Ioana Dumitriu (University of California, San Diego) (Two talks)

                                                                                Talk 1: Spectra of Random Graphs I: Shape and Connection With Random Matrices  

Abstract: The spectral study of random graphs is a well-established area at the 
intersection of random matrix theory and spectral graph theory, with applications 
ranging from machine learning to coding theory and signal processing. We will 
review several models of random graphs and survey what is known about their 
limiting spectral shapes, comparing and contrasting with similar results from 
random matrix theory, both in (relatively) dense and in sparse cases.
 

         Talk 2: Spectra of Random Graphs II: Outliers

Abstract: Outliers in the spectra of random graphs are crucial to many machine learning 
applications (clustering, matrix completion, etc.) and, especially in sparse and 
inhomogeneous cases, their existence is much harder to determine than in random 
matrix theory (where we have the famous Bai-Yin 4th moment theorem). Methods of 
approach include a generalized method of high moments, often in conjunction with 
the very promising non-backtracking operator. We will explain the basics of these 
methods, and show how they work in the case of a few graph models.
    

Robin Pemantle (University of Pennsylvania) (Two talks)

      Talk 1: A Review of Negative Dependence
    

Abstract: This talk reviews 50-60 years of the theory of negative dependence of binary
random variables, beginning with origins in mathematical statistics and statistical 
mechanics. The high point is the Borcea-Branden-Liggett theory,connecting negative 
dependence to the geometry of zero sets of polynomials, and providing a long sought 
after checkable condition with strong consequences such as negative association.  
I will end with some more recent (last ten years) work of various people, including 
concentration inequalities, Lorentzian measures, and a CLT based on the geometry of 
zeros.  The talk will end with some open problems.

      Talk 2: An Active Learning Approach to Graduate Applied Probability 

Abstract: The demand for a graduate level probability course far exceeds the population 
of Math PhD students whoplan to do research in probability theory.  I will begin by 
describing a broad population interested in and n and qualified to take a measure theoretic 
course in probability.The next question is how the needs of most of these students 
compare to the course(s) we usually teach.  This has implications for design of 
curriculum and for pedagogy. I will describe one possible solution, with many examples 
from a recent course.  Finally, I will discuss assessment, the use of final projects, 
and how the demands on instructor time compare to those of a standard course.  All examples
are drawn from a freely available set of materials.

Tom Hutchcroft (California Institute of Technology)	

	Percolation on Finite Transitive Graphs

Abstract: cI will describe recent work developing a general theory of percolation on arbitrary 
 finite transitive graphs, summarising our progress on the basic questions: When is there a phase 
transition for the emergence of a giant cluster? When is the giant cluster unique? How does this 
relate to percolation on infinite graphs? While the answers to these questions are classical in 
several well-studied examples such as tori, complete graphs, hyercubes, expanders, and so on, 
we will see that a surprisingly general and complete answer to each of these questions is possible 
starting only with the assumption of transitivity. Joint work with Philip Easo.

 
Xin Sun (University of Pennsylvania)

   Random Modulus of the Brownian Annulus and the Ghost CFT

Abstract: Brownian surfaces are the scaling limits of uniformly sampled random planar maps. 
Under conformal embedding, their geometry is described by pure Liouville quantum gravity. 
This connection is well understood when the underlying topological surface is a sphere or 
a disk. When the surface is not simply connected, the corresponding Brownian surface has a 
random conformal structure (i.e. modulus). According to Polyakov's bosonic string theory, 
it was conjectured that the law of the random modulus is described by the partition function 
of a -26 dimensional conformal field theory (CFT) called the ghost CFT. We will review this 
conjecture and report the recent proof in the annulus case. Our proof also works for 2D 
quantum gravity coupled with conformal matters, which allows us to compute modular-dependent 
observables for SLE curves on an annulus, such as the continuum limit of the annulus crossing 
probability for critical percolation conjectured by Cardy.  Based on a joint work with Ang, 
Remy and a work in progress with Xu and Zhuang.

Dapeng Zhan (Michigan State University)

       Boundary Green's function and Minkowski content for SLE$_\kappa(\rho)$