Georgia Institute of Technology College of Sciences

We will look at a system of hamiltonian equations on the torus, with an initial condition in momentum and a terminal condition in position, that arises in mean field game theory. Existence of and uniqueness of solutions will be shown, and a few remarks will be made in regard to its connection to the minimization problem of a cost functional. This is the second part of lasrt week's talk.

In a self-organizing particle system, an abstraction of programmable matter, simple computational elements called particles with limited memory and communication self-organize to solve system-wide problems of movement, coordination, and configuration. In this paper, we consider stochastic, distributed, local, asynchronous algorithms for 'shortcut bridging', in which particles self-assemble bridges over gaps that simultaneously balance minimizing the length and cost of the bridge. Army ants of the genus Eticon have been observed exhibiting a similar behavior in their foraging trails, dynamically adjusting their bridges to satisfy an efficiency tradeoff using local interactions. Using techniques from Markov chain analysis, we rigorously analyze our algorithm, show it achieves a near-optimal balance between the competing factors of path length and bridge cost, and prove that it exhibits a dependence on the angle of the gap being 'shortcut' similar to that of the ant bridges. We also present simulation results that qualitatively compare our algorithm with the army ant bridging behavior. Our work presents a plausible explanation of how convergence to globally optimal configurations can be achieved via local interactions by simple organisms (e.g., ants) with some limited computational power and access to random bits. The proposed algorithm demonstrates the robustness of the stochastic approach to algorithms for programmable matter, as it is a surprisingly simple extension of a stochastic algorithm for compression. This is joint work between myself/my professor Andrea Richa at ASU and Sarah Cannon and Prof. Dana Randall here at GaTech.

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will describe the structure of G when (G, a0, a1, a2, b1, b2) is infeasible, using frames and connectors. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

The study of nonconventional sums $S_{N}=\sum_{n=1}^{N}F(X(n),X(2n),\dots,X(\ell n))$, where $X(n)=g \circ T^n$ for a measure preserving transformation $T$, has a 40 years history after Furstenberg showed that they are related to the ergodic theory proof of Szemeredi's theorem about arithmetic progressions in the sets of integers of positive density. Recently, it turned out that various limit theorems of probabilty theory can be successfully studied for sums $S_{N}$ when $X(n), n=1,2,\dots$ are weakly dependent random variables. I will talk about a more general situation of nonconventional arrays of the form $S_{N}=\sum_{n=1}^{N}F(X(p_{1}n+q_{1}N),X(p_{2}n+q_{2}N),\dots,X(p_{\ell}n+q_{\ell}N))$ and how this is related to an extended version of Szemeredi's theorem. I'll discuss also ergodic and limit theorems for such and more general nonconventional arrays.