# Seminars & Events for 2009-2010

##### The Reflector Antenna Problem and its Connection to Mass Transport

The reflector antenna problem is the problem of constructing a reflective surface which directs a specified energy distribution on the sphere to another specified energy distribution on the so-called far-field sphere.

##### Fractional total colourings of graphs of high girth

The total graph $T(G)$ has vertex set $V(G) \cup E(G)$, in which two vertices of $T(G)$ are adjacent precisely if they arise from (i) two adjacent vertices of $G$, (ii) two incident edges of $G$, or (iii) an edge of $G$ and one of its endpoints. The (fractional) total chromatic number of $G$ is the (fractional) chromatic number of $T(G)$.

##### Some results in toric topology

##### A combinatorial approach to harmonic maps

I will discuss joint work with Hass on a combinatorial approach to harmonic maps. This work is still in progress. Such an approach has been used previously by other authors for computational reasons, but we are developing our approach with an eye to theoretical applications. Some applications in low dimensional topology will be discussed.

##### On the areas of rational triangles or How did Euler (and how can we) solve $xyz(x+y+z) = a$?

By Heron's formula there exists a triangle of area $\sqrt{a}$ all of whose sides are rational if and only if $a > 0$ and $xyz(x + y + z) = a$ for some rationals $x, y, z$. In a 1749 letter to Goldbach, Euler constructed infinitely many such $(x, y, z)$ for any rational $a$ (positive or not), remarking that it cost him much effort but not explaining his method.

##### Analogue of the Narasimhan-Seshadri theorem in higher dimensions and holonomy

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

##### The geometry of static and stationary matter configurations in relativity

There are a number of interesting questions and some recent results concerning the geometry of equilibrium matter and black hole configurations for the Einstein equations. In this talk we will give background on these questions and survey known results. We will then describe recent work with R. Beig and G. Gibbons on the problem.

##### Homotopy theory and spaces of representations

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer $q>1$ and every topological group $G$, with realizations $B(q,G)$ that filter the classifying space $BG$. In particular for $q=2$ this yields a single space $B(2,G)$ assembled from all the n-tuples of commuting elements in $G$.

##### Mirror symetry for del Pezzo surfaces

Joint Columbia-Courant-Princeton University Algebraic Geometry Seminar

##### Testable New Theory about Early-Universe Density Fluctuations and Origins of Solar Systems: Applied-Probability and Quantum-Physics Aspects

The talk will summarize, with a focus on applied-probability aspects, the main findings, testable predictions and research opportunities stemming from a new probabilistic model of how complex patterns of energy-density fluctuations may have arisen during the inflation phase of the Big Bang.

##### Astala's conjecture on Hausdorff measure distortion under planar quasiconformal mappings and related removability problems

In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994) (for which he received the Salem prize), Astala proved that if $E$ is a compact set of Hausdorff dimension $d$ and $f$ is $K$-quasiconformal, then $fE$ has Hausdorff dimension at most $d' = \frac{2Kd}{2+(K-1)d}$, and that this result is sharp.

##### Smoothing surface singularities via mirror symmetry

We use the Strominger-Yau-Zaslow interpretation of mirror symmetry to describe deformations of surface singularities in terms of counts of holomorphic curves and discs on a mirror surface. In particular we prove Looijenga's conjecture on smoothability of cusp singularities. This is joint work with Mark Gross and Sean Keel, and builds on work of Gross-Siebert and Gross-Pandharipande-Siebert.

##### The emergence of a giant vortex in a fast rotating Bose gas

A Bose gas in fast rotation normally exhibits a growing number of vortices of unit strength if the angular velocity is increased. In an anharmonic trap at sufficiently high velocity, however, a phase transition is expected: Vortices in the bulk should disappear and all vorticity become concentrated in a region where the density is very low.

##### Abelian sandpile model and self-similar groups

The sandpile model was introduced in 1987 by physisists Bak, Tang and Wiesenfeld as a tool to study what they called *the self-organized criticality*—spontaneous appearance of power laws or fractal interfaces, observed in some natural phenomena. The mathematical study of the model was initiated a couple of years later by Deepak Dhar.

##### Extend Your Function Now! Results Guaranteed or Your Money Back

A question that is often asked in Extension Theory is the following: Given $E \subset \mathbb{R}^n$, and $f:E \rightarrow \mathbb{R}$. Is it possible to extend $f$ to a function lying in the space $X(\mathbb{R}^n)$ ?. This question has been answered in the case when $X$ is the space $C^m$ of functions continuously differentiable through order $m$.

##### Scheduling, Percolation, and the Worm Order

We show that in any submodular system there is a maximal chain that is minimal, in a very strong sense, among all paths from 0 to 1. The consequence is a set of general conditions under which parallel scheduling can be done without backward steps.

##### Ergodicity of some boundary driven integrable Hamiltonian chains

Small Hamiltonian systems are connected in a chain the ends of which are coupled to unequal heat baths, forcing the system out of equilibrium. Energy exchange is of a form that leads to integrable dynamics. A proof of ergodicity of both equilibrium and nonequilibrium steady states will be presented.

##### Comultiplication in link Floer homology and transversely non-simple links

By theorems of Bennequin, Wrinkle and Orevkov-Shevchishin, transverse links in the unique tight contact structure on $R3$ may thought of as closed braids. For a word $h$ in the braid group on $n$ strands, let us denote by $T_h$ the corresponding transverse link.

##### Slope filtrations in families

In the 21st-century approach to p-adic Hodge theory, one studies local Galois representations (and related objects) by converting them into modules over certain power series rings carrying certain extra structures (Frobenius actions and derivations). A key tool in matching up the two sides is a certain classification theorem for the second class of objects, called the slope filtration theorem.

##### A brief survey of effective equidistribution results in Gamma\G

Equidistribution results for orbits and more general configurations in Gamma\G are a central focus of the theory of flows on homogeneous spaces. A notable example that comes to mind is Ratner's equidistribution theorem. I will survey some old and new quantitive equidistribution results of this flavor by several authors.