# The zeros of entire functions

Graduate student Rintaro “Yoshi” Yoshida will defend the degree of Ph.D. on Thursday May 2, 2:00pm, in Keller 301.

We invite the reader to consider the entire function
$$\varphi_{_{1/5}}(x) = \sum_{k=0}^\infty \frac{x^k}{(k!)^{(6/5)}}.$$
Do you think this function has any non-real zeros? Does it belong to the Laguerre-Pólya class? See the draft dissertation for answers.

# Nondeterministic finite state complexity

Graduate student Kayleigh Hyde will present her Master’s project on Monday April 22, 10:30am, in Shidler College of Business Room E201. Draft paper

Let $M$ be a nondeterministic finite automaton, having $q$ states and no $\epsilon$-transitions. If there is exactly one path through $M$ of length $n$ leading to an accept state, and $x$ is the string read along that path, then we say that $A_N(x)\le q$ (the NFS complexity of $x$ is at most $q$).

# Requirements for the PhD in Mathematics

(Note: the following requirements are for students beginning in Fall 2016 or later; students who arrived before then should consult with the Graduate Chair.)

The PhD program in Mathematics at UH Mānoa has four principal components:

There are further specific requirements presented below.

##### Course requirements

Students must pass 30 credit hours of mathematics courses numbered 600–699 subject to the following conditions:

• All courses that count towards the 30 credit requirement, or to any of the other requirements below, must be passed with a B− or better.
• Students must “pass” three rows in the following table:

 Algebra andnumber theory 611 612 661 Analysis 631 633 644 Applied mathematics 601 603 607 Foundations 654 655 657 Topology andgeometry 621 623 625

To pass a row, a student must either pass a course from that row, or pass the qualifying exam corresponding to the row.

• At most 6 credit hours can be numbered 649 or 699 unless authorized by the graduate chair. It is expected that the graduate chair will authorize additional courses if they are regular graduate courses that are running with a 649 number, but not otherwise.
• Courses must be from the mathematics department, unless authorized by the graduate chair in consultation with the student’s advisor. It is expected that the graduate chair will authorize graduate-level courses from other departments that are judged to be relevant to the student’s work, and to have serious mathematical content.
##### Qualifying exams

The department offers qualifying exams in four subjects:

• Algebra
• Analysis
• Applied mathematics
• Topology

Students must pass two of these exams. They must do so by the end of the fall semester of their fourth year. Additionally, students must have at least attempted two qualifying exams by the end of the fall semester of their third year. Other than the constraints imposed by these time requirements, students may attempt each qualifying exam any number of times.

The applied mathematics and topology quals are scheduled to take place each fall, while the algebra and analysis quals will be given in the spring. Four courses, numbered 601, 621, 611, and 631, respectively, are meant to cover a fair portion of the material for the qualifying exams. It is however expected that the students will need to learn some material on their own, as well. For further details on the content of the qualifying exams and to see some past exams, click here.

##### Comprehensive exam

A written or oral comprehensive examination on a subject chosen by the student and their advisor, and with the approval of the graduate chair, must be completed by the end of the student’s fourth year. Students are allowed to attempt the comprehensive exam at most twice.

##### Dissertation and defense

The writing and oral defense of a PhD thesis covering substantial original contributions to research are the hallmark of a PhD program. After passing your comprehensive exam, you can officially form a ‘dissertation committee’ chaired by your advisor. As you approach completion, make sure to discuss the timing with your advisor and your committee. You should contact the graduate chair at least a month ahead of defending so that they can ensure all the various forms and announcements go out in time.

##### Further requirements

Additional university-wide requirements can be found on the Office of Graduate Education website.