# PhD Colloquium

## Organisation

Organisation: Ruben Stienstra and Milan Lopuhaä-Zwakenberg.
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

### Past talks

February 22, 2018 - HG00.303: Bruno Eijsvoogel - A glimpse of resurgence theory
Abstract: The talk will start with a treatment of some concepts in asymptotic analysis, with the goal of describing qualitatively what resurgence theory is about. Time permitting we will consider the Stokes phenomenon and its role in resurgence theory.

February 15, 2018 - HG03.082: He-who-must-not-be-named - Harmonic analysis on quantum groups and Harish-Chandra modules
Abstract: A brief introduction to the theory of Harish-Chandra modules and its applications to harmonic analysis of semi-simple groups is given. We then present the notion of quantum symmetric pair coideal subalgebra and explain how it can be used to define Harish-Chandra modules for quantum groups. Finally, we shall enumerate some problems and conjectures in the quantum case related to the classical case.

February 7, 2018 - HG01.060: Salvatore Floccari - Topology of algebraic maps
Abstract: I will start by recalling the usual cohomological framework which arises from a smooth proper map between smooth complex algebraic varieties (decomposition theorem, poincarè duality, relative hard Lefschetz). Most of this fails when dropping the smoothness assumptions. I will then explain how looking at perverse sheaves can fix this failure.

December 6, 2017 - HG03.082: Milan Lopuhaä-Zwakenberg - What are stacks and why should you care?
Abstract: How do we classify geometric spaces? In this talk I show how the set of isomorphism classes of geometric spaces of a certain 'type' occasionally has a geometric structure itself. We call the resulting geometric structure a moduli space, and they play an important role in algebraic geometry. However, in some cases moduli spaces might fail to exist. I will talk about what might cause this failure, and how this leads to the definition of stacks.

November 15, 2017 - HG01.129: Stijn Cambie - Maximum Wiener indices of unicyclic graphs of given matching number.
Abstract: The total distance (called the Wiener index) $W(G)$ of a connected graph $G$ is the sum of distances between all unordered pairs of vertices, i.e. $W(G) = \sum_{\{u,v\} \subset V(G)} d(u,v).$ The minimum and maximum possible Wiener indices of a connected graph of order $n$ with matching number $m$ was determined by Dankelmann (1994). The graphs that attain the maximal Wiener indices are trees, while the graphs that attain the minimal Wiener indices contain many cycles. Hence two natural questions arise:
- What is the minimal Wiener index for trees?
- What is the maximal Wiener index for the graphs that are not trees?
In the latter case, the graphs will necessarily be unicyclic. For graphs that contain multiple cycles, we can take a maximal matching, and deleting an edge not contained in the matching such that the graph is still connected, will increase the Wiener index. Du and Zhou (2010) solved the first question and also determined the minimal Wiener index for unicyclic graphs.
In this talk, we sketch the proof for the remaining question What are the maximal Wiener indices for unicyclic graphs with given order $n$ and matching number $m$?'. We also give some thoughts on another problem related to the Wiener index, the conjecture of DeLaViña and Waller.

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.

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, Searles 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.