Santa Barbara,
CA,
United States
Type
Public
Degrees offered
Bachelor's Degree
PhD in Mathematics - Topology in Santa Barbara United States | University of California, Santa Barbara
University of California, Santa Barbara
|
Santa Barbara United States
Visit institution website
for more information
Qualification
Doctor of Philosophy
Languages
English
Delivery Mode
On-Campus
Tuition (2025)
USD
29,118
Attendance
Full-time
Full-time Duration
48 months
Geometric topology is commonly divided into low-dimensional (4 or fewer dimensions) and high-dimensional categories. This classification reflects the methods used, the types of questions addressed, and the current understanding in each area. The 1960s saw groundbreaking progress in high-dimensional topology, such as proving the high-dimensional Poincaré conjecture and gaining insights into how differentiability affects structures, particularly through exotic smooth structures on spheres.
Current research focuses heavily on exploring 3- and 4-dimensional manifolds. While the approaches, hypotheses, and perspectives in these two fields differ significantly, some unifying patterns have emerged. A notable discovery in the 1980s by Donaldson, Freedman, and Casson revealed that exotic smooth structures exist exclusively in 4-dimensional Euclidean space.
Thurston's Geometrization Conjecture in the late 1970s transformed 3-dimensional manifold theory. Eight homogeneous Riemannian geometries seem to play a role in three dimensions analogous to the three constant curvature geometries in two dimensions. Certain 3D problems are most effectively tackled using combinatorial and topological methods involving surfaces and their extensions. Numerous challenges in knot theory fall into this category.
Destination of Study
Subjects of Study
Language Requirements
English
IELTS 7.0
Qualification Requirements
Applicants who have conducted their post-secondary education outside of the United States must have completed an undergraduate degree program equivalent to a University of California bachelor's degree by the beginning of their graduate program at UCSB. Equivalent undergraduate degrees usually include a minimum four years of university work and above-average scholarship. The degree must be awarded by an accredited institution. To be considered for admission, you must have received a bachelor's degree or its equivalent from an accredited university prior to the quarter for which you seek admission, and have at least a B average (3.0 GPA) in your undergraduate coursework. Satisfaction of minimal standards does not, however, guarantee admission, since the number of qualified applicants far exceeds the number of spaces available. An excellent command of written and spoken English is required prior to enrollment at UCSB. Applicants whose native language is not English are required to take the Test of English as a Foreign Language (TOEFL), the International English Language Testing System (IELTS), or the Duolingo English Test (DET). TOEFL paper-based test (PBT) 550, TOEFL internet-based test (iBT) and TOEFL iBT Home Edition 80, IELTS Academic and IELTS Indicator - Overall Band Score of 7.
Tuition
USD
29,118