Past

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.

This is joint work with Angus Macintyre.

We prove that the first-order theory of a finite extension of the field of p-adic numbers is model-complete in the language of rings, for any prime p.

To prove this we prove universal definability of the valuation rings of such fields using work of Cluckers-Derakhshan-Leenknegt-Macintyre on existential

definability, quantifier elimination of Basarab-Kuhlmann for valued fields in a many-sorted language involving higher residue rings and groups,

a model completeness theorem for certain pre-ordered abelian groups which generalize Presburger arithmetic (we call finite-by-Presburger groups),

and an interpretation of higher residue rings of such fields in the higher residue groups.

Lie algebroids are geometric structures that interpolate between finite-dimensional Lie algebras and tangent bundles of manifolds. They give a useful language for describing geometric situations that have local symmetries. I will give an introduction to the basic theory of Lie algebroids, with examples drawn from foliations, principal bundles, group actions, Poisson brackets, and singular hypersurfaces.

This is a joint work with V. Lemaire

(LPMA-UPMC). We propose and analyze a Multilevel Richardson-Romberg

(MLRR) estimator which combines the higher order bias cancellation of

the Multistep Richardson-Romberg ($MSRR$) method introduced

in~[Pag\`es 07] and the variance control resulting from the

stratification in the Multilevel Monte Carlo (MLMC) method (see~$e.g.$

[Heinrich 01, M. Giles 08]). Thus we show that in standard frameworks

like discretization schemes of diffusion processes, an assigned

quadratic error $\varepsilon$ can be obtained with our (MLRR)

estimator with a global complexity of

$\log(1/\varepsilon)/\varepsilon^2$ instead of

$(\log(1/\varepsilon))^2/\varepsilon^2$ with the standard (MLMC)

method, at least when the weak error $\E Y_h-\EY_0}$ induced by the

biased implemented estimator $Y_h$ can be expanded at any order in

$h$. We analyze and compare these estimators on several numerical

problems: option pricing (vanilla or exotic) using $MC$ simulation and

the less classical Nested Monte Carlo simulation (see~[Gordy \& Juneja

2010]).

We are interested in large systems of agents collectively looking for a

consensus (about e.g. their direction of motion, like in bird flocks). In

spite of the local character of the interactions (only a few neighbours are

involved), these systems often exhibit large scale coordinated structures.

The understanding of how this self-organization emerges at the large scale

is still poorly understood and offer fascinating challenges to the modelling

science. We will discuss a few of these issues on a selection of specific

examples.

The accurate and efficient solution of linear systems Ax = b is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden.

This is joint work with Jacek Gondzio.

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.

We discuss shock reflection problem for compressible gas dynamics, various patterns of reflected shocks, and von Neumann conjectures on transition between regular and Mach reflections. Then

we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock

reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on the joint work with Gui-Qiang Chen.

The four topics are:

1. Thermal interface materials

2. Low temperature joining technology

3. Nano Ag materials

4. Status of PV technology

Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids.

In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the

relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same.

In such problems, the geometric nonlinearities of the fluid-structure interface

and the significant exchange in the energy between a moving fluid and a structure

require sophisticated ideas for the study of their solutions.

In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with

different mechanical characteristics.

No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers.

In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions

for a class of problems

describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid,

giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.}

A stable, modular, loosely coupled scheme will be presented, and an existence proof

showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed.

Our results reveal a new physical regularizing mechanism in

FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution.

All theoretical results will be illustrated with numerical examples.

This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).

Finite sharply 2-transitive groups were classified by Zassenhaus in the 1930's. It has been an open question whether infinite sharply 2-transitive group always contain a regular normal subgroup. In joint work with Rips and Segev we show that this is not the case.

Spectral networks are certain collections of paths on a Riemann surface, introduced by Gaiotto, Moore, and Neitzke to study BPS states in certain N=2 supersymmetric gauge theories. They are interesting geometric objects in their own right, with a number of mathematical applications. In this talk I will give an introduction to what a spectral network is, and describe the "abelianization map" which, given a spectral network, produces nice "spectral coordinates" on the appropriate moduli space of flat connections. I will show that coordinates obtained in this way include a variety of previously known special cases (Fock-Goncharov coordinates and Fenchel-Nielsen coordinates), and mention at least one reason why generalising them in this way is of interest.

If X/F is a smooth and proper variety over a global function field of

characteristic p, then for all l different from p the co-ordinate ring of the l-adic

unipotent fundamental group is a Galois representation, which is unramified at all

places of good reduction. In this talk, I will ask the question of what the correct

p-adic analogue of this is, by spreading out over a smooth model for C and proving a

version of the homotopy exact sequence associated to a fibration. There is also a

version for path torsors, which enables me to define an function field analogue of

the global period map used by Minhyong Kim to study rational points.

Despite the theoretical and empirical criticisms of the M-V and CAPM, they are found virtually in all curriculums. Why?

The talk is on impacts, penetrations and lift-offs involving bodies and fluids, with applications that range from aircraft and ship safety and our tiny everyday scales of splashing and washing, up to surface movements on Mars. Several studies over recent years have addressed different aspects of air-water effects and fluid-body interplay theoretically. Nonlinear interactions and evolutions are key here and these are to be considered in the presentation. Connections with experiments will also be described.

Factorization is a property of global objects that can be built up from local data. In the first half, we introduce the concept of factorization spaces, focusing on two examples relevant for the Geometric Langlands programme: the affine Grassmannian and jet spaces.

In the second half, factorization algebras will be defined including a discussion of how factorization spaces and commutative algebras give rise to examples. Finally, chiral homology is defined as a way to give global invariants of such objects.

Many successful methods in image processing and computer vision involve

parabolic and elliptic partial differential equations (PDEs). Thus, there

is a growing demand for simple and highly efficient numerical algorithms

that work for a broad class of problems. Moreover, these methods should

also be well-suited for low-cost parallel hardware such as GPUs.

In this talk we show that two of the simplest methods for the numerical

analysis of PDEs can lead to remarkably efficient algorithms when they

are only slightly modified: To this end, we consider cyclic variants of

the explicit finite difference scheme for approximating parabolic problems,

and of the Jacobi overrelaxation method for solving systems of linear

equations.

Although cyclic algorithms have been around in the numerical analysis

community for a long time, they have never been very popular for a number

of reasons. We argue that most of these reasons have become obsolete and

that cyclic methods ideally satisfy the needs of modern image processing

applications. Interestingly this transfer of knowledge is not a one-way

road from numerical analysis to image analysis: By considering a

factorisation of general smoothing filters, we introduce novel, signal

processing based ways of deriving cycle parameters. They lead to hitherto

unexplored methods with alternative parameter cycles. These methods offer

better smoothing properties than classical numerical concepts such as

Super Time Stepping and the cyclic Richardson algorithm.

We present a number of prototypical applications that demonstrate the

wide applicability of our cyclic algorithms. They include isotropic

and anisotropic nonlinear diffusion processes, higher dimensional

variational problems, and higher order PDEs.

"Show that there is a function $f$ such that for any sequence $(x_1, x_2, \dots)$ we have $x_n = f(x_{n + 1}, x_{n + 2}, \dots)$ for all but finitely many $n$."

Fred Galvin. Problem 5348. The American Mathematical Monthly, 72(10):p. 1135, 1965.\\

This quote is one of the earliest examples of an infinite hat problem, although it's not phrased this way. A hat problem is a non-empty set of colours together with a directed graph, where the nodes correspond to "agents" or "players" and the edges determine what the players "see". The goal is to find a collective strategy for the players which ensures that no matter what "hats" (= colours) are placed on their heads, they will ensure that a "sufficient" amount guess correctly.\\

In this talk I will discuss hat problems on countable sets and show that in a non-transitive setting, the relationship between existence of infinitely-correct strategies and Ramsey properties of the graph breakdown, in the particular case of the parity game. I will then introduce some small cardinals (uncountable cardinals no larger than continuum) that will be useful in analysing the parity game. Finally, I will present some new results on the sigma-ideal of meagre sets of reals that arise from this analysis.

Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G

is called p-nilpotent if S has a normal complement N in G, that is, G is

the semidirect product between S and N. The notion of p-nilpotency plays

an important role in finite group theory. For instance, Thompson's

criterion for p-nilpotency leads to the important structural result that

finite groups with fixed-point-free automorphisms are nilpotent.

By a classical result of Tate one can detect p-nilpotency using mod p

cohomology in dimension 1: the group G is p-nilpotent if and only if the

restriction map in cohomology from G to S is an isomorphism in dimension

1. In this talk we will discuss cohomological criteria for p-nilpotency by

Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the

1960s and 1970s. Finally, we will discuss how one can extend Tate's

result to study p-solvable and more general finite groups.