A review of the current state of multivariate public-key cryptosystems compares and contrasts the most promising multivariate schemes in digital signatures and public-key encryption as well as their security.

Free kicks play an important role in a modern soccer match. A skillful soccer player usually uses the opportunity of a free kick to score a goal. There are mainly two ways to give a free kick, falling ball and banana ball. The last thesis analyzes the path of falling soccer in two-dimensional plane. This thesis mainly focuses on using geometrical ways to generate a graph in three-dimensional coordinate system to analyze the path of banana ball and find out the most suitable distance to give a free kick.

We explore the higher-form generalized global symmetries and higher anomalies based on a generalized cobordism theory. Our cobordism calculation guides us to classify higher anomalies and topological terms for Yang-Mills (YM) gauge theories and sigma models in various dimensions. Some of YM gauge theories can be obtained from dynamically gauging the SU(N) time-reversal symmetric cobordism invariants (SU(N)-generalized topological superconductors/insulators [arXiv:1711.11587]). We elaborate the cases of YM theory with a compact Lie gauge group (such as SU(N) with N=2, 3 and others) particularly in a 4d spacetime. We provide the relevant homotopy and cobordism group calculations of higher classifying spaces, based on mathematical tools of algebraic topology, to support the physics stories.

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The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.