Parabolic bundles on a punctured Riemann surface were introduced by Mehta and Seshadri in the ‘80s, in relation to unitary representations of its topological fundamental group. Their definition was generalized, in several steps, to a definition over an arbitrary logarithmic scheme due to Borne and Vistoli, who also proved a correspondence with sheaves on stacks of roots. I will review these constructions, and push them further to the case of an “infinite” version of the root stacks. Towards the end I will discuss a comparison result (for log schemes over the complex numbers) between this “infinite root stack” and the so-called Kato-Nakayama space, and hint at some work in progress about relating sheaves on this latter space to parabolic sheaves with arbitrary real weights.

We shall give several examples of Yang-Mills Theories.

This talk will be a basic interactive introduction to Coq. I will start with simple operations on natural numbers, then move on to functions on arbitrary types, and illustrate how to prove a logical statement in computer.

We present the following theorem: A totally real smooth surface in $\mathbb{C}^2$ with an open Whitney umbrella at the origin, is locally polynomially convex near the singular point. This is a natural generalization of a result of Shafikov and Sukhov that addresses the same problem, but in the generic case. Our theorem establishes polynomial convexity in full generality in this context. This is joint work with Rasul Shafikov.

We shall give a motivated description of prime number theorems in general and the Sato-Tate conjecture in particular, and describe some recent joint work with Ram Murty.

We try to investigate a generalization of the Heisenberg commutation relation ${[p,q]=-i \hslash}$, introduced by Chamseddine, Connes and Mukhanov as ``the one-sided and the two-sided quantization equations'', which captures the geometry. The momentum variable $p$ is encoded by the Dirac operator and the analogue of the position variable $q$ is the Feynman slash of real scalar fields over a closed even-dimensional spin manifold. Existence of a solution of the one-sided equation implies that the manifold decomposes into a disconnected sum of spheres of unit volume which represent quanta of geometry. The two-sided equation, as the refined version of the one-sided equation by involving the real structure on a spin manifold, implies the quantization of the volume of the spin manifold.

Quaternion algebras contain quadratic field extensions of the center. Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another. If the center is of characteristic 2 then those extensions divide into two types - the separable type and the inseparable type. It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true. We shall discuss this fact and its generalization to $p$-algebras of arbitrary prime degree.