Past

I will survey the theory of quasiHamiltonian spaces, a.k.a. group valued moment maps. In rough correspondence with historical development, I will first show how they emerge from the study of loop group representations, and then how they arise as a special case of "presymplectic realizations" in Dirac geometry.

HTML clipboard /*-->*/ /*-->*/ We introduce a "Polya choice" urn model combining elements of the well known "power of two choices" model and the "rich get richer" model. From a set of $k$ urns, randomly choose $c$ distinct urns with probability proportional to the product of a power $\gamma>0$ of their occupancies, and increment one with the smallest occupancy. The model has an interesting phase transition. If $\gamma \leq 1$, the urn occupancies are asymptotically equal with probability 1. For $\gamma>1$, this still occurs with positive probability, but there is also positive probability that some urns get only finitely many balls while others get infinitely many.

In this talk I will introduce and study opers over a smooth projective curve X defined over a field of positive characteristic. I will describe a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F^*(E) under the Frobenius

map F of X has maximal Harder-Narasimhan polygon and the set of opers having zero p-curvature. These sets turn out to be finite, which allows us to derive dimensions of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X.

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Fix a positive integer $k$, and consider the class of all graphs which do not have $k+1$  vertex-disjoint cycles.  A classical result of Erdos and P\'{o}sa says that each such graph $G$ contains a blocker of size at most $f(k)$.  Here a {\em blocker} is a set $B$ of vertices such that $G-B$ has no cycles.

 

We give a minor extension of this result, and deduce that almost all such labelled graphs on vertex set $1,\ldots,n$ have a blocker of size $k$.  This yields an asymptotic counting formula for such graphs; and allows us to deduce further properties of a graph $R_n$ taken uniformly at random from the class: we see for example that the probability that $R_n$ is connected tends to a specified limit as $n \to \infty$.

 

There are corresponding results when we consider unlabelled graphs with few disjoint cycles. We consider also variants of the problem involving for example disjoint long cycles.

 

This is joint work with Valentas Kurauskas and Mihyun Kang.

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Gupta et al. introduced a very general multi-commodity flow problem in which the cost of a given flow solution on a graph $G=(V,E)$ is calculated by first computing the link loads via a load-function l, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function.

 

We show the existence of an oblivious routing scheme with competitive ratio $O(\log n)$ and a lower bound of $\Omega(\log n/\logl\og n)$ for this model when the aggregation function agg is an $L_p$-norm.

 

Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics and the work on minimum congestion oblivious. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the $L_p$-norm of the link loads. The embedding techniques of Bartal and Fakcharoenphol et al. [FRT03] can be viewed as solving this problem in the $L_1$-norm while the result on congestion minmizing oblivious routing solves it for $L_\infty$. We give a single proof that shows the existence of a good tree-based oblivious routing for any $L_p$-norm.

Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria.

We further discuss the scaling laws and $\Gamma$-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric on a bounded domain into $\mathbb R3$.

Lattices in semisimple Lie groups have been studied from the point of view of number theory, algebraic groups, topology and geometry, and geometric group theory. The Fragestellung of one line of investigation is to what extent the properties of the lattice determine, and are determined by, the properties of the group. This talk reviews a number of results about lattices, and in particular looks at Mostow--Margulis rigidity.

We address the effect of extreme geometry on a non-convex variational problem motivated by recent investigations of magnetic domain walls trapped by sharp thin necks. We prove the existence of local minimizers representing geometrically constrained walls under suitable symmetry assumptions on the domains and provide an asymptotic characterization of the wall profile. The asymptotic behavior, which depends critically on the scaling of length and width of the neck, turns out to be qualitatively different from the higher-dimensional case and a richer variety of regimes is shown to exist.

Abstract: Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is a ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last twenty years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.

Understanding the critical parameters controlling the stability of solidification interfaces in colloidal systems is a necessary step in many domains were the freezing of colloids is present, such as materials science or geophysics. What we understand so far of the solidification of colloidal suspensions is derived primarily from the analogies with dilute alloys systems, or the investigated behaviour of single particles in front of a moving interface and is still a subject of intense work. A more realistic, multi-particles model should account for the particles movement, the various possible interactions between the particles and the multiple interactions between the particles and the solid/liquid cellular interface. In order to bring new experimental observations, we choose to investigate the stability of a cellular interface during directional solidification of colloidal suspensions by using X-ray radiography and tomography. I will present recent experimental results of ice growth (ice lenses) and particle redistribution observations, their implications, and open the discussion regarding the limitations of the technique and the potential for further progress in the field using this approach.

The spectrum of the integers is an affine scheme which number theorists would like to complete to a projective scheme, adding a point at infinity. We will list some reasons for wanting to do this, then gather some hints about what properties the completed object might have. In particular it seems that the desired object can only exist in some setting extending traditional algebraic geometry. We will then present the proposals of Durov and Shai Haran for such extended settings and the compactifications they construct. We will explain the close relationship between both and, if time remains, relate them to a third compactification in a third setting, proposed by Toen and Vaquie.

We will present different ideas leading to and evolving around geometry over the field with one element. After a brief summary of the so-called numbers-functions correspondence we will discuss some aspects of Weil's proof of the Riemann hypothesis for function fields. We will see then how lambda geometry can be thought of as a model for geometry over $\mathbb{F}_\mathrm{un}$ and what some familiar objects should look like there. If time permits, we will

explain a link with stable homotopy theory.

One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability $p$, then long range connections appear if and only if $p>1/2$.  The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.

Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.

We present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the

space $\dot H^{1/2}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a different approach. We use the method of "concentration-compactness" + "rigidity theorem" which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authors' knowledge, this is the first instance in which this method has been applied to a parabolic equation. This is joint work with Carlos Kenig.

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).