Georgia Institute of Technology College of Sciences

First, we introduce a new field theoretical interpretation of quantum mechanical wave functions, by postulating that the wave function is the common wave function for all particles in the same class determined by the external potential V, of the modulus of the wave function represents the distribution density of the particles, and the gradient of phase of the wave function provides the velocity field of the particles. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. Third, we derive the field and topological phase transition equations for condensates, and make connections to the quantum phase transition, as a topological phase transition. This is joint work with Tian Ma.

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Minimum s-t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization.

Here, we are given a graph with edges labeled + or - and the goal is to produce a clustering that agrees with the labels as much as possible: + edges within clusters and - edges across clusters.

The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective, e.g., minimizing the total number of disagreements or maximizing the total number of agreements.

We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective.

This naturally gives rise to a family of basic min-max graph cut problems.

A prototypical representative is Min-Max s-t Cut: find an s-t cut minimizing the largest number of cut edges incident on any node.

In this talk we will give a short introduction of Correlation Clustering and discuss the following results:

1. an O(\sqrt{n})-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems)
2. a remarkably simple 7-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 48)
3. a (1/(2+epsilon))-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the (1/(4+epsilon))-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.

Joint work with Moses Charikar and Neha Gupta.

Linear Schur multipliers, which act on matrices by entrywisemultiplications, as well as their generalizations have been studiedfor over a century and successfully applied in perturbation theory (asdemonstrated in the previous talk). In this talk, we will discussestimates for finite dimensional multilinear Schur multipliersunderlying these applications.