My research expertise is algebraic topology and, more specifically, stable homotopy theory. Homotopy theory studies algebraic invariants of geometric objects that are unchanged under continuous deformations, while stable invariants are independent of certain spacial dimension shifts. My work on stable invariants has two distinct components: Chromatic Homotopy Theory: Chromatic homotopy theory is a framework that bridges algebraic geometry and topology, allowing the use of sophisticated algebraic tools to enable calculations of stable invariants. My work in this area has implications for structural conjectures and a large computational component whose goal is to provide data to help study these fundamental invariants. My work exploits the interplay between chromatic and equivariant homotopy theory (the study of symmetries) in innovative ways. Phases of Matter: In a collaboration with mathematicians and physicists at CU Boulder, I also study the relationship between stable homotopy theory and condensed matter physics. Our group has a unique combination of expertise that allows us to study problems relevant to physics using the rigorous framework of stable homotopy theory and algebraic quantum mechanics. Our current work is inspired by a general belief that certain phases of matter can be classified by stable homotopical invariants.
keywords
algebraic topology, homotopy theory and applications to condensed matter physics
Motivic homotopical Galois extensions.
Topology and its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology.
290-338.
2018
MATH 2001 - Introduction to Discrete Mathematics
Primary Instructor
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Spring 2018
Introduces the ideas of rigor and proof through an examination of basic set theory, existential and universal quantifiers, elementary counting, discrete probability, and additional topics. Credit not granted for this course and MATH 2002.
MATH 2002 - Number Systems: An Introduction to Higher Mathematics
Primary Instructor
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Fall 2019
Introduces the concepts of mathematical proofs using the construction of the real numbers from set theory. Topics include basic logic and set theory, equivalence relations and functions, Peano's axioms, construction of the integers, the rational numbers and axiomatic treatment of the real numbers. Credit not granted for this course and MATH 2001.
MATH 2135 - Introduction to Linear Algebra for Mathematics Majors
Primary Instructor
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Spring 2021 / Fall 2021 / Spring 2024 / Fall 2024
Examines basic properties of systems of linear equations, vector spaces, inner products, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors and diagonalization. Intended for students who plan to major in Mathematics. Degree credit not granted for this course and MATH 2130 or APPM 3310. Formerly MATH 3135.
MATH 3001 - Analysis 1
Primary Instructor
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Spring 2019 / Fall 2020 / Spring 2023
Provides a rigorous treatment of the basic results from elementary Calculus. Topics include the topology of the real line, sequences of numbers, continuous functions, differentiable functions and the Riemann integral.
MATH 4200 - Introduction to Topology
Primary Instructor
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Fall 2019 / Fall 2021 / Fall 2023
Introduces the basic concepts of point set topology. Includes topological spaces, metric spaces, homeomorphisms, connectedness and compactness. Same as MATH 5200.
MATH 5200 - Introduction to Topology
Primary Instructor
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Fall 2021 / Fall 2023
Introduces the basic concepts of point set topology. Includes topological spaces, metric spaces, homeomorphisms, connectedness and compactness. Same as MATH 4200.
MATH 6210 - Introduction to Topology 1
Primary Instructor
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Fall 2024
Introduces elements of point-set topology and algebraic topology, including the fundamental group and elements of homology. Department enforced prerequisites: MATH 2130 and MATH 3140 and MATH 4001. Instructor consent required for undergraduates.
MATH 6220 - Introduction to Topology 2
Primary Instructor
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Spring 2018 / Spring 2019 / Spring 2022 / Spring 2023 / Spring 2024
Continuation of MATH 6210. Department enforced prerequisite: MATH 6210. Instructor consent required for undergraduates.
MATH 6280 - Advanced Algebraic Topology
Primary Instructor
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Fall 2020
Covers homotopy theory, spectral sequences, vector bundles, characteristic classes, K-theory and applications to geometry and physics. Department enforced prerequisite: MATH 6220. Instructor consent required for undergraduates.
