20192020 Graduate Course Catalog [ARCHIVED CATALOG]
Mathematics, PhD


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Department Chair: Uday Banerjee, 215 Carnegie Building, banerjee@syr.edu, 3154431472
Associate Chair for Graduate Studies: Graham Leuschke, 317G Carnegie Building, gileusch@syr.edu, 3154431500
Faculty
Uday Banerjee, Pinyuen Chen, Dan Coman, J. Theodore Cox, Steven Diaz, Nicole M.L. Fonger, Jack E. Graver, Duane Graysay, Philip S. Griffin, Tadeusz Iwaniec, Lee Kennard, H. HyuneJu Kim, Mark Kleiner, Leonid Kovalev, Loredana Lanzani, Graham J. Leuschke, Wei Li, Jianxuan Liu, Adam Lutoborski, Joanna O. Masingila, Terry R. McConnell, Claudia Miller, Jani Onninen, Evgeny Poletsky, Declan Quinn, Minghao Rostami, Lixin Shen, Gregory Verchota, Andrew Vogel, Stephan Wehrli, William Wylie, Yuan Yuan, Dan Zacharia
The Department of Mathematics has 33 faculty members, with research interests in several areas of mathematics, statistics, and mathematics education, and approximately 55 graduate students. The department is housed in the recently renovated Carnegie Library building on the main campus quadrangle. Programs of study include those for M.S. and Ph.D. degrees in Mathematics, with or without a concentration in Statistics, and for M.S. and Ph.D. degrees in Mathematics Education.

Student Learning Outcomes
1. Demonstrate mastery in the core areas of algebra/topology and analysis by solving problems using advanced techniques
2. Demonstrate advanced knowledge in their chosen specialty and in an additional area of mathematics by solving problems using advanced techniques
3. Plan and successfully conduct original research, producing results worthy of publication in peer reviewed journals
4. Effectively communicate mathematical ideas
Ph.D. in Mathematics
Doctoral students are expected to have completed the requirements for a master’s degree in mathematics or the equivalent. They then take at least 60 credits of additional work including up to 30 credits of dissertation credit and at least 30 credits of coursework. All students must demonstrate a mastery of English. Students must pass preliminary examinations in analysis and algebra and qualifying examinations in a major area and a minor area chosen (subject to some restrictions), from algebra, analysis, combinatorics, numerical analysis, statistics, and topology. Students who successfully complete the qualifying examination are granted the Master’s of Philosophy (M.Phil.) degree in mathematics. Each Ph.D. student must defend in oral examination a dissertation that demonstrates ability to carry out independent investigation which makes an original contribution to mathematics. Mathematics students may write a Ph.D. dissertation under certain faculty members in computer science. Further information is available from Graham Leuschke, 215 Carnegie Building, or on our web site: math.syr.edu.
Research Areas
The department’s Colloquium series features weekly lectures by mathematicians from all over the United States and abroad in many of the areas of mathematical research represented in the department. Furthermore several of the research groups organize regular research seminars. Colloquia and seminar schedules, along with other information about our programs, courses, and events, can be found at math.syr.edu.
The following research groups are currently represented in the department.
Algebra
Algebraic geometry (moduli spaces of curves, equations defining finite sets of points), commutative algebra (homological algebra, CohenMacaulay modules, characteristic p), noncommutative algebra (representations of finitedimensional algebras, homological algebra, group actions on noncommutative rings, Hopf algebras, enveloping algebras, noncommutative algebraic geometry). Faculty: Diaz, Kleiner, Leuschke, Miller, Quinn, Zacharia
Analysis
Complex analysis (several complex variables, pluripotential theory, complex dynamics, invariant metrics, holomorphic currents, Kähler geometry, rigidity problems), geometric analysis (PDE on manifolds, geometric flows), harmonic analysis, partial differential equations (linear and nonlinear elliptic PDE, boundary value problems on nonsmooth domains), geometric function theory (quasiconformal mappings, analysis on metric spaces). Faculty: Coman, Iwaniec, Kovalev, Lanzani, Onninen, Poletsky, Verchota, Vogel, Wylie, Yuan
Applied Mathematics
Numerical analysis (approximate solutions of elliptic PDE, generalized finite element methods and meshless methods), nonlinear variational problems (microstructure in nonlinear elasticity), applied and computational harmonic analysis (wavelets, digital image processing), numerical linear algebra, computational fluid dynamics. Faculty: Banerjee, Lutoborski, Rostami, Shen
Combinatorics
Combinatorics, graph theory, rigidity theory, symmetries of planar graphs, automorphism groups of graphs. Faculty: Graver
Geometry/Topology
Lowdimensional topology and knot theory (knot concordance, Heegaard Floer homology, homology theories for knots and links), Ktheory (topological Ktheory of EilenbergMac Lane spaces, equivariant homotopy theory), Riemannian/Kähler geometry (Ricci curvature and topology, special metrics, geometric flows, rigidity problems). Faculty: Kennard, Wehrli, Wylie, Yuan
Mathematics Education
Secondary mathematics education, teacher learning, mathematical representations, outofschool mathematics practice, teacher development. Faculty: Fonger, Graysay, Masingila
Probability
Interacting particle systems, Brownian motion, random walks, probabilistic methods in mathematical finance, martingales. Faculty: Cox, Griffin, McConnell
Statistics
Ranking and selection theory (applications in radar signal processing and twostage procedures for multinomial problems), changepoint problems, sequential analysis, longitudinal analysis, neural networks. Faculty: Chen, Kim, Li, Liu
Graduate Awards
Figures for graduate appointments represent 20192020 stipends.
Graduate Scholarships:
Support graduate study for students with superior qualifications; provide, in most cases, full tuition for the academic year.
Graduate Assistantships:
Offered to most Graduate Scholarship recipients; no more than an average of 20 hours of work per week; nine months; stipend ranging from $19,000 to $22,500 in addition to tuition scholarship for 24 credits per year. Additional summer support is generally available.
Syracuse University Graduate Fellowships:
Taxfree stipends are $25,290 for nine months of fulltime study; tuition scholarship for 15 credits per semester for a total of 30 credits during the academic year.
Facilities
The mathematics collection is held within the Carnegie Library and supports mathematical research over a broad range of pure and applied mathematics, as well as mathematics education, mathematical statistics, and interdisciplinary areas. Most of the nonbook resources are online and includes an extensive collection of databases and journals supporting the mathematical sciences. In addition, the library provides a growing collection of ebooks.
Students may borrow course reserved textbooks, laptops, TI graphing calculators, and geometry kits from the Carnegie Library service desk. Students may also reserve one of three group study rooms located on the first floor of the library. A computer lab in the library provides software for programming, statistical and data analysis, video and multimedia, and access to printers.
Carnegie Library is home to collections in the sciences, including engineering and computer science, the life sciences, and the physical sciences and hosts a strong collection of databases, journals, and ebooks supporting all disciplines. The historic Reading Room gives the library a distinctive ambience and provides a quiet place for students to study.

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