Past

Mean curvature vector is the negative gradient of the area functional. Thus if we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. When the ambient manifold is Kahler-Einstein, being Lagrangian is preserved under MCF. It is thus very natural trying to construct special Lagrangian/ Lagrangian minimal through MCF. In this talk, I will make a brief introduction and report some of my recent works with my coauthors in this direction, which mainly concern the singularities of the flow.

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let

Y --> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.

We consider finite sequences $h = (h_1, . . . h_s)$ of real polynomials in $X_1, . . . ,X_n$ and assume that

the semi-algebraic subset $S(h)$ of $R^n$ defined by $h1(a1, . . . , an) \leq 0$, . . . , $hs(a1, . . . , an) \leq 0$ is

bounded. We call $h$ (quadratically) archimedean if every real polynomial $f$, strictly positive on

$S(h)$, admits a representation

$f = \sigma_0 + h_1\sigma_1 + \cdots + h_s\sigma_s$

with each $\sigma_i$ being a sum of squares of real polynomials.

If every $h_i$ is linear, the sequence h is archimedean. In general, h need not be archimedean.

There exists an abstract valuation theoretic criterion for h to be archimedean. We are, however,

interested in an effective procedure to decide whether h is archimedean or not.

In dimension n = 2, E. Cabral has given an effective geometric procedure for this decision

problem. Recently, S. Wagner has proved decidability for all dimensions using among others

model theoretic tools like the Ax-Kochen-Ershov Theorem.

A modelling framework is introduced in which there is a small agent who is more susceptible to the flow of information in the market as compared to the general market participants. In this framework market participants have access to a stream of noisy information concerning the future returns of the asset, whereas an informative trader has access to an additional information source which is also obscured by further noise, which may be correlated with the market noise. The informative trader utilises the extraneous information source to seek statistical arbitrage opportunities, in exchange with accommodating the additional risk. The information content of the market concerning the value of the impending cash flow is represented by the mutual information of the asset price and the associated cash flow. The worthiness of the additional information source is then measured in terms of the difference of mutual information between market participants and the informative trader. This difference is shown to be strictly nonnegative for all parameter values in the model, when signal-to-noise ratio is known in advance. Trading strategies making use of the additional information are considered. (Talk is based on joint work with M.H.A. Davis (Imperial) & R.L. Friedman (Imperial & Royal Bank of Scotland).

Refining Julia Robinson's 1949 work on the undecidability of the first order theory of Q, we prove that Z is definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points.

The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an

overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.

Linear operators preserving non-vanishing properties are an important

tool in e.g. combinatorics, the Lee-Yang program on phase transitions, complex analysis, matrix theory. We characterize all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing when the variables are in prescribed open circular domains, which solves the higher dimensional counterpart of a long-standing classification problem going back to P\'olya-Schur. This also leads to a self-contained theory of multivariate stable polynomials and a natural framework for dealing with Lee-Yang and Heilmann-Lieb type problems in a uniform manner. The talk is based on joint work with Petter Brändén.

In this talk I shall review analytical and numerical results on equilibrium configurations of rotating fluid bodies within Einstein's theory of gravitation.