Princeton

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.

This talk will report on the definition of some motivic cohomology classes and the proof that they satisfy the norm relations expected of Euler systems, emphasizing a connection with the local Gan-Gross-Prasad conjecture.

A "hole" in a graph is an induced subgraph which is a cycle of length > 3. The perfect graph theorem says that if a graph has no odd holes and no odd antiholes (the complement of a hole), then its chromatic number equals its clique number; but unrestricted graphs can have clique number two and arbitrarily large chromatic number. There is a nice question half-way between them - for which classes of graphs is it true that a bound on clique number implies some (larger) bound on chromatic number? Call this being "chi-bounded".

Gyarfas proposed several conjectures of this form in 1985, and recently there has been significant progress on them. For instance, he conjectured

• graphs with no odd hole are chi-bounded (this is true);
• graphs with no hole of length >100 are chi-bounded (this is true);
• graphs with no odd hole of length >100 are chi-bounded; this is still open but true for triangle-free graphs.

We survey this and several related results. This is joint with Alex Scott and partly with Maria Chudnovsky.

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the absolute values of the flows along it is at most one. We may also require the flows to be 1/2-integral, or indeed 1/p-integral for some fixed p.

If the problem is feasible (that is, the desired flows exist) then it is still feasible after contracting any edge, so let us say a flow problem is ``critical'' if it is infeasible, but becomes feasible when we contract any edge. In many special cases, all critical instances have only two vertices, but if we ask for integral flows (that is, p = 1, essentially the edge-disjoint paths problem), then there arbitrarily large critical instances, even with k = 2. But it turns out that p = 1 is the only bad case; if p>1 then all critical instances have bounded size (depending on k, but independent of p), and the same is true if there is no integrality requirement at all.

The proof gives rise to a very simple algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs.

I will discuss some p-adic (and mod p) criteria ensuring that an elliptic curve over the rationals has algebraic and analytic rank one, as well as some applications.

Composite membranes comprised of an ultra-thin coating film formed over a porous support membrane are the basis for state-of-the-art reverse osmosis (RO) and nanofiltration (NF) membranes, offering the possibility to independently optimize the support membrane and the coating film. However, limited information exists on transport through such composite membrane structures. Numerical calculations have been carried out in order to probe the impacts of the support membrane skin-layer pore size and porosity, support membrane bulk micro-porosity, and coating film thickness and morphology (i.e. surface roughness) on solvent and solute transport through composite membranes. Results suggest that the flux and rejection of a composite membrane may be fine-tuned, by adjusting support membrane skin layer porosity and pore size, independent of the properties of the coating film. Further, the water flux over the membrane surface is unevenly distributed, creating local ‘hot spots’ of high flux that may govern initial stages of membrane fouling and scaling. The analysis provides important insight on how the non-trivial interaction of support properties and film roughness may result in widely varying transport properties of the composite structure. In particular, the simulations reveal inherent trade-offs between flux, rejection and fouling propensity (the latter due to ‘hot spots’), which are purely consequences of geometrical factors, irrespective of materials chemistry.

Abstract: In this talk, I will discuss the proportionality between tree amplitudes and the ultraviolet divergences in their one-loop corrections in Yang-Mills and (N < 4) Super Yang-Mills theories in four-dimensions. From the point of view of local perturbative quantum field theory, i.e. Feynman diagrams, this proportionality is straightforward: ultraviolet divergences at loop-level are absorbed into coefficients of local operators/interaction vertices in the original tree-amplitude. Ultraviolet divergences in loop amplitudes are also calculable through on-shell methods. These methods ensure manifest gauge-invariance, even at loop-level (no ghosts), at the expense of manifest locality. From an on-shell perspective, the proportionality between the ultraviolet divergences the tree amplitudes is thus not guaranteed. I describe systematic structures which ensure proportionality, and their possible connections to other recent developments in the field.