Inspired by empirical observations on honey bee colonies, we analyze the emergence of task differentiation in a model complex system, characterized by an absence of hierarchical control, yet able to exhibit coordinated behavior and collective function. The analysis considers the steady-state response of a mechanical interaction network to exogenous resonant excitation.

Time discounting is a ubiquitous assumption in economic literature. We (re)explore the foundations of such time preferences. "Impatience" is defined as a preferences for experiencing the better states sooner rather than later, even when there is no uncertainty associated with the future. We show that, assuming consistency and some weak stationarity assumptions, impatience is incompatible with a meaningful notion of a risk-attitude (risk aversion/love/neutrality).On the other hand, if there is uncertainty associated with the future then discounting necessarily emerges.

A unified integral approach to all the calibration results in the literature -- from regular probabilistic calibration to smooth deterministic calibration -- using simple "hairy" fixed point and minimax results.

Abstract: Given two permutations A and B which "almost" commute, are they "close" to permutations A' and B' which really commute? This can be seen as a question about a property the equation XY=YX. Studying analogous problems for more general equations (or sets of equations) leads to the notion of "locally testable groups" (aka "stable groups").

G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse). A famous conjecture by Birkhoff claims that ellipses are the only domains with this

A recent result of Lubetzky and Peres showed that the random walk on a $q+1$-regular Ramanujan graph has $L^{1}$-cutoff, and that its “almost-diameter” is optimal. Similar optimal results were proven by other authors in various contexts, e.g. Parzanchevski-Sarnak for Golden Gates and Ghosh-Gorodnik-Nevo for Diophantine approximations. Those results rely in general on a naive Ramanujan conjecture, which is either very hard, unknown, or even false in some situations. We show that a general version of those results can be proven using the density hypothesis of Sarnak-Xue.

Title: Proving conformal invariance using discrete holomorphicity Abstract:

Billiards in polygons can exhibit some bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry (and in particular Hodge theory), Teichmuller theory and ergodic theory on homogeneous spaces. I will attempt to give a gentle introduction to the subject. A large part of this talk will be accessible to undergraduates.

The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate for a very wide class of transcendental sets. This result plays a key role in Pila-Zannier's proof of the Manin-Mumford conjecture, Pila's proof of the Andre-Oort conjecture for modular curves, Masser-Zannier's work on torsion anomalous points in elliptic families, and many more recent developments.

A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. We will consider the following variant of this theme: suppose a graph G is far (in some well defined sense) from satisfying property P. Must G contain a small proof of this fact? We will show that for many natural graph properties the answer is Yes. In particular, we will show that the answer is Yes whenever P is a semi-algebraic graph property, thus conforming a conjecture of Alon. Joint work with L. Gishboliner

A measure on the plane is called self-affine if it is stationary with respect to a finitely supported measure on the affine group of R^2. Under certain randomization, it is known that the Hausdorff dimension of these measures is almost surely equal to the Lyapunov dimension, which is a quantity defined in terms of the linear parts of the affine maps. I will present a result which provides conditions for equality between these two dimensions, and connects the theory of random matrix products with the dimension of self-affine measures.