The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a fundamental
question to understand the singularities which arise in the reductions
modulo $p$ of integral models of Shimura varieties. For PEL type
Shimura varieties with parahoric level structure at $p$, this question
has been studied since the 1990's. Due to the recent construction of
We present recent advances in constructions of globally consistent
F-theory compactifications with the exact chiral spectrum of the minimal
supersymmetric Standard Model. We highlight the first such example and
then turn to a subsequent systematic exploration of the landscape of
F-theory three-family Standard Models with a gauge coupling unification.
Employing algebraic geometry techniques, all global consistency
conditions of these models can be reduced to a single criterion on the
Anomalies are invariants under renormalization group flow which lead to powerful constraints on the phases of quantum field theories. I will explain how these ideas can be generalized to families of theories labelled by coupling constants like the theta angle in gauge theory. Using these ideas we will be able to prove that certain systems, such as Yang-Mills theory in 4d, necessarily have a phase transition as these parameters are varied. We will also show how to use the same ideas to constrain the dynamics of defects where coupling constants vary in spacetime.