undergraduate research experiences in computational ... · mathematics:3 semester calculus...

Undergraduate Research Experiences in ComputationalMathematics

Eric Kostelich

SCHOOL OF MATHEMATICAL & STATISTICAL SCIENCES

March 4, 2011

Fall 2005 CBMS report

Fall enrollment growth: +21% at 4-year colleges &universities since 1990 (total: 1.6 million students)Math enrollment growth: −2.3%All math/stat bachelors degrees: −2%Number of bachelor’s degrees awarded by doctoraldepartments was up by 41% over 2000 but those bymaster’s & 4-year schools declined by 19–27%Fraction of 4-year colleges offering AdvancedEngineering Mathematics once every 2 years: 7%Advanced Calculus: 57% Algebra: 52%

CSE 2011 E. Kostelich MATHEMATICAL & STATISTICAL SCIENCES 2 / 33

“Big picture” questions

Should math departments have the primaryresponsibility for teaching undergraduate courses inmathematics?Should mathematically talented undergraduates majorin mathematics?How can mathematics departments contributemeaningfully to fast-growing disciplines like genomics,climate, and materials science?How can mathematics departments contribute tostudents’ professional development besides homeworkproblem sets?


My interests and philosophy

Professional goals: Create modern, high-impactprograms for mathematics undergraduates that canserve as national modelsObjective: Educate mathematically literate scientists(not necessarily more mathematicians per se)Develop skills in scientific collaboration, publicspeaking, and technical writingUndergraduate education should be an opportunity tolearn a little about a lot of things—prematurespecialization is undesirableInterdisciplinary collaboration is essential


A sample undergraduate course catalog

Calculus I, II, III Differential Equations I, IIAdvanced Calc I, II Linear AlgebraProbability Mathematical StatisticsNumber Theory Modern AlgebraComplex Variables Vector AnalysisNumerical Analysis Engineering MathLogic Computer Programming


The post-Sputnik problem

It’s the ASU program in 1958–59!

Undergraduates see few applications of mathematicsafter 1950

Metropolis algorithmFast Fourier transformSimplex algorithmPublic-key cryptographyKalman-Bucy filterViterbi algorithmShotgun genome sequencing


The post-Sputnik problem

It’s the ASU program in 1958–59!Undergraduates see few applications of mathematicsafter 1950

Metropolis algorithmFast Fourier transformSimplex algorithmPublic-key cryptographyKalman-Bucy filterViterbi algorithmShotgun genome sequencing


The standard undergraduate curriculum fails most students

The #1 FAQ: What can I do with a math degree?Little if any exposure to:

Real dataMathematical modeling of “real world” problemsUncertainty in data and modelsComputation beyond a numerical analysis courseTeamwork and scientific collaborationTechnical writing beyond short proofs in homework

If it takes too long for students to see interestingapplications, then they lose interestCourse catalogs are opaque


Some relevant numbers

∼ 106 students enrolled in calculus courses at anygiven time∼ 103 Ph.D. degrees awarded per year in the U.S.∼ 102 tenure-track job openings per year in the U.S.Mathematics departments need to prepare students atundergraduate and graduate levels for jobs in industryand governmentpayscale.com study: the top 15 highest-payingbachelor’s degrees, both after graduation and 15 yearslater, all involve applied mathematics and statistics2009 AMS data: median starting salary for Ph.D.’s inacademia: $60 K in industry: $100 K


Flexibility in undergraduate programs is essential

Programs must give students an idea of what’s possibleThe majority of our math majors transfer from otherprograms—so programmatic requirements mustaccommodate themUndergraduate programs must provide options besidesgraduate schoolCourses in public speaking, leadership, business,finance, entrepreneurship, etc. should be allowableoptionsMentoring (by faculty) is essential


Undergraduate programs at ASU

Current enrollment: 70,000 students on four campusesAbout 52,000 undergraduatesTotal mathematics enrollments: 15,000 students in fall,13,500 in spring∼ 40% precalculus & below; ∼ 30% business math;∼ 30% in engineering calculus & aboveApproximately 4,000 engineering majorsApproximately 550 math majorsMinimum high school competency: 4 years ofmathematics through precalculus (but nobody checksthe rigor of the courses)


The mathematics programs at ASU

56 tenure-track faculty, ∼ 50 lecturers/instructors, ∼ 45TAsThree undergraduate degrees: BA Math, BS Math(with statistics option), BS in ComputationalMathematical SciencesCombined BS/MA degree optionFour new Ph.D. programs:

Applied mathematicsCore mathematicsStatisticsMathematics education


The Computational Mathematical Sciences B.S. degree

Started 10 years ago; accounts for about 1/4 of allmath majorsGoal: Create an interdisciplinary program that wouldappeal to good science and math studentsMust accommodate transfer students easilyEncourages independent study and summer internshipsTakes a broad view of “scientific computing”


Other considerations

Natural home should be mathematicsMust leverage existing courses as much as possibleAll interested faculty should be able to participateCan accommodate students with a variety of scientificcareers in mind


CMS core areas

Mathematics: 3 semester calculus sequence, ordinarydifferential equations, linear algebra (18 credits total)Science: 2 one-year sequences in life/physical sciences(16 credits)Computing: 2 semesters of introductory programmingand data structures plus Scientific Computing andIntroduction to Numerical Methods (12 credits)Advanced courses: At least 3 in math/stat and 1internship/research course (12 credits)Liberal studies: 5 courses in humanities/socialsciences, 2 “science and society” courses (21 credits)


The Scientific Computing course

Audience: advanced undergraduate and beginninggraduate students in mathematics and physical sciencesEmphasis: Software development skills for scientificand high-performance computingTopics: Scripting (shell, Python, makefiles); Fortran95/2003, C99, C++; LAPACK.Prerequisites: At least 2 semesters of programming,differential equations and/or linear algebra.Parallelism: In development as a collaboration withIntel: brief introduction to OpenMP and/or MPI


The CSUMS program at NSF

Computational Science Training for Undergraduates inthe Mathematical Sciences (CSUMS)Funded through Divisions of Mathematical Sciencesand Undergraduate Education at the National ScienceFoundationFirst call for proposals in June 2006A 5-year effort; starting year 4 nowhttp://nsf.gov/pubs/2006/nsf06559/nsf06559.htm


Quotes from the program solicitation

Goal: help prepare math/stat majors for careers andgraduate study in fields that require integrated strengthsin computation and the mathematical sciencesCore: Long-term research experiences for cohorts of atleast six undergraduatesRequirements: Projects must be genuine researchexperiences rather than rehearsals of research methods


The CSUMS project at ASU


The CSUMS project at ASU

47 students in overlapping cohorts since Jan. 2008New cohort of 10–13 students begins each year17 women, 17 first-generation college students, ages15–31Weekly “pizza seminar” during the academic year &8-week summer research projectConference presentation, honors thesis, hopefully apaperSliding window of faculty participants


CSUMS participation timeline

Basic requirements: Math major with 3.5+ GPA,differential equations, linear algebra, 2 semesters ofprogrammingYear 1: Weekly pizza seminarSummer 1: Research projectsYear 2: Weekly pizza seminar, conference, paper,graduationOption: Continue to BS/MA program


CSUMS presentations at SIAM CSE 11

John Ingraham, diffusion tensors for modeling gliomainvasionJuan Durazo, surface flows of phytoplankton in theoceanHelene Rhodes, MR image processing for tumor andedemaKeith Voytek, modeling creation of hydroxyl radicalsin radiotherapyEric Adams, simulating radiation in a brain tumormodel


CSUMS presentations at SIAM CSE 11, continued

Jordan Martel, radial basis functions for 2-d fluid flowon the sphereMaple So, glioma modeling on individual patient brainimagesRebecca Borchering, multi-species rabies models withdata from Arizona and TexasMiles Manning and April Chiu, model of a Drosophilalarva heart


Glioma “painting” with MIPAV

Goal: estimate volume of tumor and edematous regions


Standardize coordinates (Maple So)ACPC Alignment Talairach Transformation

Original Skull Stripped

ACPC Aligned Talairach

Talairach Atlas VOIsBrain Tumor Analysis Maple So MATHEMATICS AND STATISTICS 1 / 2CSE 2011 E. Kostelich MATHEMATICAL & STATISTICAL SCIENCES 24 / 33

Rabies epidemiology (Becky Borchering & Mara Steinhaus)

Rabies Model for Skunks (Mara Steinhaus & Dr. Yang Kuang, CSUMS 2010)

In the current model, susceptible skunks become exposed after coming into contact with an infected skunk. Exposed skunks are consideredinfected when they begin displaying rabies symptoms. No infected skunks recover. For low values of the unknown parameter β, the rabiesinfection eventually dies out and is contained within the initial area of introduction. For high values of β, the infection spreads out in a ring fromthe center of introduction. See diagrams below.

Parameter Set 1: Distribution of Infected:

Parameter Value InformationK 6 individuals per km2

d 0.1 km2 per yearβ 0.1 unknownr 1 2-4 births per yearm 0.5 lifespan = 1-2 yearsmr 1 100% mortalityσ 1 1 week - several years

t=0 t=1 t=2

Parameter Set 2: Distribution of Infected:

Parameter Value InformationK 6 individuals per km2

d 0.1 km2 per yearβ 1 unknownr 1 2-4 births per yearm 0.5 lifespan = 1-2 yearsmr 1 100% mortalityσ 1 1 week - several years

t=0 t=1 t=2

My Objectives for Future Work:• Identify influential parameters and incorporate them into the standing model.

• Develop a system of ODEs that model rabies dynamics for bats and add these equations to the current model. The new model would describedispersion of both infected skunks and infected bats.

• Update the diffusion coefficient, d, to be a spatially dependent function. A changing rate of diffusion would account for differences in terrain(e.g. mountainous regions vs. rivers) that hinder or ease movement.

• Verify the accuracy of my updated model by cross-referencing my results with the extensive confirmed case data from Texas.

Susceptible: St = rS(1−S/K)−βSIExposed: Et = βSI − (σ +m)EInfected: It = σE−mrI +∇ · (D∇I)


Selected outcomes to date

14 talks and 6 posters at 4 SIAM conferences (CSE,Imaging, National meetings)5 talks at undergrad conferences (SUNMARC,CSUMS), 3 talks at AAAS2 (of 15 nationally) Honorable Mentions in AppliedMathematics in the NSF Graduate FellowshipCompetition13 honors theses defended or in progress


Selected outcomes to date, continued

2 research journal publications12 alumni in Ph.D. programs in applied math, 5 ongraduate fellowships4 alumni in medical school (including MD/PhDprograms)1 in industryFollow-on internships at Max Planck Institute, HarvardSystems Biology program, Budapest Semesters inMathematics, Hispanic College Fund4 resignations


Things that have worked for us

Mathematical preparation of vector calculus, ODE, andlinear algebra suffices to do something interestingThe 8-week summer format allows students to focus ontheir projects without other distractions—yet stillleaves time for a breakThe first week includes a “boot camp” on basics ofLATEX and MATLABThe pizza seminar allows time for preparatory lecturesleading up to the summer—as well as a time for guestlectures, student talk practice, etc.


Overall philosophy

An undergraduate research project is not a miniaturePh.D. thesisPedagogical goal is for students to decide whether theyare interested in research in mathematical sciencesStudents need a couple weeks to learn about the topicand help to define a project that can be substantiallycompleted in another 6 weeksNew mathematics can be learned as needed along thewayThe computer is a tool for discovery, not the principalfocus


Recruitment challenges

Word of mouth is very helpfulOtherwise promising students are sometimes turned offby the perception of a “nerd factor”Having a critical mass of female students and facultymentors is important for recruiting womenWell motivated students are essential—and not alwaysobvious from transcripts and letters of recommendationNot all students will succeed


Faculty challenges

A program of this size is a lot of workProportionally smaller cohorts are appropriatedepending on the student and faculty poolA “sliding window” of faculty mentors helps preventburn-outResearch-active faculty are essentialThe summer format may not be practical for somefaculty


Community challenges

Devising a durable “ecosystem” that supportsundergraduate researchMaintaining funding and momentum in a period ofbudget crisesExpanding opportunities for students (and faculty!) atundergraduate institutions for summer research andprofessional development


Acknowledgments

Thanks to:National Science FoundationASU College of Liberal Arts and SciencesSchool of Mathematical and Statistical Sciences, ASU


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