Location: variable, see below.

Time: Wednesdays, 14:00 - 15:00.

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.

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.

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.

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.

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.

Abstract: We start with two simple facts of SL

- SL
_{2}(**C**) acts on the Riemann sphere**CP**^{1}as Möbius transformations. In particular, any upper triangular matrix has a fixed point. - 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: $$\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) = \left(\begin{array}{cc} c^{-1} & 0 \\ 0 & c \end{array}\right)\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} 1 & c^{-1}d \\ 0 & 1 \end{array}\right)$$

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.

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.

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.

Abstract: Chebotarev's density theorem is a striking result in number theory. Let

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.

Abstract: The goal of this talk is to introduce the notion of linear group schemes over a general commutative ring

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.