Seminars from until

This week

Conditional probability distributions, the martingale problem and the cadlag path space.

The main objective of this work is to study some classes of integral and integro-differential equations with regular and singular kernels. We introduce a wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel; we also apply a collocation method based on the airfoil polynomial to numerically solve an integro-differential equation of second order with Cauchy kernel.

By solving a singular initial value problem, we prove the existence of solutions of the wave equation $\Box_g\phi=0$ which are bounded at the Big Bang in the Friedmann-Lemaitre-Robertson-Walker cosmological models. More precisely, we show that given any function $A \in H^3(\Sigma)$ (where $\Sigma=\mathbb{R}^n$, $\mathbb{S}^n$ or $\mathbb{H}^n$ models the spatial hypersurfaces) there exists a unique solution $\phi$ of the wave equation converging to $A$ in $H^1(\Sigma)$ at the Big Bang, and whose time derivative is suitably controlled in $L^2(\Sigma)$.

We propose a comprehensive Learning Analytics methodology to investigate the level of understanding students achieve in the learning process. The goals of such methodology are

1. To identify topics in which students experience difficulties on;
2. To assess whether these difficulties are recurrent along semesters;
3. To decide if there are conceptual associations between topics in which students experience difficulties on; and, more generally,
4. To discover statistically significant groups of topics in which students show similar performance.

The proposed methodology uses statistics and data visualization techniques to address the first and the second goals, frequent itemset mining to tackle the third goal, and biclustering is proposed to find relationships within educational data, revealing meaningful and statistically significant patterns of students’ performance.

We illustrate the application of the methodology to a Computer Science course.

This week

Uniqueness for a martingale problem, Markov property and examples.

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in $P^3$ with $A_n$ singularities and of low degree (low $c_1^2/c_2$). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree $8$ (and potentially $7$). The best known result till date was degree $13$.

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 its potential in the context of qubit networks.

This week

Uniqueness for a martingale problem, Markov property and examples.

We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e., in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.

This week

Uniqueness for a martingale problem, Markov property and examples.

This week

Convergence of random paths in Skorokhod topology. Assymptotic lumpability.

This week

Convergence of random paths in Skorokhod topology. Assymptotic lumpability.

This week

Convergence of random paths in Skorokhod topology. Assymptotic lumpability.

Metrics of Poincaré type are Kähler metrics defined on the complement X\D of a smooth divisor D in a compact Kähler manifold X which near D are modeled on the product of a smooth metric on D with the standard cusp metric on a punctured disk in \mathbb{C}. In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. I will discuss that even in the toric case, this gives an obstruction to blowing up fixed points (which is unobstructed in the compact case). This additional condition is conjecturally related to ensuring the metrics remain of Poincaré type.

This week

Kingman’s process and coalescing random walks. Trace process and metastability of zero range process.

This week

Kingman’s process and coalescing random walks. Trace process and metastability of zero range process.

This week

Kingman’s process and coalescing random walks. Trace process and metastability of zero range process.

The construction of initial data for the Cauchy problem in General Relativity is an interesting problem from both the mathematical and physical points of view. As such, there have been numerous methods studied in the literature the "Conformal Method" of Lichnerowicz-Choquet-Bruhat-York and the "gluing" method of Corvino-Schoen being perhaps the best-explored. In this talk I will describe an alternative, perturbative, approach proposed by A. Butscher and H. Friedrich, and show how it can be used to construct non-linear perturbations of initial data for spatially-closed analogues of the $k = -1$ FLRW spacetime. Time permitting, I will discuss possible renements/extensions of the method, along with its generalisation to the full Conformal Constraint Equations of H. Friedrich.

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method.

We prove the following theorem: axisymmetric, stationary solutions of the Einstein field equations formed from classical gravitational collapse of matter obeying the null energy condition, that are everywhere smooth and ultracompact (i.e., they have a light ring) must have at least two light rings, and one of them is stable. It has been argued that stable light rings generally lead to nonlinear spacetime instabilities. Our result implies that smooth, physically and dynamically reasonable ultracompact objects are not viable as observational alternatives to black holes whenever these instabilities occur on astrophysically short time scales. The proof of the theorem has two parts: (i) We show that light rings always come in pairs, one being a saddle point and the other a local extremum of an effective potential. This result follows from a topological argument based on the Brouwer degree of a continuous map, with no assumptions on the spacetime dynamics, and hence it is applicable to any metric gravity theory where photons follow null geodesics. (ii) Assuming Einstein's equations, we show that the extremum is a local minimum of the potential (i.e., a stable light ring) if the energy-momentum tensor satisfies the null energy condition.

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, Vol($K$). A less studied question is whether Vol($K$) can be recovered directly from the original Jones polynomial ($N=1$). In this report we use a deep neural network to approximate Vol($K$) from the Jones polynomial.