a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollaryof the uniqueness of the trajectory of the ODE for a.e. initial datum. In this talk we give an overview of the topic and we provide a negative answer to this question. To show this result we exploit the connection with the transport equation, based on Ambrosio’s superposition principle, and a new ill-posedness result for positive solutions of the continuity equation.

## Geometria em Lisboa

Probabilistic aspects of toric Kahler geometry.

*Steve Zelditch*, Northwestern University.

## Abstract

Let $(M, \omega, L)$ be a polarized toric Kahler manifold with polytope $P$. Associated to this data is a family $\mu_k^x$ of probability measures on $P$ parametrized by $x \in P.$ They generalize the multi-nomial measures on the simplex, where $M = \mathbb{CP}^n$ and $\omega$ is the Fubini-Study measure. As is well-known, these measures satisfy a law of large numbers, a central limit theorem, a large deviations principle and entropy asymptotics. The measure of maximal entropy in this family corresponds to the center of mass $x$ of $P$. All of these results generalize to any toric Kahler manifold, except the center of mass result, which holds for Fano toric Kahler-Einstein manifolds.

Joint work with Peng Zhou and Pierre Flurin.

## QM^{3} Quantum Matter meets Maths

Turbulent hydrodynamics in strongly correlated Kagome metals.

*Johanna Erdmenger*, University of Würzburg.

## Abstract

A current challenge in condensed matter physics is the realization of strongly correlated, viscous electron fluids. These fluids are not amenable to the perturbative methods of Fermi liquid theory, but can be described by holography, that is, by mapping them onto a weakly curved gravitational theory via gauge/gravity duality. The canonical system considered for realizations has been graphene, which possesses Dirac dispersions at low energies as well as significant Coulomb interactions between the electrons. In this work, we show that Kagome systems with electron fillings adjusted to the Dirac nodes of their band structure provide a much more compelling platform for realizations of viscous electron fluids, including non-linear effects such as turbulence. In particular, we find that in stoichiometric Scandium (Sc) Herbertsmithite, the fine-structure constant, which measures the effective Coulomb interaction and hence reflects the strength of the correlations, is enhanced by a factor of about 3.2 as compared to graphene, due to orbital hybridization. We employ holography to estimate the ratio of the shear viscosity over the entropy density in Sc-Herbertsmithite, and find it about three times smaller than in graphene. These findings put, for the first time, the turbulent flow regime described by holography within the reach of experiments.

**See also**: Reference

## LisMath

Quiver Representations.

*Javier Orts*, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

## Abstract

A quiver is a directed graph where multiple arrows between two vertices and loops are allowed. A representation of a quiver $Q$, over a field $K$, is an assignment of a finite dimensional $K$-vector space $V_i$ to each vertex $i$ of $Q$ and a linear map $f_a:V_i\rightarrow V_j$ to each arrow $a:i\rightarrow j$. Given a quiver $Q$, the set of all representations of $Q$ forms a category, denoted by $\mathrm{Rep}(Q)$. A connected quiver is said to be of finite type if it has only finitely many isomorphism classes of indecomposable representations.

Quiver representations have remarkable connections to other algebraic topics, such as Lie algebras or quantum groups, and provide important examples of moduli spaces in algebraic geometry [3].

The main goal of this work would consist, first, of good comprehension of the category $\mathrm{Rep}(Q)$. Then, the student would cover the basics on quiver representations to be able to prove Gabriel's theorem [1], following a modern approach, as in [2]:

*A connected quiver is of finite type if and only if its underlying graph is one of the ADE Dynkin diagrams $A_n$, $D_n$, for $n \in \mathbb N$, $E_6$, $E_7$ or $E_8$. Moreover, the indecomposable representations of a given quiver of finite type are in one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.*

These basic concepts involve topics such as the Jacobson radical, Dynkin diagrams or homological algebra of quiver representations.

**Bibliography:**

[1] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Mathematica **6**, pp. 71–103 (1972).

[2] H. Derksen and J. Weyzman, An Introduction to Quiver Representations, Graduate Studies in Mathematics 184, American Mathematical Society (2017).

[3] A. Soibelman, Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry, arXiv:1909.03509 (2019)

## Mathematics, Physics & Machine Learning

Computation, Statistics, and Optimization of random functions.

*Afonso Bandeira*, ETH Zurich.

## Abstract

When faced with a data analysis, learning, or statistical inference problem, the amount and quality of data available fundamentally determines whether such tasks can be performed with certain levels of accuracy. Indeed, many theoretical disciplines study limits of such tasks by investigating whether a dataset effectively contains the information of interest. With the growing size of datasets however, it is crucial not only that the underlying statistical task is possible, but also that is doable by means of efficient algorithms. In this talk we will discuss methods aiming to establish limits of when statistical tasks are possible with computationally efficient methods or when there is a fundamental *Statistical-to-Computational gap* in which an inference task is statistically possible but inherently computationally hard.

This is intimately related to understanding the geometry of random functions, with connections to statistical physics, study of spin glasses, random geometry; and in an important example, algebraic invariant theory.

## Topological Quantum Field Theory

Bundle Gerbes on Supermanifolds.

*John Huerta*, Instituto Superior Técnico and CAMGSD.

## Abstract

Bundle gerbes are a generalization of line bundles that play an important role in constructing WZW models with boundary. With an eye to applications for WZW models with superspace target, we describe the classification of bundle gerbes on supermanifolds, and sketch a proof of their existence for large families of super Lie groups.

## String Theory

New boundary conditions for $AdS_2$.

*Charles Marteau*, Institut Polytechnique de Paris.

## Abstract

We describe new boundary conditions for $AdS_2$ in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to $Diff(S^1) \ltimes C^{\infty}(S^1)$, whose breaking to $SL(2, R) \times U(1)$ controls the near-$AdS_2$ dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory. It can be interpreted as the coadjoint action of the warped Virasoro group. We show that this theory is holographically dual to the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

## Lisbon WADE — Webinar in Analysis and Differential Equations

To be announced.

*Lucio Boccardo*, Università di Roma La Sapienza.

## Geometria em Lisboa

Partial orders on contactomorphism groups and their Lie algebras.

*Kai Cieliebak*, Augsburg University.

## Abstract

Eliashberg, Kim and Polterovich constructed nontrivial partial orders on contactomorphism groups of certain contact manifolds. After recalling their results, the subject of this talk will be the remnants of these partial orders on the orbits of the coadjoint action on their Lie algebras.

## QM^{3} Quantum Matter meets Maths

Quantum many-body scars: a new form of weak ergodicity breaking in constrained quantum systems.

*Zlatko Papic*, University of Leeds.

## Abstract

Recent experiments on large chains of Rydberg atoms [1] have demonstrated the possibility of realising one-dimensional, kinetically constrained quantum systems. It was found that such systems exhibit surprising signatures of non-ergodic dynamics, such as robust periodic revivals in global quenches from certain initial states. This weak form of ergodicity breaking has been interpreted as a manifestation of "quantum many-body scars" [2], i.e., the many-body analogue of unstable classical periodic orbits of a single particle in a chaotic stadium billiard. Scarred many-body eigenstates have been shown to exhibit a range of unusual properties which violate the Eigenstate Thermalisation Hypothesis, such as equidistant energy separation, anomalous expectation values of local observables and subthermal entanglement entropy. I will demonstrate that these properties can be understood using a tractable model based on a single particle hopping on the Hilbert space graph, which formally captures the idea that scarred eigenstates form a representation of a large $\operatorname{SU}(2)$ spin that is embedded in a thermalising many-body system. I will show that this picture allows to construct a more general family of scarred models where the fundamental degree of freedom is a quantum clock [3]. These results suggest that scarred many-body bands give rise to a new universality class of constrained quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

## Mathematics, Physics & Machine Learning

Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau.

*Marcelo Pereyra*, Heriot-Watt University.

## Abstract

This talk summarises some new developments in Bayesian statistical methodology for performing inference in high-dimensional inverse problems with an underlying convex geometry. We pay particular attention to problems related to imaging sciences and to new stochastic computation methods that tightly combine proximal convex optimisation and Markov chain Monte Carlo sampling techniques. The new computation methods are illustrated with a range of imaging experiments, where they are used to perform uncertainty quantification analyses, automatically adjust regularisation parameters, and objectively compare alternative models in the absence of ground truth.

## Topological Quantum Field Theory

Conformal Bootstrap for Liouville Theory.

*Antti Kupiainen*, University of Helsinki.

## Lisbon WADE — Webinar in Analysis and Differential Equations

To be announced.

*Riccardo Adami*, Politecnico di Torino.

## Geometria em Lisboa

Quasi-parabolic Higgs bundles and null hyperpolygon spaces.

*Alessia Mandini*, PUC-Rio.

## LisMath

Nonabelian Cohomology.

*Arber Selimi*, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

## Abstract

For a given groupoid $G$ and $M$ a $G$-module, the $n$-th cohomology is defined as the set of homotopy classes $H^n(G,M)=[F_{\star}^{st} (G), K_n(M,G);\phi ]$, where $F_{\star}^{st} (G)$ is the free crossed resolution of $G$, and $\phi : F_1^{st} (G)\to G$ is the standard morphism.

In this talk we assign a free crossed complex to a cover $\mathcal{U}$ of the topological space $X$, so we get the notion of nonabelian cohomology.

We finish our talk by introducing a long exact sequence for nonabelian cohomology.

**Bibliography:**

[1] R. Brown , P. Higgins and R. Sivera. Nonabelian algebraic topology. European Mathematical Society, 2010

[2] T. Nikolaus and K. Waldorf. Lifting problems and transgression for non-abelian gerbes, Advances in Mathematics 242, pp. 50–79, 2013

[3] L. Breen. Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, pp. 401–476, 2007.

## Probability and Statistics

Investment problem with switching modes.

*Igor Kravchenko*, CEMAT-IST.

## Abstract

In this talk we will look at the optimal control problem of a firm that may operate in two different modes, one being more risky than the other, in the sense that in case the demand decreases, the return of the risky mode is lower than with the more conservative mode. On the other side, in case the demand increases, the opposite holds. The switches between these two alternative modes have associated costs. In both modes, there is the option to exit the market.

We will focus on two different parameter scenarios, that describe particular (and somehow extreme) economic situations. In the first scenario, we assume that the market is expected to increase in such a way that once the firm is producing in the more risky mode, it is never optimal to switch to the more conservative one. In the second scenario, there is a hysteresis region, where the firm is waiting in the more risky mode, in production, until some drop or increase in the demand leads to an exit or changing to the more conservative mode. This hysteresis region cannot be attained under continuous production.

We then address the problem of the optimal time to invest under each situation. Depending on the relation between the switching costs (equal or different from one mode to another), it may happen that the firm invests in the hysteresis region.

Joint work with Cláudia Nunes and Carlos Oliveira

## Mathematics, Physics & Machine Learning

Wave dynamics for manifold learning.

*Akshat Kumar*, Clarkson University.

## Geometria em Lisboa

Gauge theory for string algebroids.

*Mario Garcia-Fernandez*, ICMAT and Universidad Autónoma de Madrid.

## Abstract

In this talk I will overview recent joint work with Roberto Rubio and Carl Tipler in arXiv:2004.11399. We introduce a moment map picture for string algebroids, a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329. An interesting feature of our construction is that the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of the moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions. Assuming a technical condition, we prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the *dilaton functional*, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system.

## QM^{3} Quantum Matter meets Maths

Topological theory of non-Hermitian photonic systems.

*Mário Silveirinha*, Instituto Superior Técnico.

## Abstract

Recently, topological materials and topological effects have elicited a great interest in the photonics community [1]. While condensed-matter phenomena are traditionally described by Hermitian operators, the same is not true in the context of macroscopic electrodynamics where a dissipative response is the rule, not the exception. In this talk, I will discuss how to determine the topological phases of dissipative (non-Hermitian) photonic structures from first principles using a gauge-independent Green function [2, 3]. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. I apply the developed theory to nonreciprocal electromagnetic continua and photonic crystals, with lossy and or gainy elements. Furthermore, I discuss the validity of the bulk-edge correspondence in the non-Hermitian case.

[1] L. Lu, J. D. Joannopoulos, M. Soljačić, "Topological photonics", Nat. Photonics, 8, 821, (2014).

[2] M. G. Silveirinha, "Topological theory of non-Hermitian photonic systems", Phys. Rev. B, 99, 125155, 2019.

[3] F. R. Prudêncio, M. G. Silveirinha, First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals, arXiv:2003.01539

## LisMath

From racks to pointed Hopf algebras.

*António Lages*, LisMath, Instituto Superior Técnico, Universidade de Lisboa.

## Abstract

A walk from racks to pointed Hopf algebras with a view on solutions of the Yang-Baxter equations.

**Bibliography:**

[1] Andruskiewitsch and Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003) 177-243

## Mathematics, Physics & Machine Learning

To be announced.

*Csaba Szepesvári*, University of Alberta and DeepMind.

## String Theory

Complexity in the presence of a boundary.

*Aldo Cotrone*, University of Florence.

## Abstract

The effects of a boundary on the circuit complexity are studied in two dimensional theories. The analysis is performed in the holographic realization of a conformal field theory with a boundary by employing different proposals for the dual of the complexity.

## QM^{3} Quantum Matter meets Maths

The insulating state of matter: a geometrical theory.

*Raffaele Resta*, Democritos, IOM-CNR, Trieste, Italy.

## Probability and Statistics

COVID, uncertainty and clinical trials.

*Joaquim Ferreira*, FMUL and IMM.

## Mathematics, Physics & Machine Learning

Climate action and cooperation dynamics under uncertainty.

*Francisco C. Santos*, Instituto Superior Técnico and INESC-ID.

## Geometria em Lisboa

Geometry of symplectic rational surfaces.

*Tian-Jun Li*, University of Minnesota.

## QM^{3} Quantum Matter meets Maths

Four-dimensional semimetals with tensor monopoles: from surface states to topological responses.

*Giandomenico Palumbo*, Université Libre de Bruxelles.

## Abstract

Quantum anomalies offer a useful guide for the exploration of transport phenomena in topological semimetals. A prominent example is provided by the chiral magnetic effect in three-dimensional Weyl semimetals, which stems from the chiral anomaly. Here, we reveal a distinct quantum effect, coined "parity magnetic effect", which is induced by the parity anomaly in a four-dimensional topological semimetal. Upon preserving time-reversal symmetry, the spectrum of our model is doubly degenerate and the nodal (Dirac) points behave like Z2 monopoles. When time-reversal symmetry is broken, while preserving the sublattice (chiral) symmetry, our system supports spin- 3/2 quasiparticles and the corresponding Dirac-like cones host tensor monopoles characterized by a Z number, the Dixmier-Douady invariant. In both cases, the semimetal exhibits topologically protected Fermi arcs on its boundary. Besides its theoretical implications in both condensed matter and quantum field theory, the peculiar 4D magnetic effect revealed by our model could be measured by simulating higher-dimensional semimetals in synthetic matter.

## Probability and Statistics

Elements of Bayesian geometry.

*Miguel de Carvalho*, University of Edinburgh.

## Abstract

In this talk, I will discuss a geometric interpretation to Bayesian inference that will yield a natural measure of the level of agreement between priors, likelihoods, and posteriors. The starting point for the construction of the proposed geometry is the observation that the marginal likelihood can be regarded as an inner product between the prior and the likelihood. A key concept in our geometry is that of compatibility, a measure which is based on the same construction principles as Pearson correlation, but which can be used to assess how much the prior agrees with the likelihood, to gauge the sensitivity of the posterior to the prior, and to quantify the coherency of the opinions of two experts. Estimators for all the quantities involved in our geometric setup are discussed, which can be directly computed from the posterior simulation output. Some examples are used to illustrate our methods, including data related to on-the-job drug usage, midge wing length, and prostate cancer.

Joint work with G. L. Page and with B. J. Barney

## Mathematics, Physics & Machine Learning

Reinforcement Learning and Adaptive Control.

*João Miranda Lemos*, Instituto Superior Técnico and INESC-ID.

## Abstract

The aim of this seminar is to explain, to a wide audience, how to combine optimal control techniques with reinforcement learning, by using approximate dynamic programming, and artificial neural networks, to obtain adaptive optimal controllers. Although with roots since the end of the XX century, this problem has been the subject of an increasing attention. In addition to the promising tools that it offers to tackle difficult nonlinear problems with major engineering importance (ranging from robotics to biomedical engineering and beyond), it has the charm of creating a meeting point between the control and machine learning research communities.

## Topological Quantum Field Theory

Rational and algebraic links and knots-quivers correspondence.

*Marco Stošić*, Instituto Superior Técnico and CAMGSD.

## Topological Quantum Field Theory

Homotopy Quantum Field Theories.

*Alexis Virelizier*, Université de Lille.