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This paper is concerned with the classical motion of a string in global AdS3. The initially static string stretches between two antipodal points on the boundary circle. Both endpoints are perturbed which creates cusps at a steady rate. The cusps propagate towards the interior where they collide. The behavior of the string depends on the strength of forcing. Three qualitatively different phases can be distinguished: transparent, gray, and black. The transparent region is analogous to a standing wave. In the black phase, there is a horizon on the worldsheet and cusps never reach the other endpoint. The string keeps folding and its length grows linearly over time. In the gray phase, the string still grows linearly. However, cusps do cross to the other side. The transparent and gray regions are separated by a transition point where a logarithmic accumulation of cusps is numerically observed. A simple model reproduces the qualitative behavior of the string in the three phases.

We consider the mixed q-Gaussian algebras introduced by Speicher which are generated by the variables Xi = li + l ∗ i , i = 1, . . . , N, where l ∗ i lj − qij lj l ∗ i = δi,j and −1 < qij = qji < 1. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed q-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra L(FN ), provided that maxi,j |qij | is small enough. Similar results hold in the reduced C ∗ -algebra setting. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case qij ≡ q.

We study novel variations of Voronoi games (and associated random processes) that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios in a geometric setting. Our work can be seen as a natural variation of the classical framework of Hotelling. While games based on Voronoi diagrams have appeared recently in the literature, our variations appear to be the first that take into account a notion of limited choices and the corresponding asymmetries in player strategies. As a problem example, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points in the unit square (or, for symmetry, the unit torus), so the kth miner will have associated points pk1, pk2, . . . , pkm. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Generally the points will be distinct for each miner. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player’s goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games? We provide several results on games in this setting. For example, we show that a correlated equilibrium can be found in polynomial time for the natural 1, 2, and 3-dimensional variations of the problem. For the one-dimensional version of the problem on the unit circle, we show that a pure Nash equilibrium exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists when each player owns the arc starting from their point clockwise to the next point. We also consider a random version of the game, where players have m points chosen uniformly at random. We derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game in this setting using interesting properties of random arc lengths on circles.

We use Coulomb branch indices of Argyres-Douglas theories on S1×L(k,1) to quantize moduli spaces H of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of H, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in H under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform STkS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.