# PhD/PostDoc colloquium — spring 2016

The new PhD colloquium is organized by Ruben Stienstra and Milan Lopuhaä. The new website can be found here.

Organization
Johan Commelin

## Description

The PhD colloquium is meant as a colloquium where PhD students can introduce others to their research, where they can practice their presentation skills, and where we can learn from other fields. The talks should be understandable for a general audience, with backgrounds ranging from Stochastics to Topology, and Mathematical Physics to Algebra. Some tips and tricks on how to give a good colloquium talk can be found at: http://ams.org/profession/leaders/workshops/gcoll.pdf. The old website of the PhD colloquium can be found here: PhD Colloquium (organised together with Rutger Kuyper)

## Schedule and location

Thu, 16 Apr (HG03.085)
Julius WitteLattices and representations of $\textrm{GL}_2$. Let $p$ be a prime. Let $\mathbb{Q}_p$ be the $p$-adic field corresponding with $p$, i.e. the completion of the rational numbers with respect to a norm related to $p$. First we will look at lattices in the 2-dimensional vector space over $\mathbb{Q}_p$. Then we will apply the theory about lattices to study certain irreducible representations of $\textrm{GL}_2(\mathbb{Q}_p)$.
Thu, 21 Apr (HG03.085)
Abel SternAF algebras and discrete approximations of (smooth) manifolds. Motivated by regularization of quantum gravity, we aim to view a (smooth, Riemannian) manifold as a limit of finite-dimensional spectral triples. This means in particular that the topology of said manifold should be accurately captured in an AF algebra. In the process we encounter a way to (functorially) associate an AF algebra to a countable poset.
Thu, 31 Mar (HG03.085)
Peter HochsIntroduction to index theory. Index theory has its origins in Atiyah and Singer’s index theorem from the 1960s. This states that analytic information, related to solution spaces of differential equations on manifolds, equals topological/geometrical information. An example is the Gauss-Bonnet theorem, which states that the Euler characteristic of a compact surface equals its average scalar curvature (modulo a constant factor). If a group acts on all relevant structure, this leads to a link with representation theory. I will give an informal overview of this area, and focus on the basics.
Thu, 17 Mar (HG03.085)
Giovanni CavigliaOperads and homotopy theory. Operads are mathematical objects that are used to present and to handle algebraic structures in symmetric monoidal categories. In this introductory talk on operad theory I will recall the basic definitions, give some examples and present some applications to homotopy theory.
Thu, 03 Mar (HG01.139)
Frank RoumenCircles.
Mon, 14 Dec (HG03.085)
Bert LindenhoviusCommutative C*-subalgebras and Boolean subalgebras. Given a unital C*-algebra A, we discuss the poset C(A) of all commutative C*-subalgebras of A ordered by inclusion. In a similar way, we can consider the poset B(P) of Boolean subalgebras of an orthomodular poset P. We will compare Gelfand duality (between commutative C*-algebras and compact Hausdorff spaces) and Stone duality (between Boolean algebras and totally disconnected compact Hausdorff spaces) in order to obtain a relation between C(A) and B(P), which allows us to reconstruct the projections of a C*-algebra A from C(A).
Mon, 7 Dec (HG03.085)
Joost NuitenLie algebroids in deformation theory. Given any sort of geometric object (e.g. a curve in the plane), one usually likes to study what can happen if one slightly perturbs this geometric structure. In this talk I want to sketch how (infinitesimally) small deformations of such geometric objects are controlled by a certain algebraic gadget called a (derived) Lie algebroid.
Mon, 30 Nov (HG02.028)
Johan CommelinPeriods (and why the fundamental theorem of calculus conjecturely is a fundamental theorem). In 2001 M. Kontsevich and D. Zagier posed a conjecture on algebraic integrals, which rougly says that the theorem of Stokes (the fundamental theorem of calculus in higher dimensions) is the only non-trivial relation between such integrals. In this talk I will formulate this conjecture, and indicate how it relates to conjectures and research in other fields.
Mon, 23 Nov (HG03.085)
Ruben Stienstra — Connection and Gauge theory: a long introduction
Mon, 16 Nov (HG03.085)
Ruben Stienstra — Connection and Gauge theory: an introduction
Mon, 2 Nov (HG03.632)
Ruben Stienstra — Connection and Gauge theory: a brief introduction
Mon, 12 Oct (HG03.632)
Peter Badea — An Introduction to the Local Langlands Conjecture
Mon, 28 Sep (HG03.632)
Milan LopuhaäThe number of field topologies on a given field. Let F be a field. A field topology on F is a topology such that addition, multiplication and inversion are continuous. In 1973 (Podewski, Kiltinen) it was proven that an infinite field F admits exactly 2^(2^#F) field topologies. The countable case is radically different from the uncountable case. In this talk I will sketch the proofs of both of these cases and talk about the number of Galois invariant topologies.