The Department of Mathematics and Statistics and the Center for the Relativity and Cosmology jointly hosts a master's in applied mathematical sciences in Mathematical Physics. This program offers a rigorous curriculum that includes a study of global Lorentzian geometry, causal structure, singularity theorems, the initial value problem of the Einstein equation, quantum fields in curved spacetimes and Hawking radiation from black holes.

The Basic Curriculum

1st Semester

2nd Semester
MTH 6640 - Advanced Concepts of Analysis I MTH 6633 - Advanced Linear Algebra
MTH 6615 - Advanced Topology
 MTH 6613 - Differential Geometry I
PHY 5560 - Relativity I PHY 5578 - Relativity II

3rd Semester

4th Semester
MTH 6623 - Differential Geometry II MTH 6620 - Advanced Concepts of Algebra (Lie Groups)
MTH 6692 - Research and Thesis* MTH 6692 - Research and Thesis*

* Research is undertaken under the guidance of a Center’s faculty member. Our current research interests include (but are not limited to) mathematical relativity, quantum fields in curved spacetimes, force-free electrodynamics, and exact solutions.

Use this link to apply for the graduate program. Please select options as shown below.

Highly competitive students may be considered for Graduate Teaching Assistantship (GTA) positions. These roles involve instructional responsibilities; however, recipients are awarded a comprehensive package that includes a full tuition scholarship and an annual stipend of $15,000. Interested students should reach out to relativity-at-troy.edu with a brief cover letter expressing their interest in the GTA position, their academic goals and a description of their background in mathematics and physics.

FAQ

  • For international students only: The deposit for the I-20 will be waived for international students who are recomended for a GTA position.
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    Core Courses

  • PHY 5560 Relativity I: Topics included Lorentz transformation, inertial coordinates, causal structure of spacetime, equivalence principle, gravitational effects in special relativity, curved spacetime, and introduction to black holes. This course contains additional graduate-level content which will further investigate extensions of the topics discussed in the course.

  • PHY 5578 Relativity II:Topics include tensor calculus, Einstein field equations, rotating black holes and cosmology. This course contains additional graduate-level content which will further investigate extensions of the topics discussed in the course.

  • MTH 6615 Advanced Topology: Generalization of such topics as functions, continuous functions, open, closed, compact and connected sets, arbitrary topological spaces.

  • MTH 6613 Differential Geometry I: This course introduces Semi-Riemannian Geometry with an emphasis on its applications to General Relativity. Topics to cover include mathematical structures underlying spacetime, including manifolds, tensors, curvature, and geodesics.

  • MTH 6623 Differential Geometry II: This is the second part of MTH661. In this course the following topics will be covered: Jacobi Fields, Calculus of Variations, Causality and Singularity Theorems.

  • MTH 6620 Advanced Concepts of Algebra: The topics covered will depend on student interest. It could range from groups, rings and Galois theory to the theory of Lie Groups.

  • MTH 6633 Advanced Linear Algebra: A study of linear and orthogonal transformations, orthogonal and unitary matrices, numerical linear algebra, and applications. Spectral theory and duality.

  • MTH 6640 Advanced Concepts of Analysis I: The rigorous course in analysis covers continuous functions, differentiation, the Reimann-Stieltjes integral, sequences and series of function (uniform convergence), integration and differentiation in more than one-dimension, with discussion of the implicit and inverse function theorems.

  • MTH 6692 Research and Thesis: Under the guidance of the student’s adviser and the chair of the department, the student may pursue original research or project in a particular area of mathematics. The completion of a thesis is required. The results and conclusions must be successfully defended before the student’s graduate committee. Grading System is a Pass/Fail.

    • Elective Courses

    (Offered if there is sufficient interest)

  • MTH 6651 Advanced Concepts of Analysis II: Topics include integration and measure, 𝐿𝑝 spaces, Hilbert space theory and derivatives of measures. Prerequisites: MTH 6640.

  • MTH 6619 Partial Differential Equations I:This course focuses on linear partial differential equations, beginning with a review of fundamental results for ordinary differential equations. We will then examine basic constant coefficient solutions for the transport, heat, wave, and Laplace equations. The core of the course covers weak solutions, energy methods, and local well-posedness of second-order linear PDEs with non-constant coefficients.

  • MTH 6629 Partial Differential Equations II: This course explores linear and nonlinear wave equations in curved spacetime. The first part reviews well-posedness using energy estimates and establishing local well-posedness for nonlinear wave equations. The second part of the course deals with global theory.

  • MTH 6645 Algebraic Quantum Field Theory: The course offers an introduction to Algebraic Quantum Field Theory (AQFT). It starts with a review of Hamiltonian mechanics and canonical quantization. The student is then introduced to the axioms of AQFT and some of its main results, including the GNS construction and the Reeh-Schlieder theorem. The formalism allows the study of quantum fields in curved spacetimes. After an introduction to renormalization, the course concludes with the Unruh effect as an example application.