We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve".  In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

Fireshape is a shape optimization library based on finite elements. In this talk I will describe how Florian Wechsung and I developed Fireshape and will share my reflections on lessons learned through the process.

Topological quantum field theories (TQFTs) are an extensively studied scheme for constructing invariants of manifolds, inspired by physics. In this talk, we will discuss a particular flavour of TQFT, where we equip our manifolds with principal bundles for some finite group. After introducing TQFTs and this particular flavour, I will discuss games one can play with these TQFTs, and a possible strategy for classifying equivariant TQFTs in three dimensions. 

A cohomology class on the diffeomorphism group Diff(M) of a manifold M

can be thought of as a characteristic class for smooth M-bundles.
I will survey a technique for producing examples of such classes,
and then explain how the signature (of 4-manifolds) provides an
obstruction to this technique in dimension 3.

I will define Miller-Morita-Mumford classes and explain how we can
think of them as coming from classes on the cobordism category.
Madsen and Weiss showed that for a surface S of genus g all cohomology
classes
of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.
This technique has been successfully ported to higher even dimensions d= 2n,
but it cannot possibly work in odd dimensions:
a theorem of Ebert says that for d=3 all MMM-classes are trivial.
In the second part of my talk I will sketch a new proof of (a part of)
Ebert's theorem.
I first recall the definition of the signature sign(W) of a 4 manifold W,
and some of its properties, such as additivity with respect to gluing.
Using the signature and an idea from the world of 1-2-3-TQFTs,
I then go on to define a 'central extension' of the three dimensional
cobordism category.
This central extension corresponds to a 2-cocycle on the 3d cobordism
category,
and we will see that the construction implies that the associated MMM-class
has to vanish on all 3-dimensional manifold bundles.

Commuting concerns people’s spatial behaviour resulting from the geographic separation of home and workplace and is connected with their willingness to seek economic opportunities outside their place of residence (Rouwendal J., Nijkamp P., 2004). Such opportunities are usually found in the urban areas, so this phenomenon is often a subject of urban studies or research focusing on city centres (Drejerska N., Chrzanowska M., 2014). In literature, commuting patterns are used to determine the boundaries of local and regional labour markets. Furthermore, labour market is one of the most important features for the delimitation of functional regions, as commuting involves not only working outside one’s place of residence but also, among other things, using various services offered there, from shopping to health or cultural services. Taking this into account, it can be stated that commuting is an important characteristic of relations between territories, and these relations form complex networks.

People decide to commute to work for various reasons. Most commuters travel from a small town, village or rural area to a city or town where they have a wider range of employment opportunities. However, people differ in their attitudes toward commuting. While some people find it troublesome, others enjoy their daily travel. There are also people who regard commuting as the necessary condition for supporting themselves and their families. Therefore, commuting is an important factor that should be taken into account in the research on the quality of life and quality of work.

The main goals of this presentation is to identify and analyse relations between communities (municipalities) from the perspective of labour market, especially commuting in the vicinity of Warsaw, Data on the number of commuters come from the Central Statistical Office of Poland and cover the year 2011.

 Bibliography

Drejerska N., Chrzanowska M., 2014: Commuting in the Warsaw suburban area from a spatial perspective – an example of empirical research, Acta Universitatis Lodziensis. Folia Oeconomica 2014, Vol. 6, no 309, pp. 87-96.

Rouwendal J., Nijkamp P., 2004: Living in Two Worlds: A Review of Home-to-Work Decisions, Growth and Change, Volume 35, Issue 3, p. 287.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

Abstract:  As is well known, every rational representation of a finite group $G$ can be realized over $\mathbb{Z}$, that is, the corresponding $\mathbb{Q}G$-module $V$ admits a $\mathbb{Z}$-form. Although $\mathbb{Z}$-forms are usually far from being unique, the famous Jordan--Zassenhaus Theorem shows that there are only finitely many $\mathbb{Z}$-forms of any given $\mathbb{Q}G$-module, up to isomorphism. Determining the precise number of these isomorphism classes or even explicit representatives is, however, a hard task in general. In this talk we shall be concerned with the case where $G$ is the symmetric group $\mathfrak{S}_n$ and $V$ is a simple $\mathbb{Q}\mathfrak{S}_n$-module labelled by a hook partition. Building on work of Plesken and Craig we shall present some results as well as open problems concerning the construction of the
integral forms of these modules. This is joint work with Tommy Hofmann from Kaiserslautern.

Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves.