Seminars and Colloquia by Series

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, WangSchool 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 FriezeCarnegie 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 LlaveGeorgia 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.

Duality in Convex Algebraic Geometry

Series
Algebra Seminar
Time
Monday, September 21, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rainer SinnGeorgia Tech
Duality is an important feature in convexity and in projective algebraic geometry. We will discuss the interplay of these two dualities for the cone of sums of squares of ternary forms and its dual cone, the Hankel spectrahedron.

A Birman-Hilden theorem for free groups

Series
Geometry Topology Seminar
Time
Monday, September 21, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Neil FullartonRice University
The Birman-Hilden theorem relates the mapping class groups of two orientable surfaces S and X, given a regular branched covering map p from S to X. Explicitly, it provides an isomorphism between the group of mapping classes of S that have p-equivariant representatives (mod the deck group of the covering map), and the group of mapping classes of X that have representatives that lift to homeomorphisms of S. We will translate these notions into the realm of automorphisms of free group, and prove that an obvious analogue of the Birman-Hilden theorem holds there. To indicate the proof of this, we shall explore in detail several key examples, and we shall describe some group-theoretic applications of the theorem. This is joint work with Rebecca Winarski, John Calabrese, and Tyrone Ghaswala

A geometric mechanism for Arnold diffusion in the a priori stable case

Series
CDSNS Colloquium
Time
Monday, September 21, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marian GideaYeshiva University
We prove the existence of diffusion orbits drifting along heteroclinic chains of normally hyperbolic 3-dimensional cylinders, under suitable assumptions on the dynamics on the cylinders and on their homoclinic/heteroclinic connections. These assumptions are satisfied in the a priori stable case of the Arnold diffusion problem. We provide a geometric argument that extends Birkhoff's procedure for constructing connecting orbits inside a zone of instability for a twist map on the annuls. This is joint work with J.-P. Marco.

The Symmetric Rendezvous Problem: Codes and Lower Bounds

Series
Combinatorics Seminar
Time
Friday, September 18, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom HayesThe University of New Mexico
In the Rendezvous problem on the complete graph, two parties are trying to meet at some vertex at the same time, despite starting out with independent random labelings of the vertices. It is well known that the optimal strategy is for one player to wait at any vertex, while the other visits all n vertices in consecutive steps, which guarantees a rendezvous within n steps and takes (n + 1)/2 steps on average. This strategy is very far from being symmetric, however. E. Anderson and R. Weber presented a symmetric algorithm that achieves an expected meeting time of 0.829n, which has been conjectured to be optimal in the symmetric setting. We change perspective slightly: instead of trying to minimize the expected meeting time, we try to maximize the probability of successfully meeting within a specified number of timesteps. In this setting, for all time horizons that are linear in n, the Anderson-Weber strategy has a constant probability of failure. Surprisingly, we show that this is not optimal: there exists a different symmetric strategy that almost surely guarantees meeting within 4n timesteps. This bound is tight, in that the factor 4 cannot be replaced by any smaller constant. Our strategy depends on the construction of a new kind of combinatorial object that we dub”rendezvous code.”On the positive side, for T < n, we show that the probability of meeting within T steps is indeed (approximately) maximized by the Anderson-Weber strategy. Our results imply new lower bounds on the expected meeting time for any symmetric strategy, which establishes an asymptotic difference between the best symmetric and asymmetric strategies. Finally, we examine the symmetric rendezvous problem on other vertex-transitive graphs.

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