丁香园AV

Maxwell鈥檚 equations successfully model electromagnetic fields, and their mathematical study presents fascinating challenges. During the summer of 2023, Grayson studied a decades-old open question: is it al- ways possible for a bounded region to have electromagnetic resonances in which certain boundary components of non-zero electric and magnetic fields are proportional to each other? Does this question even permit a well-posed mathematical formulation? Grayson carefully investigated this truly open-ended problem. He then designed a structure-preserving discretization to find approximate solutions and then visualize them. His results have motivated a program of research involving several researchers, and we anticipate a publication by the end of this academic year.

Quantum computers cannot be built until the problem of how to protect extremely sensitive quantum data from environmental noise is solved.  To this end, quantum versions of error-correcting codes are used; the task of manipulating encoded data using imperfect hardware is the subject of quantum fault-tolerance.  The overhead added by these methods renders fault-tolerant quantum computation using near-term technology infeasible.  Thus, theoretical developments to improve efficiency are urgently needed.  Oscar unified two conjectures made in 2019 concerning families of quantum logic gates that can be performed with significant efficiency gains over known methods.  These families are indexed by natural numbers (n,k,d) with n = 1 or 2, k > 2, and d an odd prime.  This conjecture had been previously established for (n, 3, d).  Under his supervisor's guidance, Oscar developed novel algebraic techniques and established the conjecture for (1, k, 3), going beyond k = 3 for the first time.  The (1, k, d) case is nearly fully established and its extension to the complete (n, k, d) conjecture is in progress.