Seminars from until

We develop a model of one-dimensional (Conformal) Quantum Gravity. By discussing the connection between Goldstone and Gauge theories, we establish that this model effectively computes the partition function of the Schwarzian theory where the $\operatorname{SL}(2,\mathbb{R})$ symmetry is realized on the base space. The computation is straightforward, involves a local quantum measure and does not rely on localization arguments. Non-localities in the model are exclusively related to the value of fixed gauge invariant moduli. Furthermore, we study the properties of these models when all degrees of freedom are allowed to fluctuate. We discuss the UV finiteness properties of these systems and the emergence of a Planck's length.

We study the existence of approximate pure Nash equilibria (PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d. Previously it was known that d-approximate equilibria always exist, while nonexistence was established only for small constants, namely for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have √d-approximate PNE, which provides the first super-constant lower bound.

Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an √d-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games.

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Our homogenization method is based on the notion of stochastic two-scale Young measures, whose existence we establish.

In this talk we introduce models of short wave-long wave interactions in the relativistic setting. In this context the nonlinear Schrödinger equation is no longer adequate for describing short waves and is replaced by a nonlinear Dirac equation. Two specific examples are considered: the case where the long waves are governed by a scalar conservation law; and the case where the long waves are governed by the augmented Born-Infeld equations in electromagnetism. This is a joint work with João Paulo Dias.

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.

The cylinder’s percolation model arises from a Poissonian soup of infinite lines in $R^d$ and it is a stationary process under the isometries of the underlying space. Each such line is then thickened, becoming the axis of a cylinder of radius one. The associated percolation picture exhibits long range correlations and the rigidity of the underlying objects hampers direct attempts at proving decorrelation inequalities via sprinkling of the intensity parameter. We obtain such inequalities by exploiting the continuity of the process, taking the radii of the cylinders as a parameter and using it in a sprinkling argument. As an application, we prove that for small intensities of the cylinder’s process the simple random walk on the vacant set is transient. The talk will also go over similar decoupling inequalities for other models, their applications and open problems.
This talk is based on a joint work with Caio Alves.

The problem of determining the density distribution of celestial bodies from the induced gravitational pull is of great importance in astrophysics as well as space engineering (thinking of situations where spacecraft need to perform orbital and surface proximity operations). Knowledge of a body density distribution provides also great insights on the body's origin and composition. In practice, the state-of-the-art approaches for modelling the gravity field of extended bodies are spherical harmonics models, mascon models and polyhedral gravity models. All of these, however, while being widely studied and developed since the early works from Laplace, introduce requirements such as knowledge of a shape model, assumption of a homogeneous internal density, being outside the Brillouin sphere, etc...

In this talk, we introduce and explain Neural Density Fields, a new approach to represent the density of extended bodies and learn its accurate form inverting data from gravitational accelerations, orbits or the gravity potential. The resulting deep learning model, called geodesyNets is able to compete with classical approaches while solving most of their limitations. We also introduce eclipseNets, a deep learning model based on related ideas and able to learn the eclipse shadow cones of irregular bodies, thus allowing highly precise propagation and stability studies.

I will present our theory of the Regularity Transformation (RT-)equations, an elliptic system of partial differential equations which determines coordinate and gauge transformations that remove apparent singularities in spacetime by establishing optimal regularity for general connections. This gain of one derivative for the connections above their L^p curvature then suffices to establish Uhlenbeck compactness. By developing an existence theory for the RT-equations we prove optimal regularity and Uhlenbeck compactness in Lorentzian geometry, including general affine connections and connections on vector bundles with both compact and non-compact gauge groups. As an application in General Relativity, our optimal regularity result implies that the Lorentzian metrics of shock wave solutions of the Einstein-Euler equations are non-singular---geodesic curves, locally inertial coordinates and the Newtonian limit all exist in a classical sense---, resolving a longstanding open problem in the field.

We present a new infinite class of gravitational observables in asymptotically Anti-de Sitter space living on codimension-one slices of the geometry, the most famous of which is the volume of the maximal slice. We show that these observables display universal features for the thermofield-double state: they grow linearly in time at late times and reproduce the switch-back effect in shock wave geometries. We argue that any member of this class of observables is an equally viable candidate as the extremal volume for a gravitational dual of complexity.

It is well known that for surfaces the positivity property of the cotangent bundle $\Omega^1_X$ called bigness implies hyperbolic properties. We give a criterion for bigness of $\Omega^1_X$ involving the singularities of the canonical model of $X$ and compare it with other criterions. The criterion involves invariants of the canonical singularities whose values were unknown. We describe a method to find the invariants and obtain formulas for the $A_n$ singularities. An application of this work is to determine for which degrees do hypersurfaces in $\mathbb {P}^3$ have deformations with big cotangent bundles and have symmetric differentials of low degrees.

Learning a behaviour to conduct a given task can be achieved by interacting with the the environment. This is the crux of reinforcement learning (RL), where an (automated) agent learns to solve a problem through an iterative trial-and-error process. More specifically, an RL agent can interact with the environment and learn from these interactions by observing a feedback on the goal task. Therefore, these methods typically require to be able to intervene on the environment and make (possibly a very large number of) mistakes. Although this can be a limiting factor in some applications, simple RL settings, such as bandit settings, can still host a variety of problems for interactively learning behaviours. In other situations, simulation might be the key.

In this talk, we will show that RL can be used to formulate and tackle data acquisition (imaging) problems in neurosciences. We will see how bandit methods can be used to optimize super-resolution imaging by learning on real devices through an actual empirical process. We will also see how simulation can be leveraged to learn more sequential decision making strategies. These applications highlight the potential of RL to support expert users on difficult task and enable new discoveries.

Lecture 1: From one-dimensional interacting particle systems to hyperbolic systems of conservation laws.

Lecture 2: Eigenmodes, fluctuation fields and dynamical universality.

Lecture 3: Mode coupling theory.

Software engineers and analysts traditionally focus on cyber systems as technical systems, which are built only from software processes, communication protocols, crypto algorithms, etc. They often neglect, or choose not, to consider the human user as a component of the system’s security as they lack the expertise to fully understand human factors and how they affect security. However, humans should not be designed out of the security loop. Instead, we must deal with security assurance as a true socio-technical problem rather than a mere technical one, and consider cyber systems as socio-technical systems with people at their hearts. The main goal of this talk is to advocate the use of formal methods to establish the security of socio-technical systems, and to discuss some of the most promising approaches, including those that I have helped develop. I will also discuss my recent work on “Cybersecurity Show and Tell”, namely how different kinds of artworks can be used to explain cybersecurity and how telling (i.e., explaining notions in a formal, technical way) can be paired with showing through visual storytelling or other forms of storytelling.

Dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems, especially when the governing equations are not known. The Buckingham Pi theorem provides a procedure for finding a set of dimensionless groups from given measurements, although this set is not unique. We propose an automated approach using the symmetric and self-similar structure of available measurement data to discover the dimensionless groups that best collapse this data to a lower dimensional space according to an optimal fit. We develop three data-driven techniques that use the Buckingham Pi theorem as a constraint: (i) a constrained optimization problem with a nonparametric function, (ii) a deep learning algorithm (BuckiNet) that projects the input parameter space to a lower dimension in the first layer, and (iii) a sparse identification of nonlinear dynamics (SINDy) to discover dimensionless equations whose coefficients parameterize the dynamics. I discuss the accuracy and robustness of these methods when applied to known nonlinear systems.

Lecture 4: The Fibonacci family of dynamical universality classes.

We consider the equations of a non-Newtonian incompressible fluid in a general time space cylinder $Q_{T}= \Omega \times (0,T) \subset \mathbb{R}^{n} \times \mathbb{R}, n \geq 2$. We assume that the rheology of the fluid is changing with respect to time and space and satisfies for each $(x,t) \in Q_{T}$ the associated power law $ |D|^{p(x,t) } D $. Under the assumption that $ \frac{2n}{n+2} < p_{0} \le p(x,t) \leq p_{1} < +\infty$ and the set of discontinuity of $p$ is closed and of measure zero we show the existence of a weak solution to the corresponding equations of PDEs for any given initial velocity in $L^{2}_{\sigma } (\Omega) $. Joint work with Prof. H-O. Bae (Ajou University, Suwon).

Deep neural networks have drastically changed the landscape of several engineering areas such as computer vision and natural language processing. Notwithstanding the widespread success of deep networks in these, and many other areas, it is still not well understood why deep neural networks work so well. In particular, the question of which functions can be learned by deep neural networks has remained unanswered.

In this talk we give an answer to this question for deep residual neural networks, a class of deep networks that can be interpreted as the time discretization of nonlinear control systems. We will show that the ability of these networks to memorize training data can be expressed through the control theoretic notion of controllability which can be proved using geometric control techniques. We then add an additional ingredient, monotonicity, to conclude that deep residual networks can approximate, to arbitrary accuracy with respect to the uniform norm, any continuous function on a compact subset of $n$-dimensional Euclidean space by using at most $n+1$ neurons per layer. We will conclude the talk by showing how these results pave the way for the use of deep networks in the perception pipeline of autonomous systems while providing formal (and probability free) guarantees of stability and robustness.

Living beings do an extraordinary thing. By being alive they are resisting the second law of thermodynamics. This law stipulates that open, living systems tend to dissipation by the increase of entropy or chaos. From minimal cognitive organisms like plants to more complex organisms equipped with nervous systems, all living systems adjust and adapt to their environments, thereby resisting the second law. Impressively, while all animals cognitively enact and survive their local environments, more complex systems do so also by actively constructing their local environments, thereby not only defying the second law, but also (evolution) selective properties. Because all living beings defy the second law by adjusting and engaging with the environment, a prominent question is how do living organisms persist while engaging in adaptive exchanges with their complex environments? In this talk I will offer an overview of how the Free Energy Principle (FEP) offers a principled solution to this problem. The FEP prescribes that living systems maintain themselves by remaining in non-equilibrium steady states by restricting themselves to a limited number of states; it has been widely applied to explain neurocognitive function and embodied action, develop artificial intelligence and inspire psychopathology models.

Gromov-Witten invariants of a given Kahler target space are defined as suitable intersection numbers in moduli spaces of stable maps of complex curves into the target space. Their K-theoretic analogues are defined as holomorphic Euler characteristics of suitable vector bundles over these moduli spaces.

We will describe how the Kawasaki-Riemann-Roch theorem expressing holomorphic Euler characteristics in cohomological terms leads to the adelic formulas for generating functions encoding K-theoretic Gromov-Witten invariants.