In recent work (see https://arxiv.org/abs/1711.06951), Craig Huneke, Ilya Smirnov, and Javid Validashti study conjectured generalisations of an inequality of Lech, relating the multiplicity and the colength of an ideal I of finite colength in a Noetherian local ring (R,m,k). Using mixed multiplicities, they show that one of these conjectures regarding the multiplicity of mI is true. In this series of talks, we will discuss some of the results in their paper.

Lag windows are commonly used in the time series, steady state simulation, and Markov chain Monte Carlo (MCMC) literature to estimate the long range variances of ergodic averages. We propose a new lugsail lag window specifically designed for improved finite sample performance. We use this lag window for batch means and spectral variance estimators in MCMC simulations to obtain strongly consistent estimators that are biased from above in finite samples and asymptotically unbiased. This quality is particularly useful when calculating effective sample size and using sequential stopping rules where they help avoid premature termination. Further, we calculate the bias and variance of lugsail estimators and demonstrate that there is little loss compared to other estimators. We also show mean square consistency of these estimators under weak conditions. Our results hold for processes that satisfy a strong invariance principle, providing a wide range of practical applications of the lag windows outside of MCMC. Finally, we study the finite sample properties of lugsail estimators in various examples.

Let G be a Lie group. For a natural number k, we denote by P_k, the k-th power map, defined by g\to g^k for all g\in G. In this talk, we discuss the question as to when the individual power map P_k has a dense image or a surjective image in a Lie group. The problem is closely related to the study of exponentiality of Lie group.

In this lecture, we are interested in the study of an inverse problem for an integral equation arising from the biology of the human olfaction system. The transduction of an odour into an electrical signal is accomplished by a depolarising influx of ions through cyclic-nucleotide-gated (CNG for short) channels on the cilium membrane. The inverse problem studied is to determine the spatial distribution of these ion channels from measurements of the electrical current at the ends of the cilium. The Mellin transform will allow us to write an explicit formula for its solution, so obtaining observability or continuity results will then amounts to punctual estimates of the kernel of this integral equation on vertical lines of the complex plane. These will be accomplished using arguments in the spirit of the stationary phase method.

We will show how one can determine properties of linear codes, by studying algebraic properties of matroids, which in a natural way are associated to the codes. Concretely, one studies resolutions of Stanley-Reisner rings associated to the simplicial independence complexes of the matroids involved. From these resolutions one may extract generalized Hamming weights and higher weight spectra, of the codes in question.

Then we will outline a proof of Jones Isomorphism Theorem using simplicial sets and hence obtain the homology of the loop space of spheres.

In recent work (see https://arxiv.org/abs/1711.06951), Craig Huneke, Ilya Smirnov, and Javid Validashti study conjectured generalisations of an inequality of Lech, relating the multiplicity and the colength of an ideal I of finite colength in a Noetherian local ring (R,m,k). Using mixed multiplicities, they show that one of these conjectures regarding the multiplicity of mI is true. In this series of talks, we will discuss some of the results in t

The geometry of a Riemann surface is captured by the ways in which it can manifest as a projective variety, or more precisely, by the geometry of spaces parameterizing embeddings of the curve into projective space. These “Brill-Noether varieties” of a curve are well understood in two cases. On one end, work of Clifford gives a complete understanding of hyperelliptic curves. On the other end, a curve that is general in moduli exhibits expected behaviour. In recent joint work with Dave Jensen, building on previous work of Nathan Pflueger, we determine formulas for the dimensions of the Brill-Noether varieties for the intermediate cases, i.e. general curves of a fixed gonality. Our methods blend the combinatorics of the sandpile model on graphs with methods from non-archimedean analysis and deformation theory. I will give an overview of ideas surrounding the theorem and its proof, and try to give a sense of the link between algebraic and combinatorial geometry.

In this talk, we first define the dynamical setting of a non-autonomous system and its relation to Bedford conjecture. Further, we will discuss about Short Ck's that arises as basins of attraction of a fixed point in the non-autonomous setting. Lastly, we will see some methods to construct short Ck's with pathological properties and discuss some related problems.