Seminars and Colloquia by Series

Sampling on lattices with free boundary conditions using randomized extensions

Series
ACO Student Seminar
Time
Friday, September 25, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sarah Cannon – Georgia Institute of Technology
Many statistical physics models are defined on an infinite lattice by taking appropriate limits of the model on finite lattice regions. A key consideration is which boundary to use when taking these limits, since the boundary can have significant influence on properties of the limit. Fixed boundary conditions assume that the boundary cells are given a fixed assignment, and free boundary conditions allow these cells to vary, taking the union of all possible fixed boundaries. It is known that these two boundary conditions can cause significant differences in physical properties, such as whether there is a phase transition, as well as computational properties, including whether local Markov chain algorithms used to sample and approximately count are efficient. We consider configurations with free or partially free boundary conditions and show that by randomly extending the boundary by a few layers, choosing among only a constant number of allowable extensions, we can generalize the arguments used in the fixed boundary setting to infer bounds on the mixing time for free boundaries. We demonstrate this principled approach using randomized extensions for 3-colorings of regions of Z2 and lozenge tilings of regions of the triangle lattice, building on arguments for the fixed boundary cases due to Luby et.al. Our approach yields an efficient algorithm for sampling free boundary 3-colorings of regions with one reflex corner, the first result to efficiently sample free boundary 3-colorings of any nonconvex region. We also consider self-reducibility of free boundary 3-colorings of rectangles, and show our algorithm can be used to approximately count the number of free-boundary 3-colorings of a rectangle.

Critical exponents in the Abelian sandpile

Series
Stochastics Seminar
Time
Thursday, September 24, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jack Hanson – School of Mathematics, Georgia Tech and CUNY
The Abelian sandpile was invented as a "self-organized critical" model whose stationary behavior is similar to that of a classical statistical mechanical system at a critical point. On the d-dimensional lattice, many variables measuring correlations in the sandpile are expected to exhibit power-law decay. Among these are various measures of the size of an avalanche when a grain is added at stationarity: the probability that a particular site topples in an avalanche, the diameter of an avalanche, and the number of sites toppled in an avalanche. Various predictions about these exist in the physics literature, but relatively little is known rigorously. We provide some power-law upper and lower bounds for these avalanche size variables and a new approach to the question of stabilizability in two dimensions.

Cyclic polynomials in two variables

Series
Analysis Seminar
Time
Wednesday, September 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alan Sola – University of South Florida
In my talk, I will discuss coordinate shifts acting on Dirichlet spaces on the bidisk and the problem of finding cyclic vectors for these operators. For polynomials in two complex variables, I will describe a complete characterization given in terms of size and nature of zero sets in the distinguished boundary.

Spatial epidemic models: lattice differential equation analysis of wave behavior

Series
Research Horizons Seminar
Time
Wednesday, September 23, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chi-Jen, Wang – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar.

Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We analyze this stochastic model by applying truncation approximations to the exact master equations describing the evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis.

Probabilistic analysis of some combinatorial optimization problems

Series
Joint ACO and ARC Colloquium
Time
Wednesday, September 23, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Klaus 1116 E
Speaker
Alan Frieze – Carnegie Mellon University
We consider the following probabilistic model. The edges of a (complete) graph have unknown random edge weights. We want to build a minimum cost structure. We can ask for the weight of an edge and then accept or reject the edge. Once rejected, the edge cannot be accepted later. We must accept enough edges to support a structure and we are charged for all the edges accepted, even if not used. We give results in this model for minimum spanning tree, perfect matching and shortest path. Joint work with Colin Cooper and Wesley Pegden.

Whitney differentiability in KAM theory

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 22, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Rafael de la Llave – Georgia Institute of Technology
We will review the notion of Whitney differentiability and the Whitney embedding theorem. Then, we will also review its applications in KAM theory.

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