PhD in Mathematics - Logic and Foundations in Irvine United States | University of California Irvine

The Mathematics Department faculty at UC Irvine are distinguished scholars dedicated to teaching, conducting groundbreaking research in both theoretical and practical mathematics, and serving the community through STEM outreach initiatives. The department comprises 34 tenured faculty members along with 4 permanent adjunct instructors, supported by an equal number of postdocs, lecturers, and visiting professors who contribute significantly to both instruction and research. Established in 1965, UC Irvine merges top-tier academic research with Southern California's ideal setting, featuring excellent climate and a vibrant, diverse campus. Our programs equip undergraduate and graduate students with professional expertise in teaching, mathematical research, or applied fields. We also fulfill the mathematical course requirements for all UCI students, serving approximately 110 graduate students, more than 615 undergraduate majors, and many others pursuing minors or specialized tracks.

Upon admission to the Ph.D. program, students begin a structured curriculum of coursework, seminars, and independent study to prepare for written qualifying exams, candidacy oral examinations, and dissertation work. First-year students must pass at least two M.S.-level exams in Real Analysis, Complex Analysis, or Algebra by their second year. By year three, they must pass two Ph.D.-level exams in these subjects. Students can fulfill these requirements through either Core Assessment Exams (held each spring) or Qualifying Exams (administered before fall quarter). Each subject exam may be attempted no more than three times. Passing a Qualifying Exam before taking the related course exempts students from that course requirement.

UC Irvine's Logic and Foundations of Mathematics research group specializes in set theory and model theory. Set theory research concentrates on forcing techniques, large cardinals, inner model theory, fine structure, cardinal combinatorics, and descriptive set theory - particularly its intersections with ergodic theory and connections to determinacy and large cardinals. Model theory research emphasizes practical applications across mathematics, frequently employing nonstandard analysis, along with specialized work on metric structures like operator algebras.