It is well known that the theory of differential-difference fields in characteristic zero has a model companion. Here by a differential-difference field I mean a field with a differential and a difference structure where the operators commute (in other words the difference structure is a differential-endomorphism). The theory DCFA_0 was studied in a series of papers by Bustamante. In this talk I will address the case of positive characteristic.

There is a mystery at the heart of operator learning: how can one recover a non-self-adjoint operator from data without probing the adjoint? Current practical approaches suggest that one can accurately recover an operator while only using data generated by the forward action of the operator without access to the adjoint. However, naively, it seems essential to sample the action of the adjoint for learning time-dependent PDEs. 

In this talk, we will first explore connections with low-rank matrix recovery problems in numerical linear algebra. Then, we will show that one can approximate a family of non-self-adjoint infinite-dimensional compact operators via projection onto a Fourier basis without querying the adjoint.

Realizing chiral global symmetries on a finite lattice is a long-standing challenge in lattice gauge theory, with potential implications for non-perturbative regularization of the Standard Model. One of the simplest examples of such a symmetry is the axial U(1) symmetry of the 1+1d massless Dirac fermion field theory: it acts by equal and opposite phase rotations on the left- and right-moving Weyl components of the Dirac field. This field theory also has a vector U(1) symmetry which acts identically on left- and right-movers. The two U(1) symmetries exhibit a mixed anomaly, known as the chiral anomaly. In this talk, we will discuss how both symmetries are realized as ordinary U(1) symmetries of an "ultra-local" lattice Hamiltonian, on a finite-dimensional Hilbert space. Intriguingly, the anomaly of the Abelian U(1) symmetries in the infrared (IR) field theory is matched on the lattice by a non-Abelian Lie algebra. The lattice symmetry forces the low-energy phase to be gapless, closely paralleling the effects of the anomaly in the field theory.