Past

Last week in the Kinderseminar I talked about a rough estimate on volumes of certain hyperbolic 3-manifolds. This time I will describe a different approach for similar estimates (you will not need to remember that talk, don't worry!), which is, in some sense, complementary to that one, as it regards mapping tori. A theorem of Jeffrey Brock provides bounds for their volume in terms of how the monodromy map acts on the pants graph (a relative of the better known curve complex) of the base surface. I will describe the setting and the relevance of this result (in particular the one it has for me); hopefully, I will also tell you part of its proof.

Berz used the Hahn-Banach Theorem over Q to prove that the graph of a measurable subadditive function that is non-negatively Q-homogeneous consists of two lines through the origin. I will give a proof using the density topology and Steinhaus’ Sum-set Theorem. This dualizes to a much simpler category version: a `Baire-Berz Theorem’. I will give the broader picture of this using F. Burton Jones’ analysis of additivity versus linearity. Shift-compactness and special subsets of R will be an inevitable ingredient. The talk draws on recent work with Nick Bingham and separately with Harry I. Miller.

(A simplified version of) Ax-Grothendieck Theorem states that every injective polynomial map from some power of complex numbers into itself is surjective. I will present a simple model-theoretical proof of this fact. All the necessary notions from model theory will be introduced during the talk. The only prerequisite is basic field theory.

Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the

number of isomorphism classes of irreducible $n$-dimensional complex

characters of $\Gamma$. Representation growth is the study of the

behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of

representation growth.

We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is

bounded by a polynomial in $n$. I will discuss a question posed by

Brent Everitt: can a group with polynomial representation growth have

the alternating group $A_n$ as a quotient for infinitely many $n$?

The Ramsey number $R(K_s, Q_n)$ is the smallest integer $N$ such that every red-blue colouring of the edges of the complete graph $K_N$ contains either a red $n$-dimensional hypercube, or a blue clique on $s$ vertices. Note that $N=(s-1)(2^n -1)$ is not large enough, since we may colour in red $(s-1)$ disjoint cliques of cardinality $2^N -1$ and colour the remaining edges blue. In 1983, Burr and Erdos conjectured that this example was the best possible, i.e., that $R(K_s, Q_n) = (s-1)(2^n -1) + 1$ for every positive integer $s$ and sufficiently large $n$. In a recent breakthrough, Conlon, Fox, Lee and Sudakov proved the conjecture up to a multiplicative constant for each $s$. In this talk we shall sketch the proof of the conjecture and discuss some related problems.

(Based on joint work with Gonzalo Fiz Pontiveros, Robert Morris, David Saxton and Jozef Skokan)

In this talk we explore continuous analogues of matrix factorizations.  The analogues we develop involve bivariate functions, quasimatrices (a matrix whose columns are 1D functions), and a definition of triangular in the continuous setting.  Also, we describe why direct matrix algorithms must become iterative algorithms with pivoting for functions. New applications arise for function factorizations because of the underlying assumption of continuity. One application is central to Chebfun2. 

We prove existence of solution for evolutionary variational and quasivariational inequalities defined by a first order quasilinear operator and a variable convex set, characterized by a constraint on the absolute value of the gradient (which, in the quasi-variational case, depends on the solution itself). The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates.

Uniqueness of solution is proved for the variational inequality. We also obtain existence of stationary solutions, by studying the asymptotic behaviour in time. We shall illustrate a simple “sand pile” example in the variational case for the transport operator were the problem is equivalent to a two-obstacles problem and the solution stabilizes in finite time. Further remarks about these properties of the solution will be presented.This is a joint work with Lisa Santos.

If times allows, using similar techniques, I shall also present the existence, uniqueness and continuous dependence of solutions of a new class of evolution variational inequalities for incompressible thick fluids. These non-Newtonian fluids with a maximum admissible shear rate may be considered as a limit class of shear-thickening or dilatant fluids, in particular, as the power limit of Ostwald-deWaele fluids.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

I will describe how a sieve method can be used to establish the Hasse principle for the variety

$$f(t)=N(x_1,\ldots,x_k),$$

where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.

The vanishing mean curvature flow in Minkowski space is the

natural evolutionary generalisation of the minimal surface equation,

and has applications in cosmology as a model equation for cosmic

strings and membranes. The equation clearly admits initial data which

leads to singularity formation in finite time; Nguyen and Tian have

even shown stability of the singularity formation in low dimension. On

the other hand, Brendle and Lindblad separately have shown that all

"nearly flat" initial data leads to global existence of solutions. In

this talk, I describe an intermediate regime where global existence

of solutions can be proven on a codimension 1 set of initial data; and

where the codimension 1 condition is optimal --- The

catenoid, being a minimal surface in R^3, is a static solution to the

vanishing mean curvature flow. Its variational instability as a

minimal surface leads to a linear instability under the flow. By

appropriately "modding out" this unstable mode we can show the

existence of a stable manifold of initial data that gives rise to

solutions which scatters toward to the

catenoid. This is joint work with Roland Donninger, Joachim Krieger,

and Jeremy Szeftel. The preprint is available at http://arxiv.org/abs/1310.5606v1

Abstract: The expected signature is often viewed as a direct analogue of the Laplace transform, and as such it has been asked whether, under certain conditions, it may determine the law of a random signature. In this talk we first introduce a meaningful topology on the space of (geometric) rough paths which allows us to study it as a well-defined probability space. With the help of compact symplectic Lie groups, we then define a set of characteristic functions and show that two random variables in this space are equal in law if and only if they agree on each characteristic function. We finally show that under very general boundedness conditions, the value of each characteristic function is completely determined by the expected signature, giving an affirmative answer to the aforementioned question in many cases. In particular, we demonstrate that the Stratonovich signature is completely determined in law by its expected signature, and show how a similar technique can be used to demonstrate convergence in law of random signatures.

Background material: http://arxiv.org/abs/1307.3580

Factorization homology is an invariant of an n-manifold M together with an n-disk algebra A. Should M be

a circle, this recovers the Hochschild complex of A; should A be a commutative algebra, this recovers the

homology of M with coefficients in A. In general, factorization homology retains more information about

a manifold than its underlying homotopy type.

In this talk we will lift Poincare' duality to factorization homology as it intertwines with Koszul

duality for n-disk algebras -- all terms will be explained. We will point out a number of consequences

of this duality, which concern manifold invariants as well as algebra invariants.

This is a report on joint work with John Francis.

A review of a valuation strategy to price American-style option contracts in a “limited information” framework is discussed where sequential Monte Carlo (SMC) techniques, as presented in Doucet, de Freitas, and Gordon’s text Sequential Monte Carlo Methods in Practice, and the least–squares Monte Carlo (LSM) approach of Longstaff and Schwartz (Review of Financial Studies 14:113-147, 2001), are used as part of the valuation methodology. We utilize a risk–neutralized version of a mean-reverting model to model the volatility process. We assume that volatility is a latent stochastic process, and we capture information about it using “summary vectors” based on sequential Monte Carlo posterior filtering distributions. Of primary interest in this work is an empirical assessment of American options governed by a stochastic volatility model where the focus is on the market price of volatility risk (or the volatility risk premium). We discuss statistical modeling of the market price of volatility risk as our current evidence reveals interesting nuances about the volatility risk premium, and we hypothesize that switching models or more sophisticated time-series models could be of value to understand the empirical observations we found on the market price of volatility risk. Prior studies have shown that the magnitude of the volatility risk premium changes markedly when an American index option (NYSE Arca Oil Index Options) is in its expiration month relative to prior months, or that the magnitude varies across equities. Our objective is to study if useful information can be extracted from the volatility risk premium process, and how this information can better inform holders of American options when making decisions under uncertainty.

Key words: American options, stochastic volatility, volatility risk, sequential, Monte Carlo, risk premium, decisions, uncertainty

Disclaimer: The views expressed in this abstract (and the paper that will accompany it) are solely those of the authors and do not, in any way, reflect the opinions of the Office of the Comptroller of the Currency (OCC).

Raushan Buzyakova asked if a space is hereditarily D provided 
that the extent and Lindelöf numbers coincide for every subspace. We 
will introduce interval topologies on trees and present Nyikos' 
counterexample to this conjecture.

We consider over-the-counter (OTC) transactions with bilateral default risk, and study the optimal design of the Credit Support Annex (CSA). In a setting where agents have access to a trading technology, default penalties and collateral costs arise endogenously as a result of foregone investment opportunities. We show how the optimal CSA trades off the costs of the collateralization procedure against the reduction in exposure to counterparty risk and expected default losses. The results are used to provide insights on the drivers of different collateral rules, including hedging motives, re-hypothecation of collateral, and close-out conventions. We show that standardized collateral rules can have a detrimental impact on risk sharing, which should be taken into account when assessing the merits of standardized vs. bespoke CSAs in non-centrally cleared OTC instruments. This is joint work with D. Bauer and L.R. Sotomayor (GSU).