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Unit: College of Arts and Sciences

Department: Department of Mathematics

##### Office location and address

321 Kerchof Hall

141 Cabell Dr

Charlottesville,
Virginia
22904
##### Publications

##### Sponsored Awards

Topology of 4-manifolds

Source: The Simons Foundation, Inc.

January 01, 2020 – December 31, 2021

AS-MATH Topology of 4-manifolds, links and Engel groups

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

July 01, 2016 – June 30, 2019

AS-MATH Geometric and Quantum Topology in Low Dimensions

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

October 01, 2013 – September 30, 2016

AS-MATH Geometric Topology of 4-Manifolds

Source: The Simons Foundation, Inc.

June 01, 2014 – January 09, 2015

AS-MATH Geometric Topology of 4-Manifolds

Source: William E. Simon Foundation

July 01, 2014 – July 01, 2014

Low-dimensional topology and topological methods in condensed matter physics

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

July 01, 2010 – March 31, 2014

##### 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: 3

Geometric study of curves/surfaces/their higher-dimensional analogues. Topics vary and may include curvature/vector fields and the Euler characteristic/the Frenet theory of curves in 3-space/geodesics/the Gauss-Bonnet theorem/and/or an introduction to Riemannian geometry on manifolds. Prerequisites: MATH 2310, MATH 3250 and MATH 3351 or instructor permission.

Credits: 3

Examines the knotting and linking of curves in space. Studies equivalence of knots via knot diagrams and Reidemeister moves in order to define certain invariants for distinguishing among knots. Also considers knots as boundaries of surfaces and via algebraic structures arising from knots. Prerequisites: MATH 2310 and MATH 3351 and MATH 3354 or instructor permission.

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: 3

A rigorous program of supervised study designed to expose the student to a particular area of mathematics. Prerequisite: Instructor permission and graduate standing.

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: 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.