# Seminars & Events for 2011-2012

##### Bubbles and Onis

This is joint work with Peter Albers. We study a Floer gradient equation on a very negative line bundle over a symplectic manifold. If the line bundle is negative enough there are generically no holomorphic spheres. We explain the metamorphosis of bubbles to Onis and then point out a weakness of the Onis - namely a free involution on them...

##### Heights, discriminants and conductors

In this talk we consider the problem of giving explicit inequalities which relate heights, discriminants and conductors of a curve defined over a number field. We present such inequalities for some curves, including all curves of genus one or two and we discuss Diophantine applications.

##### Vanishing/Blow up at infinity for the 3-d critical focussing NLW

##### On how the first term of an arithmetic progression can influence the distribution of an arithmetic sequence

In this talk we will show that many arithmetic sequences have asymmetries in their distribution amongst the progressions mod q.

##### Graph Gauge Theory and Vector Diffusion Maps

We consider a generalization of graph Laplacian which acts on the space of functions which assign to each vertex a point in $d$-dimensional space. The eigenvalues of such connection Laplacian are useful for examining vibrational spectra of molecules as well as vector diffusion maps for analyzing high dimensional data.

##### The Hodge theorem as a derived self-intersection

The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic.

##### Nonintersecting Random Walkers with a Staircase Initial Condition

We study a model of one dimensional particles, performing geometrically weighted random walks that are conditioned not to intersect. The walkers start at equidistant points and end at consecutive integers. A naturally associated tiling model can be viewed as one of placing boxes on a staircase.

##### Congruent Numbers and Heegner Points

An anonymous Arab manuscript, written before 972, contains a `` problem of congruent numbers": given an integer $n$, to find a rational square $x^2$ such that $x^2+n and x^2-n$ are both rational squares. For example 1, 2, 3 are not congruent numbers but 5, 6, 7 are. A modern equivalence of this problem is to find a point with infinite order on the elliptic curve: $y^2=x^4-n^2$.

##### Sup-norms, Whittaker Periods and Hypergeometric Sums

We begin with a survey of recent results on the problem of bounding the sup-norm of automorphic forms. If f is a cuspidal automorphic forms on a reductive group G it is classical to study its value distribution and in the particular the maximum of |f(g)| for all g in G. Then we will explain an approach to this problem via Whittaker periods.

##### New Classes of Tournaments Satisfying the Erdos-Hajnal Conjecture

The Erdos-Hajnal conjecture states that for every graph $H$ there exists a constant $c>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|G|^c$. The conjecture is still open. However some time ago a version for tournaments was proven to be equivalent to the original.

##### Piecewise Laurent Polynomials and (Operational) Equivariant K-theory of Toric Varieties

For a smooth compact toric variety $X$, results of Bifet-de Concini-Procesi and Brion show that the equivariant cohomology of $X$ is identified with the ring of piecewise polynomials on the associated fan. In 2006, Payne extended this to arbitrary toric varieties, identifying the ring of piecewise polynomials with the operational equivariant Chow cohomology of $X$.

##### H-Projective Geometry on Compact Kähler Manifolds

The basic geometric structure in h-projective geometry is the family of h-planar curves, associated to a given Kähler metric. Such curves can be seen as generalisations of geodesics on Kähler manifolds. In this context, one problem of interest is the investigation of Kähler manifolds admitting another Kähler metric having the same h-planar curves as the given one.

##### Description of the blow-up for the semi-linear wave equation

We study the blow-up curve of a (blow-up) solution to the semi-linear wave equation in 1D with power nonlinearity: $u_tt - u_xx = |u|^{p-1} u$. The blow-up curve is a priori 1-Lispchitz. On this curve, we distinguish (geometrically) characteristic points and non-characteristic points.

##### Computability and Complexity of Julia Sets

Studying dynamical systems is key to understanding a wide range of phenomena ranging from planetary movement to climate patterns to market dynamics. Various computational and numerical tools have been developed to address specific questions about dynamical systems, such as predicting the weather or planning the trajectory of a satellite.

##### The Stacks Project

The stacks project is a long term open source, collaborative project documenting and developing theory on algebraic stacks. I will spend a bit of time talking about what it is, who it is for, what its goals are and how it is supposed to work. More information can be found here The Stacks Project

##### How to Raise Harmonic Families?

The talk will present a step towards answering the delicate question in the title! In the colloqium lunch I describe the work of Montgomery on the pair correlation of zeros of the Riemann zeta function. The result is beautifully connected with the eigenvalue statistics of random matrices studied by Gaudin, Mehta and Dyson.

##### The Traveling Salesman Problem: A Blueprint for Optimization

Given a list of cities along with the cost of travel between each pair of them, the traveling salesman problem is to find the cheapest way to visit them all and return to your starting point. Easy to state, but

##### Cohomological Rigidity Problems in Toric Topology

As is well-known, cohomology ring does not distinguish closed smooth manifolds up to diffeomorphism or homeomorphism in general. However, it does if we restrict our attention to a reasonably small class of objects. For instance, it is known that simply connected closed smooth 4-manifolds are classified up to homeomorphism using their integral cohomology rings.

##### Arithmetic Fake Compact Hermitian Symmetric Spaces

A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David Mumford in 1978 using p-adic uniformization.

##### Representation Theory and Homological Stability

Homological stability is the remarkable phenomenon where for certain sequences $X_n$ of groups or spaces -- for example $SL(n,Z)$, the braid group $B_n$, or the moduli space $M_n$ of genus $n$ curves -- it turns out that the homology groups $H_i(X_n)$ do not depend on n once n is large enough.