Past

I shall outline a general method for finding upper bounds on the diameters of finite groups, based on the Solovay-Kitaev procedure from quantum computation. This method may be fruitfully applied to groups arising as quotients of many familiar pro-p groups. Time permitting, I will indicate a connection with weak spectral gap, and give some applications.

Waldhausen defined higher K-groups for categories with certain extra structure. In this talk I will define categories with cofibrations and weak equivalences, outline Waldhausen's construction of the associated K-Theory space, mention a few important theorems and give some examples. If time permits I will discuss the infinite loop space structure on the K-Theory space.

 A continuum is a non-empty compact connected metric space. Given a continuum X let P(X) be the power set of X. We define the following set functions:
T:P(X) to P(X) given by, for each A in P(X), T(A) = X \ { x in X : there is a continuum W such that x is in Int(W) and W does not intersect A}
K:P(X) to P(X) given by, for each A in P(X), K(A) = Intersection{ W : W is a subcontinuum of X and A is in the interior of W}
S:P(X) to P(X) given by, for each A in P(X), S(A) = { x in T(A) : A intersects T(x)}
Some properties and relations between these functions are going to be presented.

A word w has finite width n in a group G if each element in the subgroup generated by the w-values in G can be written as the product of at most n w-values. A group G is called verbally elliptic if every word has finite width in G. In this talk I will present a proof for the fact that every finitely generated virtually nilpotent group is verbally elliptic.

The Springer Correspondence relates irreducible representations of the Weyl group of a semisimple complex Lie algebra to the geometry of the cone of nilpotent elements of the Lie algebra. The zeroth Poisson homology of a variety is the quotient of all functions by those spanned by Poisson brackets of functions. I will explain a conjecture with Proudfoot, based on a conjecture of Lusztig, that assigns a grading to the irreducible representations of the Weyl group via the Poisson homology of the nilpotent cone. This conjecture is a kind of symplectic duality between this nilpotent cone and that of the Langlands dual. An analogous statement for hypertoric varieties is a theorem, which relates a hypertoric variety with its Gale dual, and assigns a second grading to its de Rham cohomology, which turns out to coincide with a different grading of Denham using the combinatorial Laplacian.

The goal of this second talk is to study the existence of global jet differentials. Thanks to the algebraic Morse inequalities, the problem reduces to the computation of a certain Chern number on the Demailly tower of projectivized jet bundles. We will describe the significant simplification due to Berczi consisting in integrating along the fibers of this tower by mean of an iterated residue formula. Beside the original argument coming from equivariant geometry, we will explain our alternative proof of such a formula and we will particularly be interested in the interplay between the two approaches.

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. The sequence of maps has an almost sure limit G; we show that G is the distributional local limit of large, uniformly random 3-connected graphs.
A classical result of Brooks, Smith, Stone and Tutte associates squarings of rectangles to edge-rooted planar graphs. Our map growth procedure induces a growing sequence of squarings, which we show has an almost sure limit: an infinite squaring of a finite rectangle, which almost surely has a unique point of accumulation. We know almost nothing about the limit, but it should be in some way related to "Liouville quantum gravity".
Parts joint with Nicholas Leavitt.

We present our recent results  on the fluctuation of Optimal Alignments of random sequences and Longest Common Subsequences (LCS). We show how OA and LCS are special cases of certain Last Passage Percolation models which can also be viewed as growth models. this is joint work with Saba Amsalu, Raphael Hauser and Ionel Popescu.

Hyperbolicity is the study of the geometry of holomorphic entire curves $f:\mathbb{C}\to X$, with values in a given complex manifold $X$. In this introductary first talk, we will give some definitions and provide historical examples motivating the study of the hyperbolicity of complements $\mathbb{P}^{n}\setminus X_{d}$ of projective hypersurfaces $X_{d}$ having sufficiently high degree $d\gg n$.

Then, we will introduce the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy due to Bloch, Demailly, Siu, that relies in an essential way on the relation between entire curves and jet differentials vanishing on an ample divisor.

Incomplete Cholesky (IC) factorizations have long been an important tool in the armoury of methods for the numerical solution of large sparse symmetric linear systems Ax = b. In this talk, I will explain the use of intermediate memory (memory used in the construction of the incomplete factorization but is subsequently discarded)  and show how it can significantly improve the performance of the resulting IC preconditioner. I will then focus on extending the approach to sparse symmetric indefinite systems in saddle-point form. A limited-memory signed IC factorization of the form LDLT is proposed, where the diagonal matrix D has entries +/-1. The main advantage of this approach is its simplicity as it avoids the use of numerical pivoting.  Instead, a global shift strategy is used to prevent breakdown and to improve performance. Numerical results illustrate the effectiveness of the signed incomplete Cholesky factorization as a preconditioner.

Networks provide a convenient way to represent complex systems of interacting entities. Many networks contain "communities" of nodes that are more strongly connected to each other than to nodes in the rest of the network. Most methods for detecting communities are designed for static networks. However, in many applications, entities and/or interactions between entities evolve in time. To incorporate temporal variation into the detection of a network's community structure, two main approaches have been adopted. The first approach entails aggregating different snapshots of a network over time to form a static network and then using static techniques on the resulting network. The second approach entails using static techniques on a sequence of snapshots or aggregations over time, and then tracking the temporal evolution of communities across the sequence in some ad hoc manner. We represent a temporal network as a multilayer network (a sequence of coupled snapshots), and discuss  a method that can find communities that extend across time. 

We investigate the dynamics of a class of tumor growth

models known as mixed models. The key characteristic of these type of

tumor growth models is that the different populations of cells are

continuously present everywhere in the tumor at all times. In this

work we focus on the evolution of tumor growth in the presence of

proliferating, quiescent and dead cells as well as a nutrient.

The system is given by a multi-phase flow model and the tumor is

described as a growing continuum such that both the domain occupied by the tumor as well as its boundary evolve in time. Global-in-time weak solutions

are obtained using an approach based on penalization of the boundary

behavior, diffusion and viscosity in the weak formulation.

Further extensions will be discussed.

This is joint work with D. Donatelli.

Since their introduction, Shimura varieties have proven to be important landmarks sitting right at the crossroads between algebraic geometry, number theory and representation theory. In this talk, starting from the yoga of motives and Hodge theory, we will try to motivate Deligne's construction of Shimura varieties, and briefly survey some of their zoology and basic properties. I may also say something about the links to automorphic forms, or their integral canonical models.

We consider the numerical approximation of Kolmogorov equations arising in the context of option pricing under L\'evy models and beyond in a multivariate setting. The existence and uniqueness of variational solutions of the partial integro-differential equations (PIDEs) is established in Sobolev spaces of fractional or variable order.

Most discretization methods for the considered multivariate models suffer from the curse of dimension which impedes an efficient solution of the arising systems. We tackle this problem by the use of sparse discretization methods such as classical sparse grids or tensor train techniques. Numerical examples in multiple space dimensions confirm the efficiency of the described methods.

In this talk I will discuss a deformation principle for cohomology with coefficients in representations on Banach spaces. The

main idea is that small, metric perturbations of representations do not change the vanishing of cohomology in degree n, provided that

we have additional information about the cohomology in degree n+1. The perturbations considered here happen only on the generators of a

group and even for isometric representations give rise to unbounded representations. Applications include fixed point properties for

affine actions and strengthening of Kazhdan’s property (T). This is joint work with Uri Bader.

This work is joint with Peter Sarnak.

It is shown that the topologies and nestings of the zero and nodal sets of random (Gaussian) band limited functions have universal laws of distribution. Qualitative features of the supports of these distributions are determined. In particular the results apply to random monochromatic waves and to random real algebraic hyper-surfaces in projective space.

We will discuss recent results regarding generation and propagation of summability of moments to solution of the Boltzmann equation for variable hard potentials.
These estimates are in direct connection to the understanding of high energy tails and decay rates to equilibrium.