Title: On the Minimal Model Program. Abstract: The aim of the Minimal Model Program is to generalize the classification of complex projective surfaces, known in the early 20th century, to higher dimensional varieties. Besides providing a historical introduction, we will discuss some recent results and new aspects of this Program. Remark: Cascini is an aP candidate.

Let be an algebraic number, $h(\alpha)$ the logarithmic Weil height of , and the minimal polynomial of over ${\mathbb Q}$. Schinzel has proven that if is such that $f(0)=\pm 1$, $\ f(\pm 1) \neq 0$, and all roots of real, then $h(\alpha) \geq \log \Biggl ( {\dfrac{1+\sqrt{5}}{2}} \Biggr )^{1/2} $. This article establishes the following generalization of Schinzel's result. THEOREM 1: Let be an algebraic number, different from $\pm 1$. Let be the set of Galois conjugates of that are real and suppose that $\vert \Lambda \vert \neq 0$. Let $d=[{\mathbb Q}(\alpha):{\mathbb Q}]$ and let $R_{\alpha} \equiv \vert \Lambda \vert / d$. Let $\beta = 1-1 / R_{\alpha}$. Then $h(\alpha) \geq \log \Biggl (\dfrac{ 2^{\beta}+\sqrt{4^{\beta}+4}}{2} \Biggr )^{R_{\alpha}/ 2}$. We also establish the following result concerning the Weil Height of the Centralizer of Complex Conjugation, THEOREM 2: Let ${\mathbb K} / {\mathbb Q}$ be a Galois extension of finite degree. Let $G \equiv Aut({\mathbb K} / {\mathbb Q})$. Let $\alpha \in {\mathbb K}^{\times}$ have a Galois conjugate not on the archimedean unit circle. Let $\sigma: {\mathbb K} \hookrightarrow {\mathbb C}$ be an embedding. Let $\xi \in G$ correspond to complex conjugation with respect to . Let $n=[G:C_G(\xi)]$. Let $\theta(\alpha)=1$ if has a real Galois conjugate and let $\theta(\alpha)=2$ if does not have a real Galois conjugate. Then $h(\alpha) \geq \log \Biggl (\dfrac{ 2^{1-n}+\sqrt{4^{1-n}+4}}{2} \Biggr )^{1/ \theta(\alpha) n}.$

When studying the representation theory of Lie groups, the flag manifold (or Grassmannian) arises naturally and often. This talk will discuss a much larger space attached to any reductive group G, which, by a theorem of Luzstig, Ginzburg, Mirkovic-Vilonen, has encoded in its geometry all of the representation theory of the Langlands dual group. I will begin with definitions and examples, hopefully ending with a statement (and perhaps even sketch) of the theorem alluded to above.

This lecture is also background material for this week's Geometry Seminar.

In this paper, we prove that any nonconstant, solution of the infinity Laplacian equation $u_{x_i}u_{x_j}u_{x_ix_j}=0$ can not have interior critical points. This result was first proved by G. Aronsson in two dimensions. If the solution is $C^4$, L. C. Evans established a harnack inequality for $|Du|$, which implies that non-constant $C^4$ solutions have no interior critical points for any dimension. Our proof is strongly motivated by Evans's method.

I'll give a gentle introduction to the HOMFLY homology theory introduced by Khovanov and Rozansky.

Hartshorne's conjecture states that a smooth projective variety over an algebraically closed field with ample tangent bundle is necessarily a projective space. Mori gave an algebraic proof in 1979 which works over all ground fields of any characteristic. You will see Break and Bend technique, method of reduction to characteristic p and an elegant theorem of the existence of rational cuves. This paper together with some subsequent papers of Mori started the Minimal Model Program. I'll split the proof into two talks as follows, I: Some preliminaries on deformation of Hom scheme and a theorem of existence of rational curves. II: Proof of Hartshorne's conjecture.

A. Schinzel has proven that if all the conjugates of all algebraic number (different from 1 and -1) are real then the height of the number is bounded from below by the square root of the golden ratio. I show this result to be an extreme case of a new formula for the heights of algebraic numbers. I then discuss an interpretation of this result in terms of Galois Group structure and connect this to a generalization of a result of F. Amoroso and R. Dvornicich. Using a different set of methods, I will then derive the lowest Mahler measure (other than 1) in Dihedral extensions of the rationals.

Abstract: I will define rank 1 D-branes in generalized complex manifolds and focus attention on the case of holomorphic Poisson varieties, where many interesting phenomena arise. I will then describe some new constructions of such branes.

The volatile economic environment of the 1970s precipitated a rapid expansion in corporate risk management and derivatives markets. Notwithstanding the number and scope of recent derivatives debacles, the use of risk management by corporations generates benefits throughout the economy in terms of efficiency and stability.