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QM3 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 $\mathbb{Z}_2$ 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 $\mathbb{Z}$ 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.
Lisbon WADE — Webinar in Analysis and Differential Equations
Vertex coupling and spectra of periodic quantum graphs.
Pavel Exner, Doppler Institute for Mathematical Physics and Applied Mathematics, Prague.
Abstract
The talk focuses on the influence of the vertex coupling on spectral properties of periodic quantum graphs. Specifically, two questions will be addressed. The first concerns the number of open spectral gaps; it is shown that graphs with a nontrivial $\delta$ coupling can have finite but nonzero number of them. Secondly, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. For the simplest coupling of this type we show that its high-energy properties depend on the parity of the lattice vertices, and discuss various consequences of this property.
Geometria em Lisboa
Symplectic rational $G$-surfaces and the plane Cremona group.
Tian-Jun Li, University of Minnesota.
Abstract
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{CP}^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to the one in algebraic geometry. The connection with the symplectic mapping class group will be mentioned.
This is a joint work with Weimin Chen and Weiwei Wu (and partly with Jun Li).
Topological Quantum Field Theory mini workshop
Topological Links and Quantum Entanglement.
Gonçalo Quinta & Rui André, Physics of Information and Quantum Technologies Group - IST (GQ); Center for Astrophysics and Gravitation - IST (RA).
Abstract
We present a classification scheme for quantum entanglement based on topological links. This is done by identifying a nonrigid ring to a particle, attributing the act of cutting and removing a ring to the operation of tracing out the particle, and associating linked rings to entangled particles. This analogy naturally leads us to a classification of multipartite quantum entanglement based on all possible distinct links for any given number of rings. We demonstrate the use of this new classification scheme for three and four qubits, and introduce a new class of states inspired by topological symmetries of links.
Topological Quantum Field Theory mini workshop
Symplectic geometry, geometric quantization and the HOMFLYPT polynomial.
Mauro Spera, Università Cattolica del Sacro Cuore, Brescia, Italy.
Abstract
(joint work with Antonio Michele Miti)
In this talk we survey the symplectic geometric approach to knot framing via Lagrangian submanifold theory and geometric quantization developed in Besana&S. (2006) and extend it, taking inspiration from Liu&Ricca (2012,2015), to achieve a geometric interpretation of the HOMFLYPT polynomial based on helicity only.
Besana A. and Spera M., On some symplectic aspects of knots framings,
J. Knot Theory Ram. 15 (2006), 883-912.
Liu X. and Ricca R.L., The Jones polynomial for fluid knots from helicity,
J. Phys A: Math. Theor. 45 (2012), 205501 (14pp).
Liu X. and Ricca R.L., On the derivation of the HOMFLYPT polynomial invariant for fluid knots,
J. Fluid Mechanics 773 (2015), 34-48.
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
Galois symmetries of knot spaces.
Pedro Boavida de Brito, Instituto Superior Técnico and CAMGSD.
Abstract
I’ll describe how the absolute Galois group of the rationals acts on a space which is closely related to the space of all knots. The path components of this space form a finitely generated abelian group which is, conjecturally, a universal receptacle for integral finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteristic zero. I will then discuss some work in progress concerning higher-dimensional variants.
This is joint work with Geoffroy Horel.
QM3 Quantum Matter meets Maths
A new point of view on topological phase transitions.
Christophe Garban, Université Lyon 1.
Abstract
Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless in the 70's. They describe intriguing phase transitions for classical spins systems such as the plane rotator model (or $XY$ model). I will start by reviewing how this phase transition arises in cases such as:
- the $XY$ model (spins on $\mathbb{Z}^2$ with values in the unit circle)
- the integer-valued Gaussian Free Field (or $\mathbb{Z}$-ferromagnet)
- Abelian Yang-Mills on $\mathbb{Z}^4$
I will then connect topological phase transitions to a statistical reconstruction problem concerning the Gaussian Free Field and will show that the feasibility of the reconstruction undergoes a KT transition.
This is a joint work with Avelio Sepúlveda (Lyon) and the talk will be based mostly on the preprint: https://arxiv.org/abs/2002.12284
Geometria em Lisboa
To be announced.
Colin Guillarmou, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud.
Topological Quantum Field Theory mini workshop
2-representation theory of Soergel bimodules.
Vanessa Miemitz, University of East Anglia, UK.
Abstract
I will explain how to reduce the classification of ‘simple’ 2-representations of the 2-category of Soergel bimodules in many (most) cases to the known problem of the same classification for certain fusion categories.
Topological Quantum Field Theory mini workshop
Adjunction in the absence of identity.
Walter Mazorchuk, Uppsala University, Sweden.
Abstract
In this talk I plan to present and discuss a rather weak bicategorical setup in which one can talk about genuine adjoint 1-morphisms. I will describe the main motivation from representation theory of finitary 2-categories (or bicategories) and make some parallells with the stucture theory of finite semigroups. I will also try to show how this approach simplifies formulation of some results from 2-representation theory, but also how it makes some other “classical” results much more difficult.
Joint work with Hankyung Ko and Xiaoting Zhang.
Probability and Statistics
COVID, uncertainty and clinical trials.
Joaquim Ferreira, Laboratório de Farmacologia Clínica e Terapêutica, Faculdade de Medicina, Universidade de Lisboa.
Abstract
The current COVID-19 pandemic is putting an enormous pressure not just in the society but also in all the scientific community.
If we want to follow a scientific approach to respond to the doubts and challenges that were generated, we need to find a balance between the most robust data, the best experimental methodologies to address the new problems and all the uncertainty associated.
In this presentation we will try to address this balance between best available data, clinical research methodology and uncertainty applied to what we know about pandemics, vaccine development and clinical trials. There will be a particular focus on the COVID-19 pandemic data and current research efforts for the development of vaccines and efficacious treatments.
Mathematics, Physics & Machine Learning
Progress and hurdles in the statistical mechanics of deep learning.
Marylou Gabrié, Center for Data Science, NYU and Flatiron Institute, CCM.
Abstract
Understanding the great performances of deep neural networks is a very active direction of research with contributions coming from a wide variety of fields. The statistical mechanics of learning is a theoretical framework dating back to the 80s studying learning problems from a physicist viewpoint and using tools from the physics of disordered systems. In this talk, I will first go over this traditional framework, which relies on the teacher-student scenario, bayesian analysis and mean-field approximations. Then I will discuss some recent advances in the corresponding analysis of modern deep neural network, and highlight remaining challenges.
Topological Quantum Field Theory
Gauge fixing in supersymmetric field theories with topological terms.
Ezra Getzler, Northwestern University.
QM3 Quantum Matter meets Maths
To be announced.
Raquel Queiroz, Weizmann Institute of Science.
Geometria em Lisboa
Intrinsic Mirror Symmetry.
Mark Gross, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge.
Abstract
I will talk about joint work with Bernd Siebert, proposing a general mirror construction for log Calabi-Yau pairs, i.e., a pair $(X,D)$ with $D$ a “maximally degenerate” boundary divisor and $K_X+D=0$, and for maximally unipotent degenerations of Calabi - Yau manifolds. We accomplish this by constructing the coordinate ring or homogeneous coordinate ring respectively in the two cases, using certain kinds of Gromov-Witten invariants we call “punctured invariants”, developed jointly with Abramovich and Chen.
Topological Quantum Field Theory
Homotopy Quantum Field Theories.
Alexis Virelizier, Université de Lille.
Geometria em Lisboa
An invitation to Kähler-Einstein metrics and random point processes.
Robert Berman, Chalmers University of Technology.
Topological Quantum Field Theory
Poisson sigma model and integrable systems.
Nicolai Reshetikhin, University of California, Berkeley.
Topological Quantum Field Theory
Relative mapping class group representations via conformal nets.
André Henriques, University of Oxford.
Geometria em Lisboa
To be announced.
Gonçalo Oliveira, Universidade Federal Fluminense, Brasil.
Probability and Statistics
From high dimensional space to a random low dimensional space.
Conceição Amado, Instituto Superior Técnico and CEMAT.
Topological Quantum Field Theory
A solution of the Riemann-Hilbert problem on the $A_2$ quiver.
Davide Masoero, Group of Mathematical Physics, University of Lisbon.
Geometria em Lisboa
To be announced.
Éveline Legendre, Université Paul Sabatier.
Probability and Statistics
To be announced.
Manuel Scotto, Instituto Superior Técnico and CEMAT.
Topological Quantum Field Theory
Cluster realization of quantum groups and higher Teichmüller theory.
Alexander Shapiro, UC Berkeley.
Geometria em Lisboa
To be announced.
Silvia Anjos, IST/CAMGSD.