This is an elementary talk introducing the rational Cherednik algebra and its representations. Especially, we are interested in the case of complex reflection group. A tool called the Dunkl-Opdam subalgebra is used to decompose the standard modules into weight spaces and to construct the correspondence with the partitions of integers. If time allows, we might explore the concept of unitary representation and what condition a representation needs to satisfy to be qualified as one.

The well-known Schur-Weyl duality provides a link between the representation theories of the general linear group $GL_n$ and the symmetric group $S_r$ by studying tensor space $(\mathbb{C}^n)^{\otimes r}$ as a ${(GL_n,S_r)}$-bimodule. We will discuss a few variations of this idea which replace $GL_n$ with some other interesting algebraic object (e.g. O$_n$ or $S_n$) and $S_r$ with a so-called diagram algebra. If time permits, we will also briefly look at how this can be used to define Deligne's category which 'interpolates' Rep($S_t$) for any complex number $t \in \mathbb{C}$.

How can a random walker on a network be helpful for patients suffering from amyotrophic lateral sclerosis (ALS)? Clinical trials in ALS continue to rely on survival or clinical functional scales as endpoints, since anatomical patterns of disease spread in ALS are poorly characterized in vivo. In this study, we generated individual brain networks of patients and controls based on cerebral magnetic resonance imaging (MRI) data. Then, we applied a computational model with a random walker to the brain MRI scan of patients to simulate this progressive network degeneration. We observe that computer‐simulated aggregation levels of the random walker mimic true disease patterns in ALS patients. Our results demonstrate the utility of computational network models in ALS to predict disease progression and underscore their potential as a prognostic biomarker.

After presenting this study on characterizing the structural changes in neurodegenerative diseases with network science, I will give an outlook on my new work on characterizing the dynamic changes in brain networks for Parkinson’s disease and counteracting these with (simulated) deep brain stimulation using the neuroinformatics platform The Virtual Brain (www.thevirtualbrain.org) .

Article link: https://onlinelibrary.wiley.com/doi/full/10.1002/ana.25706

The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)
 

We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.