# PhD Colloquium

## Organisation

Organisation: Ruben Stienstra and Milan Lopuhaä.
Location: variable, see below.
Time: Wednesdays, 14:00 - 15:00.

## Description

This is a colloquium for PhD students in mathematics at the Radboud Universiteit. The goal is to give PhD students the possibility to introduce others to their research, practice their presentation skills, and learn from other fields of mathematics. The talks should be understandable for a general mathematical audience, with backgrounds ranging from stochastics to topology and from mathematical physics to algebra.

Some tips and tricks on how to give a good colloquium talk can be found
here. The old website of the PhD colloquium, when it was organised by Johan Commelin, can be found here.

## Schedule and location

### Upcoming talks

October 18, 2017 - room TBA Florian Zeiser - Morse Theory.
Abstract: Morse theory is a way to study the topological behaviour of manifolds using functions with a specific type of critical points. We will introduce the basic notions of the theory which will lead us all the way to Morse homology. This is a homology theory which recovers under certain assumptions (e.g. compactness of the space,...) the standard homology of the space.This theory has applications in various areas of mathematics. In this talk we will see a few of them (e.g. Morse inequalities,...) as far as time allows it.

### Past talks

June 20, 2017 - HG01.057: Peter Badea - Hecke Algebras.
Abstract: In this talk, we will begin by introducing the historical notion of a Hecke algebra (of Hecke operators) as one of many related possible constructions. We quickly specialize to the case of the Iwahori-Hecke algebra of a reductive group over a non-archimedean local field (with special treatment being given to the cases of the general linear, as well as unitary groups). We examine two different definitions of this Hecke algebra as a convolution algebra of locally constant compactly supported functions on certain double cosets of $G$, and as a deformation of the group algebra of the affine Weyl group, and if time permits, we show that these two notions coincide. In regards to applicability, we allude to the fact that the Hecke algebra is a useful object in studying representations of $G$, but do not expand upon this idea.

Tuesday April 18, 2017 - HG00.633: Yongqi Feng - Linear Algebra done deep.
1. SL2(C) acts on the Riemann sphere CP1 as Möbius transformations. In particular, any upper triangular matrix has a fixed point.
2. If $g = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \in \mathrm{SL}_2(\mathbf{C})$ and if $c \neq 0$, then we can write $g$ as a product as follows: $$\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) = \left(\begin{array}{cc} c^{-1} & 0 \\ 0 & c \end{array}\right)\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} 1 & c^{-1}d \\ 0 & 1 \end{array}\right)$$
These two facts are reflections of the so-called Bruhat decomposition (of linear algebraic groups). In the case of the general linear group, the Bruhat decomposition is a deep generalisation of the fact that every non-degenerate group can be transformed into a premutation matrix by only elementary row and colum transformations. In this talk, we will prove the Bruhat decomposition, and discuss its geometric meaning (mainly for GL).

April 3, 2017 - HG01.057: Ruben Stienstra - Quantization of effective theories: episode II.

March 27, 2017 - HG01.057: Ruben Stienstra - Quantization of effective theories.
Abstract: Effective theories are used in statistical mechanics and quantum field theory to give descriptions of systems at different length/energy/momentum scales, and are connected to each other through renormalization group (RG) transformations. After giving a very brief overview of the Hamiltonian formulations of both classical and quantum mechanics in which I will focus on the similarities between their formulations, I will discuss the notions of a collection of effective theories and the RG (which, despite its name, is not a group). I will then point out some of the difficulties in defining the RG for quantum systems.

March 13, 2017 - HG03.054: Norbert Mikolajewski - Characterization, adaptiveness, and optimality of the least squares estimator.
Abstract: We will look at a set points that describes the least squares estimator of a convex regression function in the white noise model as the best piecewise linear interpolation of the given data. This set of points may be useful in proving that the least squares estimator adapts to the underlying function and attains the optimal rate of convergence in a minimax sense.

January 18, 2017 - HG01.057: Henrique Tavares - A (very) soft introduction to Philosophy of Mind.
Abstract: We will present a summary of the history of Philosophy of Mind with focus on the twentieth century, and discussion of models. Whenever possible, arguments for why these models are not sufficient will be presented. On the second part of the talk, Searle`s arguments against pure functionalism will be studied in some detail as well as his counter-proposed model. We will then argue that this model is nigh-unverifiable and at least presents serious, possibly impossible to overcome, problems for the creation of a proper science of psyche. Finally, I shall present a naive argument for unsolubility of the Problems of Consciousness.

December 15, 2016 - HG01.139: Johan Commelin - Чеботарёв/Чоботарьов/Čebotarëv/Čobotar'ov/Chebotaryov/Chebotarov/Chebotarev/Tschebotareff's Density Theorem.
Abstract: Chebotarev's density theorem is a striking result in number theory. Let f be an irreducible polynomial with integral coefficients. We will attach a finite group to f. On the other hand, we can study the way that f factors into irreducible polynomials modulo p. Somewhat surprisingly, these two are related via the somewhat analytical/probabilistic notion of density.

November 17, 2016 - HG00.065: Francesca Arici - C*-algebras from symbolic dynamics.
Abstract: In this talk I will describe some easy but rich examples of noncommutative algebras associated to dynamical systems. I will start by describing the dynamical systems known as shifts and subshifts of finite type, and I will show how one can obtain information about the dynamics by constructing a groupoid that encodes the dynamical information, together with the corresponding C*-algebra. Notes.

October 20, 2016 - HG01.058: Milan Lopuhaä - Linear group schemes.
Abstract: The goal of this talk is to introduce the notion of linear group schemes over a general commutative ring R. Furthermore, it will be shown that in the case that R is a number field, representations of linear group schemes naturally lead to models of these group schemes over the integers, and properties of these integral models are discussed.

October 13, 2016 - HG03.632: Johan Commelin - An introduction to abelian varieties and the Mumford-Tate conjecture: from Kepler's laws to number theory.
Abstract: After a prelude consisting of a medley of ellipses, integrals, and the occasional historical interlude, we expose parts of the wonderful world of elliptic curves and abelian varieties. We will see both complex analytic and number theoretical aspects. Although it might seem that complex analysis and number theory are quite unrelated, in the end we build a canon out of these two melodies. A canon, pointing to a hidden motif. Notes.