Georgia Institute of Technology College of Sciences

An r-unform n-vertex hypergraph H is said to have the Manickam-Miklos-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this talk I will show that for n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. An immediate corollary of this result is the vector space Manickam-Miklos-Singhi conjecture which states that for n>=4k and any weighting on the 1-dimensional subspaces of F_q^n with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. I will also discuss two additional generalizations, which can be regarded as analogues of the Erdos-Ko-Rado theorem on k-intersecting families. This is joint work with Benny Sudakov.

I introduce a class of dynamical systems which exhibit motion in their lowest-energy states and thus spontaneously break time-translation symmetry. Their Lagrangians have nonstandard kinetic terms and their Hamiltonians are multivalued functions of momentum, yet they are perfectly consistent and amenable to quantization. Possible applications to condensed matter systems and cosmology will be discussed.

A comprehensive mathematical model of β1-adrenergic signaling system for mouse ventricular myocytes is developed. The model myocyte consists of three major compartments (caveolae, extracaveolae, and cytosol) and includes several modules that describe biochemical reactions and electrical activity upon the activation of β1-adrenergic receptors. In the model, β1-adrenergic receptors are stimulated by an agonist isoproterenol, which leads to activation of Gs-protein signaling pathway to a different degree in different compartments. Gs-protein, in turn, activates adenylyl cyclases to produce cyclic AMP and to activate protein kinase A. Catalytic subunit of protein kinase A phosphorylates cardiac ion channels and intracellular proteins that regulate Ca2+ dynamics. Phosphorylation is removed by the protein phosphatases 1 and 2A. The model is extensively verified by the experimental data on β1-adrenergic regulation of cardiac function. It reproduces time behavior of a number of biochemical reactions and voltage-clamp data on ionic currents in mouse ventricular myocytes; β1-adrenergic regulation of the action potential and intracellular Ca2+ transients; and calcium and sodium fluxes during action potentials. The model also elucidates the mechanism of action potential prolongation and increase in intracellular Ca2+ transients upon stimulation of β1-adrenergic receptors.

Given f: \C \times T^1 to itself, an analytic perturbation of a fibered rotation map , we will present two proofs of existence of an analytic conjugation of f to the fibered rotation , on a neighborhood of {0} \times T^1. This talk will be self- contained except for some usual "tricks" from KAM theory and which will be explained better in another talk. In the talk we will discuss carefully the number theoretic conditions on the fibered rotation needed to obtain the theorem.