TM

Unit: College of Arts and Sciences

Department: Department of Mathematics

##### Office location and address

327 Kerchof Hall

141 Cabell Dr

Charlottesville,
Virginia
22904
##### Publications

##### Sponsored Awards

RTG: Research Training in Topology and Geometry at the University of Virginia

Source: U.S. National Science Foundation (NSF)

July 01, 2019 – June 30, 2024

AS-MATH Low-dimensional contact and symplectic topology

Source: The Simons Foundation, Inc.

September 01, 2017 – August 31, 2022

Virginia Topology Conference 2018

Source: U.S. National Science Foundation (NSF)

September 01, 2018 – August 31, 2019

AS-MATH Low DImensional Contact and Symplectic Topology

Source: U.S. NSF - Directorate Math. & Physical Sciences

August 15, 2013 – July 31, 2017

Fibrations and the topology of low-dimensional manifolds

Source: U.S. NSF - Directorate Math. & Physical Sciences

July 15, 2009 – June 30, 2013

##### Courses

Credits: 4

A continuation of Calc I and II, this course is about functions of several variables. Topics include finding maxima and minima of functions of several variables/surfaces and curves in three-dimensional space/integration over these surfaces and curves. Additional topics: conservative vector fields/Stokes' and the divergence theorems/how these concepts relate to real world applications. Prerequisite: MATH 1320 or the equivalent.

Credits: 4

A rigorous development of the properties of the real numbers and the ideas of calculus including theorems on limits, continuity, differentiability, convergence of infinite series, and the construction of the Riemann integral. Students without prior experience constructing rigorous proofs are encouraged to take Math 3000 before or concurrently with Math 3310. Prerequisite: MATH 1320.

Credits: 3

Differential and integral calculus in Euclidean spaces. Implicit and inverse function theorems, differential forms and Stokes' theorem. Prerequisites: MATH 2310 or MATH 2315; MATH 3351 or MATH 4651 or APMA 3080; and MATH 3310 or MATH 4310

Credits: 3

This course provides a framework for the completion of a Distinguished Major Thesis, a treatise containing an exposition of a chosen mathematical topic. A faculty advisor guides a student through the beginning phases of the process of research and writing. Prerequisite: Acceptance into the Distinguished Major Program.

Credits: 3

This is the second semester of a two semester sequence for the purpose of the completion of a Distinguished Major Thesis. A faculty member guides the student through all phases of the process which culminates in an open presentation of the thesis to an audience including a faculty evaluation committee. Prerequisite: MATH 4900.

Credits: 1–3

Reading and study programs in areas of interest to individual students. For third- and fourth-years interested in topics not covered in regular courses. Students must obtain a faculty advisor to approve and direct the program.

Credits: 1–3

This seminar discusses the issues related to research in Mathematics. There are speakers from the different areas of mathematics represented at the University of Virginia. Credit may not be used towards a Master's degree. Prerequisite: Graduate standing in mathematics.

Credits: 3

Topics include the fundamental group, covering spaces, covering transformations, the universal covering spaces, graphs and subgroups of free groups, and the fundamental groups of surfaces. Additional topics will be from homology, including chain complexes, simplicial and singular homology, exact sequences and excision, cellular homology, and classical applications. Prerequisite: MATH 5352, 5770, or equivalent.

Credits: 3

Topics include smooth manifolds and functions, tangent bundles and vector fields, embeddings, immersions, transversality, regular values, critical points, degree of maps, differential forms, de Rham cohomology, and connections. Prerequisite: MATH 5310, 5770, or equivalent.

Credits: 3

Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity. Prerequisite: MATH 7800.

Credits: 3

Studies regular and critical values, gradient flow, handle decompositions, Morse theory, h-cobordism theorem, Dehn's lemma in dimension 3, and disk theorem in dimension 4. Prerequisite: Math 5770.

Credits: 1–3

Discusses subjects from geometry and topology.

Credits: 3–9

Independent Research

Credits: 1–12

The Mathematics Colloquium is held weekly, the sessions being devoted to research activities of students and faculty members, and to reports by visiting mathematicians on current work of interest. For doctoral dissertation, taken under the supervision of a dissertation director.