Brian Conrad

Brian Conrad
Name	Brian Conrad
Nationality	American
Fields	Number theory, Arithmetic geometry, Algebraic number theory
Workplaces	Columbia University, Princeton University, Harvard University
Alma mater	Harvard University, Princeton University
Doctoral advisor	Andrew Wiles
Known for	Proofs concerning the modularity of Galois representations, work on Fermat's Last Theorem extensions

Brian Conrad is an American mathematician known for his contributions to number theory, arithmetic geometry, and the study of Galois representations. He has made influential advances related to modularity theorems, p-adic Hodge theory, and the interplay between automorphic forms and elliptic curves. Conrad's work connects strands from the traditions of Andrew Wiles, Jean-Pierre Serre, and the Langlands program to modern developments in algebraic geometry and Iwasawa theory.

Early life and education

Conrad completed his undergraduate studies at Harvard University before pursuing graduate work at Princeton University under the supervision of Andrew Wiles. During his doctoral studies at Princeton, he engaged with problems linked to the proof of Fermat's Last Theorem and the broader context of modularity lifting theorems. His formative years brought him into contact with prominent mathematicians and institutions such as Harvard University, Institute for Advanced Study, and collaborators connected to the Mazur–Wiles circle.

Academic career

Conrad has held academic positions at prestigious institutions including Princeton University, Harvard University, and Columbia University. At Columbia he served as a professor in the Department of Mathematics, supervising research that bridged elliptic curves, Galois representations, and the Langlands reciprocity framework. He has been an invited speaker at gatherings organized by bodies like the American Mathematical Society and the International Congress of Mathematicians, and has collaborated with researchers affiliated with the Courant Institute, the Institute for Advanced Study, and international centers such as the Mathematical Sciences Research Institute.

Research contributions

Conrad's research focuses on the arithmetic of elliptic curves, the theory of modular forms, and the deformation theory of Galois representations. He contributed to the refinement and extension of modularity results that trace lineage to Gerhard Frey, Ken Ribet, and Andrew Wiles, elaborating on the modularity of two-dimensional Galois representations over Q and other number fields. His work explores interactions among p-adic Hodge theory frameworks developed by Jean-Marc Fontaine and structural aspects of local Langlands correspondence approaches influenced by Pierre Deligne and Robert Langlands.

Conrad also made advances in the theory of component groups for Néron models, connecting geometric properties studied by Alexander Grothendieck and Jean-Pierre Serre to arithmetic invariants relevant for the study of abelian varieties and the Shafarevich–Tate group. His analyses of finite flat group schemes and moduli stacks draw on techniques from scheme theory initiated by Grothendieck and developed in modern accounts by authors connected to Harvard University and Princeton University mathematics. Further contributions include clarifications of lifting theorems for residual Galois representations and explicit criteria used in modularity-lifting strategies that complement work by Richard Taylor, Fred Diamond, and Christophe Breuil.

His collaborative papers often synthesize perspectives from algebraic number theory classrooms at Cambridge University and research seminars at the Mathematical Institute, Oxford, while leveraging tools from the Étale cohomology tradition stemming from Grothendieck and Alexander Grothendieck's school. Conrad has worked with coauthors linked to institutions such as Yale University, Stanford University, and the University of Chicago.

Awards and honors

Conrad's scholarship garnered recognition through invitations to deliver plenary and sectional lectures at forums organized by the American Mathematical Society and other national academies. He has received fellowships and research support associated with bodies such as the National Science Foundation and held visiting appointments at the Institute for Advanced Study and the Mathematical Sciences Research Institute. His election to professional societies and named lectures reflect esteem from communities centered at Harvard University, Princeton University, and peer institutions in North America and Europe.

Selected publications

- Conrad, B.; Diamond, F.; Taylor, R. "Modularity of certain potentially Barsotti–Tate Galois representations." Annals of Mathematics (series associated with Princeton University Press). - Conrad, B. "Finite flat group schemes, Cartier duality, and applications." Studies originating in seminars linked to Institute for Advanced Study and Harvard University. - Conrad, B.; Edixhoven, B.; Stein, W. "Weights in Serre's conjecture." Papers associated with collaborative research at University of Washington and Universität Bonn. - Conrad, B.; Diamond, F.; Taylor, R. "On the modularity of supersingular elliptic curves." Contributions circulated among seminars at Courant Institute and Mathematical Institute, Oxford. - Conrad, B. "Néron models and component groups in arithmetic geometry." Lecture notes influenced by courses at Columbia University and Princeton University.

Category:American mathematicians Category:Number theorists