Following the recent paper of Ogasa, we attempt to construct Boy's surface using only paper and tape. If this is successful we hope to address such questions as:
Is that really Boy's surface?
Why should we care?
Do we have any more biscuits?
Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.
A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.
The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.
The recently completed auction for 4G mobile spectrum was the most importantcombinatorial auction ever held in the UK. In general, combinatorial auctions allow bidders to place individual bids on packages of items,instead of separate bids on individual items, and this feature has theoretical advantages for bidders and sellers alike. The accompanying challenges of implementation have been the subject of intense work over the last few years, with the result that the advantages of combinatorial auctions can now be realised in practice on a large scale. Nowhere has this work been more prominent than in auctions for radio spectrum. The UK's 4G auction is the most recent of these and the publication by Ofcom (the UK's telecommunications regulator) of the auction's full bidding activity creates a valuable case study of combinatorial auctions in action.
Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.
A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.
The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.
Conventional decoherence (usually called 'Environmental
Decoherence') is supposed to be a result of correlations
established between some quantum system and the environment.
'Intrinsic decoherence' is hypothesized as being an essential
feature of Nature - its existence would entail a breakdown of
quantum mechanics. A specific mechanism of some interest is
'gravitational decoherence', whereby gravity causes intrinsic
decoherence.
I will begin by discussing what is now known about the mechanisms of
environmental decoherence, noting in particular that they can and do
involve decoherence without dissipation (ie., pure phase decoherence).
I will then briefly review the fundamental conflict between Quantum
Mechanics and General Relativity, and several arguments that suggest
how this might be resolved by the existence of some sort of 'gravitational
decoherence'. I then outline a theory of gravitational decoherence
(the 'GR-Psi' theory) which attempts to give a quantitative discussion of
gravitational decoherence, and which makes predictions for
experiments.
The weak field regime of this theory (relevant to experimental
predictions) is discussed in detail, along with a more speculative
discussion of the strong field regime.
A time-invariant level surface is a (codimension one)
spatial surface on which, for every fixed time, the solution of an
evolution equation equals a constant (depending on the time). A
relevant and motivating case is that of the heat equation. The
occurrence of one or more time-invariant surfaces forces the solution
to have a certain degree of symmetry. In my talk, I shall present a
set of results on this theme and sketch the main ideas involved, that
intertwine a wide variety of old and new analytical and geometrical
techniques.
We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.
In many models of Applied Probability, the distributional limits of recursively defined quantities satisfy distributional identities that are reminiscent of equations of stability. Therefore, there is an interest in generalized concepts of equations of stability.
One extension of this concept is that of random variables ``stable by random weighted mean'' (this notion is due to Liu).
A random variable $X$ taking values in $\mathbb{R}^d$ is called ``stable by random weighted mean'' if it satisfies a recursive distributional equation of the following type:
\begin{equation} \tag{1} \label{eq:1}
X ~\stackrel{\mathcal{D}}{=}~ C + \sum_{j \geq 1} T_j X_j.
\end{equation}
Here, ``$\stackrel{\mathcal{D}}{=}$'' denotes equality of the corresponding distributions, $(C,T_1,T_2,\ldots)$ is a given sequence of real-valued random variables,
and $X_1, X_2, \ldots$ denotes a sequence of i.i.d.\;copies of the random variable $X$ that are independent of $(C,T_1,T_2,\ldots)$.
The distributions $P$ on $\mathbb{R}^d$ such that \eqref{eq:1} holds when $X$ has distribution $P$ are called fixed points of the smoothing transform
(associated with $(C,T_1,T_2,\ldots)$).
A particularly prominent instance of \eqref{eq:1} is the {\texttt Quicksort} equation, where $T_1 = 1-T_2 = U \sim \mathrm{Unif}(0,1)$, $T_j = 0$ for all $j \geq 3$ and $C = g(U)$ for some function $g$.
In this talk, I start with the {\texttt Quicksort} algorithm to motivate the study of \eqref{eq:1}.
Then, I consider the problem of characterizing the set of all solutions to \eqref{eq:1}
in a very general context.
Special emphasis is put on \emph{endogenous} solutions to \eqref{eq:1} since they play an important role in the given setting.
Abstract: Non-geometric rough paths arise
when one encounters stochastic integrals for which the the classical
integration by parts formula does not hold. We will introduce two notions of
non-geometric rough paths - one old (branched rough paths) and one new (quasi
geometric rough paths). The former (due to Gubinelli) assumes one knows nothing
about products of integrals, instead those products must be postulated as new
components of the rough path. The latter assumes one knows a bit about
products, namely that they satisfy a natural generalisation of the
"Ito" integration by parts formula. We will show why they are both
reasonable frameworks for a large class of integrals. Moreover, we will show
that Ito's formula can be derived in either framework and that this derivation
is completely algebraic. Finally, we will show that both types of non-geometric
rough path can be re-written as geometric rough paths living above an extended
version of the original path. This means that every non-geometric rough
differential equation can be re-written as a geometric rough differential
equation, hence generalising the Ito-Stratonovich correction formula.
A workshop on different aspects of deformation theory in various fields
It is well known that the trapezoid rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with powerful algorithms all across scientific computing, including double exponential and Gauss quadrature, computation of inverse Laplace transforms, special functions, computational complex analysis, the computation of functions of matrices and operators, rational approximation, and the solution of partial differential equations.
This talk represents joint work with Andre Weideman of the University of Stellenbosch.
A workshop on different aspects of deformation theory in various fields
A workshop on different aspects of deformation theory in various fields
A workshop on different aspects of deformation theory in various fields
To follow
********** PLEASE NOTE THE SPECIAL TIME **********
Total generalised variation (TGV) was introduced by Bredies et al. as a high quality regulariser for variational problems arising in mathematical image processing like denoising and deblurring. The main advantage over the classical total variation regularisation is the elimination of the undesirable stairscasing effect. In this talk we will give a small introduction to TGV and provide some properties of the exact solutions to the L^{2}-TGV model in the one dimensional case.
Abstract: Burgers equation is a quasilinear partial differential equation (PDE), proposed in 1930's to model the evolution of turbulent fluid motion, which can be linearized to the heat equation via the celebrated Cole-Hopf transformation. In the first part of the talk, we study in detail general versions of stochastic Burgers equation with random coefficients, in both forward and backward sense. Concerning the former, the Cole-Hopf transformation still applies and we reduce a forward stochastic Burgers equation to a forward stochastic heat equation that can be treated in a “pathwise" manner. In case of deterministic coefficients, we obtain a probabilistic representation of the Cole-Hopf transformation by associating the backward Burgers equation with a system of forward-backward stochastic differential equations (FBSDEs). Returning to random coefficients, we exploit this representation in order to establish a stochastic version of the Cole-Hopf transformation. This generalized transformation allows us to find solutions to a backward stochastic Burgers equation through a backward stochastic heat equation, subject to additional constraints that reflect the presence of randomness in the coefficients. In both settings, forward and backward, stochastic Feynman-Kac formulae are derived for the solutions of the respective stochastic Burgers equations, as well. Finally, an application that illustrates the obtained results is presented to a pricing/hedging problem arising from mathematical finance.
In the second part of the talk, we study a class of stochastic saddlepoint systems, represented by fully coupled FBSDEs with infinite horizon, that gives rise to a continuous time rational expectations / consol rate model with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness and dependence on a parameter of adapted solutions. Making further the connection with quasilinear backward stochastic PDEs (BSPDEs), we are led to the notion of stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of a large investor is also provided as an application to this framework.
This is joint work with N. Frangos, X.- I. Kartala and A. N. Yannacopoulos*
The standard Taylor series approach to the higher-order approximation of vector SDEs requires simulation of iterated stochastic integrals, which is difficult. The talk will describe an approach using methods from optimal transport theory which avoid this difficulty in the case of non-degenerate diffusions, for which one can attain arbitrarily high order pathwise approximation in the Vaserstein 2-metric, using easily generated random variables.
Please note the unusual day of the week for this workshop (a Monday) and also the unusual location.