# Number Theory Seminars 2/24 Visiting mathematicians from UC San Diego, Alina Bucur and Kiran Kedlaya, will give two number theory talks on Thursday, February 24 in Keller 301.

Schedule:
3-3:45 PM Kiran Kedlaya
The relative class number one problem for function fieldsAbstract: Gauss conjectured that there are nine imaginary quadratic fields of class number 1; this was resolved in the 20th century by work of Baker, Heegner, and Stark. In between, Artin had introduced the analogy between number fields and function fields, the latter being finite extensions of the field of rational functions over a finite field. In this realm, the class number 1 problem admits multiple analogues; we recall some of these, one of which was “resolved” in 1975 and then falsified (and corrected) in 2014, and another one of which is a brand-new theorem in which computer calculations (in SageMath and Magma) play a pivotal role.

3:45-4:15 PM
Q&A, break, refreshments

4:15-5 PM Alina Bucur
Counting points on curves over finite fields

Abstract: A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. This is joint work with Chantal David, Brooke Feigon, Kiran S. Kedlaya, and Matilde Lalin.

# Qualifying exams in algebra

##### Algebra exam
The algebra qualifying exam covers several standard topics in abstract algebra.
###### Material
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
###### Textbooks
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.