All posts by mmanes

Qualifying exams in algebra

Algebra exam
The algebra qualifying exam covers several standard topics in abstract algebra.
  1. Group theory: basics of group actions, semidirect products, class equation, Sylow theorems, applications, solvable groups, Jordan–Hölder theorem
  2. Field and Galois theory: finite fields, separable and normal extensions, Fundamental theorem of Galois theory, applications (e.g. solvability by radicals, constructions by straightedge and compass, …), determining Galois groups
  3. Ring theory: factorization in domains, simplicity of matrix algebras
  4. Module theory: basics, projectivity, injectivity, tensor products, flatness, Noetherian property, exact sequences, commutative diagrams, structure theory of modules over a PID, consequences for canonical forms of matrices and other linear algebra
  5. Language of category theory: objects, arrows, Hom, functors, natural transformations, universal objects, products, coproducts, Yoneda lemma
  6. Multilinear algebra: pairings, wedge products, symmetric products, multilinear forms over rings
  7. Basic commutative algebra: local rings and localization, integral extensions, Hilbert Basis Theorem, Noether Normalization, Hilbert’s Nullstellensatz
The following textbooks are recommended:
  • Dummit and Foote, Abstract algebra
  • Lang, Algebra
  • Rotman, Advanced Modern Algebra
  • Hungerford, Algebra
  • Jacobson, Basic Algebra I & II
Sample Exams

Manes joins National Science Foundation

The National Science Foundation (NSF) has selected UH Math Professor Michelle Manes as a Program Director in the Division of Mathematical Sciences (DMS). She will work with a team to make funding recommendations in the areas of Algebra and Number Theory, and she will serve on several other special programs at NSF.

From the NSF website: “DMS supports research in mathematics and statistics, training through research involvement of the next generation of mathematical scientists, conferences and workshops, and a portfolio of national mathematical sciences research institutes.”

Manes’s appointment began in September, 2018 and may be renewed for a second year. During her appointment, she will be living in Washington, DC.



Mathematics and Motherhood

Two members of the Mathematics Department — new assistant professor Elizabeth Gross and postdoc Piper Harron — have papers in the current issue of the  Journal of Humanistic Mathematics (, a Special Issue on Mathematics and Motherhood .

Dr. Gross wrote “PB&J,”  which explores pregnancy in graduate school, parenting on the tenure-track, division of household labor, and sandwiches.

In “On Contradiction,” Dr. Harron exposes the challenges faced by marginalized parents and the problems inherent in separating our lives into “mathematics” and “everything else” (including the problems inherent in separate journals, special issues like this one, and dedicated panels for the “life” parts).

Strong parallels between the very different pieces emerge, including the incredible time demands of parenthood, the importance of supportive colleagues, and hilarious stories from the trenches (including the photographic reenactment shown from Harron’s piece).

Read Dr. Gross’s essay here:

Read Dr. Harron’s essay here: