Abstract

In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the degree k rational cohomology of P\Sigma_n tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n -- that is, in some precise sense, the description of the decomposition of these cohomology groups into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.

Abstract

The Kodaira vanishing theorem and its generalizations are extremely important tools in higher dimensional geometry and the failure of these theorems in positive characteristic causes great difficulties in extending the existing theories to that realm. In this talk I will discuss new results about cases where an appropriate vanishing theorem holds and cases where the expected one fails even in characteristic zero. These results are joint works (separately) with Christopher Hacon and with János Kollár.

Abstract

High contrast in electrical conductivity motivates the investigation of electromagnetic methods in geophysics, for instance, for hydrocarbon and mineral exploration. A straightforward approach to modelling the spatial distribution of this parameter within the earth is the assumption of piecewise constant values, defined on a moderately fine tessellation of the volume under investigation by hexahedra or tetrahedra. We study here the solution of the 3-D forward problem for time-harmonic electromagnetic fields using finite elements, based on the above mentioned tessellation. Furthermore, we seek to reconstruct the spatial distribution of conductivity of an overparameterized model by a regularised output least squares approach. Our model assumption, a piecewise constant coefficient, allows for simplifications of the forward solver which eventually lead to an overall faster imaging algorithm. The model assumption also requires special care when the regularisation operator is derived for unstructured meshes within the finite element framework.

Abstract

We consider a nonlinear Schrödinger equation with a general nonlinearity. In space dimension 2 or higher, this equation admits solitons (standing/traveling waves) with a fixed profile which is not a ground state. These types of profiles are called excited states. Due to instability, excited solitons are singular objects for the dynamics of NLS. Nevertheless, we will show in this talk how to exhibit solutions of NLS behaving in large time like a sum of excited solitons with high relative speeds.

Abstract

We compute the exact value of essential dimension for split spinor groups and split even Clifford groups. I will also discuss some applications of these results to quadratic forms.

This talk is based on joint work with A. Merkurjev; see
www.math.uni-bielefeld.de/lag/man/455.html

Abstract

The notions of covering maps, Galois theory and representations, Laplacians, L^2 Betti numbers, and sheaves are well known in many areas of mathematics. Despite the fact that these notions are easy to describe in the context of graph theory, these tools seem to be greatly underutilized there.  Furthermore, such ideas can be used to prove theorems that have no obvious connection to sheaf theory, or even to graph theory.

We describe the above notions, and briefly describe the applications of sheaves and covering maps to (1) a solution of the Hanna Neumann Conjecture of the 1950's, and (2) an equivalence of two notions of "2-independence," which are generalizations of ordinary linear independence (which has no obvious connection to graph theory).  We also describe our original motivation to study sheaves on discrete structures that arose from complexity theory.

Application (2) seems to make essential use of a result regarding covering maps of graphs and "scaled Abelian limits" of Betti numbers, that may be described as a "baby version" of l-adic cohomology.

Abstract

Abstract

Every transverse knot in a contact 3-manifold is represented as a closed braid in an open book. In this talk, based on a new technique called an open book foliation, we give a formula of self-linking number in terms of braids and open books. Surprisingly, our self-linking number formula essentially uses Johnson's homomorphism. This is a joint work with Keiko Kawamuro (Univ. Iowa).

Abstract

A famous result by Steven Smale states that we can turn the sphere inside-out through immersions: this is called the eversion of the sphere. We will explain this result and the strategy of its proof which is a "cut-and-paste" strategy quite standard in algebraic topology. This approach allows us to understand globally the space of all immersions of a given manifold in another one, like the space of all immersion of the sphere in R^3 in the case of Smale's eversion. This theory has been enhanced by Goodwillie in the 1990's to understand spaces of embeddings. We will explain how this can be applied to understand spaces of knots, that is the spaces of all embeddings of a circle into a fixed euclidean space.

Details

A richly illustrated talk about Goedel's time in Vienna. These fifteen years cover his brilliant start at the University and his discovery of the incompleteness theorems, the confused period between Hitler's rise to power in Germany and the annexation of Austria, and Goedel's desperate struggle to leave national-socialist Vienna.

Abstract

• Developing notions of infinity (pdf)
• Layers of abstraction: theory and design for the instruction of limits concepts (pdf)
• Divisibility and transparency of number representations (pdf)

Abstract

Abstract

Evolutionary game theory helps to investigate the role of incentives in promoting cooperative behavior in joint enterprises. In particular, this lecture deals with the surprising effects of optional participation in collaborative enterprises. Coercion works better for voluntary rather than compulsory collaboration. A social contract need not be based on rational deliberation or the command of a higher authority. It can emerge spontaneously through social learning of individuals guided by no more than their myopic self-interest. 

Abstract

In this talk I explain a promising and previously unnoticed link between electronic structure of molecules and optimal transportation (OT), and I give some
first results. The `exact' mathematical model for electronic structure, the many-electron Schroedinger equation, becomes computationally unfeasible for more than a dozen or so electrons. For larger systems, the standard model underlying a huge literature in computational physics/chemistry/materials science is density functional theory (DFT). In DFT, one only computes the single-particle density instead of the full many-particle wave function. In order to obtain a closed equation, one needs a closure assumption which expresses the pair density in terms of the single-particle density rho.

We show that in the semiclassical Hohenberg-Kohn limit, there holds an exact closure relation, namely the pair density is the solution to a optimal transport problem with Coulomb cost. We prove that for the case with $N=2$ electrons this problem has a unique solution given by an optimal map; moreover we derive an explicit formula for the optimal map in the case when $\rho$ is radially symmetric (note: atomic ground state densities are radially symmetric for many atoms such as He, Li, N, Ne, Na, Mg, Cu).

In my talk I focus on how to deal with its main mathematical novelties (cost decreases with distance; cost has a singularity on the diagonal). I also discus the derivation of the Coulombic OT problem from the many-electron Schroedinger equation for the case with $N\ge 3$ electrons, and give some results and explicit solutions for the many-marginals OT problem.

Joint works with Gero Friesecke (TU Munich) and Claudia Klueppelberg (TU Munich).

Details

We provide results related to the study of prime points on level sets of homogeneous integral forms which are linear or quadratic. In the linear case we present an extension of the Green-Tao Theorem, which finds affine copies of finite intervals in relatively dense subsets of the primes, to a higher dimensional setting in which one finds affine copies of suitably generic point configurations in relatively dense subsets of the Cartesian product of the primes. For general integral quadratic forms we present a result which is a Birch-Goldbach type theorem for a single quadratic form with sufficient rank. This guarantees solubility among the primes on the level set of a quadratic form subject to local conditions. This is an extension of a well known result due to Hua.

Abstract

From flapping birds to space telescopes:  The modern science of Origami.

The last decade of this past century has been witness to a revolution in the development
 and application of mathematical techniques to origami, the centuries-old Japanese art
of paper-folding.  The techniques used in mathematical origami design range
from the abstruse to the highly approachable.  In this talk, I will describe how geometric
concepts led to the solution of a broad class of origami folding problems - specifically,
the problem of efficiently folding a shape with an arbitrary number and arrangement of
flaps, and along the way, enabled origami designs of mind-blowing complexity and
realism, some of which you'll see too.  As often happens in mathematics, theory originally
developed for its own sake has led to some surprising practical applications.  The
algorithms and theorems of origami design have shed light on long-standing
mathematical questions and have solved practical engineering problems.  I will discuss
examples of how origami has enabled safer airbags, robdingnagian space telescopes,
and more.

 

Abstract

We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general non-linear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of “buy and hold”, “cut loss or take profit”, “cut loss and let profit run”, and “sell on a percentage of historical high”.

Abstract

Abstract

Quantum Unique Ergodicity has been a widely studied conjecture of Rudnick and Sarnak (1994), concerning the distribution of large frequency eigenstates on a negatively curved manifold. Arithmetic Quantum Unique Ergodicity (AQUE) restricts the problem to arithmetic manifolds, such as SL(2,Z) \ H, the classical modular surface. Work of Lindenstrauss (2006) combined with the elimination of escape of mass proved by Soundararajan (2010) confirmed AQUE for the classical modular surface. This talk is concerned with AQUE for Hilbert modular surfaces, and in particular, my thesis work involving the elimination of escape of mass in this case.