# Seminars & Events for 2007-2008

##### On singularity formation for certain geometric wave equations

We dicuss recent results, obtained jointly with W. Schlag and D. Tataru, on a new kind of singularity formation for certain critical nonlinear geometric wave equations.

##### Stochastic programming, progressive hedging, and projective splitting methods

This talk presents two optimization problems arising from the same applied probability application, in which we monitor an information source to which data are added at times modelled by a nonhomogeneous Poisson process. In the first application, we suppose that the arrival rate function is known and we may poll the information source a limited number of times over some planning horizon.

##### Long range order for lattice dipoles

We consider a system of classical Heisenberg spins on a cubic lattice in dimensions three or more, interacting via the dipole-dipole interaction. We prove that at low enough temperature the system displays orientational long range order, as expected by spin wave theory. The proof is based on reflection positivity methods.

##### Grothendieck duality via the homotopy category of flat modules

We will discuss a novel perspective of dualizing complexes which has been discovered in the last three years. We will review three recent articles, by Jorgensen, Krause and Iyengar-Krause, before coming to recent work by myself and by Murfet.

##### Automorphic lifts of prescribed type

##### The classical and quantum geometry of polyhedral singularities and their resolutions

Let $G$ be a finite subgroup of $SO(3)$. Such groups admit an ADE classification: they are the cyclic groups, the dihedral groups, and the symmetries of the platonic solids. The singularity $C^3/G$ has a natural Calabi-Yau resolution $Y$ given by Nakamura's $G$-Hilbert scheme. The classical geometry of $Y$ (its cohomology) can be described in terms of the representation theory of $G$.

##### Secant Varieties and Applications

The notion of secant varieties in algebraic geometry is classical, but not much is known about these objects, even for many simple cases. Surprisingly, it is possible to translate certain questions in fields as disparate as complexity theory, statistics, and biology into straightforward (but often unsolved) problems about secant varieties.

##### On the quantitative equidistribution of nilfows and Weyl sums

It is know since the work of Furstenberg that the equidistribution of the fractional parts of polynomials sequences with irrational leading coeeficient can be derived from the unique ergodicity of (certain) nilflows.

##### The convexity of length functions on Fenchel-Neilsen coordinates for Teichmuller space

##### Explicit reduction modulo p of certain crystalline representations

We use the $p$-adic local Langlands correspondence for $GL_2(Q_p)$ to explicitly compute the reduction modulo $p$ of crystalline representations of small slope, and give applications to modular forms.

Joint work with Kevin Buzzard.

##### Algebraic properties of quantum homology

In this talk we discuss certain algebraic properties of the quantum homology algebra of toric Fano manifolds. In particular, we describe an easily-verified sufficient condition for the semi-simplicity of the quantum homology. Moreover, we provide some examples of monotone toric Fano manifolds for which the quantum homology is not semi-simple.

This is joint work with Ilya Tyomkin.

##### An Elegant and Insightful Direct Combinatorial Proof of the Arithmetical Identity 4+5=2+7

There are no trivial theorems, only trivial mathematicians (those who believe that there exist trivial theorems). Being a non-trivial mathematician myself, I will present a new, elegant, and very insightful direct combinatorial proof of the seemingly (to most people) "trivial" arithmetical theorem that states that four plus five equals two plus seven.

##### Knots and Topological Growth Laws in the Faddeev Model

In this talk, I present some joint work with Fanghua Lin on the existence of knotted solitons realized as the energy-minimizing configurations in the Faddeev field-theoretical model and the associated universal topological growth laws which relate the knot energy to knot topological charge defined by the Hopf invariant.

##### Martin boundary of non-positive curved manifolds

##### Active and Semi-Supervised Learning Theory

Science is arguably the pinnacle of human intellectual achievement, yet the scientific discovery process itself remains an art. Human intuition and experience is still the driving force of the high-level discovery process: we determine which hypotheses and theories to entertain, which experiments to conduct, how data should be interpreted, when hypotheses should be abandoned, and so on.

##### Efficient pricing of American options in models with stochastic volatility and jumps

##### Behavioral Portfolio Choice in Continuous Time

In this talk I shall report recent progress on continuous-time behavioural portfolio choice under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility functions and probability distortions. It is shown that the model well-posedness becomes a prominent issue in such a behavioural model.

##### Paint-by-numbers: pattern formation in two-dimensional sheets of cells

One of the basic mechanisms responsible for the formation of three-dimensional organs relies on the regulated folding of epithelia (two-dimensional sheets of cells). This process is driven by the spatially nonuniform and dynamic distribution of multiple chemical components (products of gene expression) across the epithelia that prepare for folding.

##### The sixth-sphere

Kirchhoff's problem (1947) asks whether $S^6$ is a complex manifold. We will review basic notions in complex and almost-complex geometry and discuss a bit of what is known regarding this problem, making contact with the octonions, $G_2$, Lie brackets, and curvature.

##### The structure of bullfree graphs

The bull is a graph consisting of a triangle and two disjoint pendant edges. Obvious examples of bullfree graphs are graphs with no triangle, or graphs with no stable set of size three. But there are others (for example, substituting one bullfree graph into another produces a bullfree graph).