Close Window Posting Number: 120549 Job Title: Faculty-Mathematics Personal Information Last Name: Parry First Name: Alan Middle Name: Reid Home Phone: 4357608361 Work Phone: Present Address: 311 S Lasalle St. Apt. 21F City: Durham State: NC Zip Code: 27705 Permanent Address: 311 S Lasalle St. Apt. 21F City: Durham State: NC Zip Code: 27705 International Address: City: Province/State: Country: Email Address: [email protected]Do you have the legal right to work in the U.S.? Yes Education University: Degree Received: Date Received: Major: Minor: Utah State University BS 05-2006 Mathematics N/A Duke University MA 12-2008 Mathematics N/A Duke University PhD 05-2013 Mathematics N/A References Name of Reference: His/Her Position: Organization: Phone Number and Email Address: Relationship to you: Hubert Bray Professor Duke University Mathematics Department, Duke University, Box 90320 Durham , NC , 27708 9197578428 [email protected]PhD Advisor Pengzi Miao Associate Professor University of Miami University of Miami Department of Mathematics, PO Box 249085 Coral Gables , FL , 33124 3052842859 [email protected]Colleague
Academic Rank HistoryAcademic Rank: Granting Institution: Year Granted:
Religious and CivicAre you or have you ever been amember of The Church of JesusChrist of Latter-day Saints?Yes
If not, please list your religiousaffiliation:
Current Ward/Branch:
Durham 1st Ward
Current Stake/Mission:
Durham North CarolinaState
Bishop/Branch President's Name:Christopher Kelsey
Work Phone:
Home Phone:
9194525171
Stake/Mission President's Name:Laurin Milton
Home Phone:9193091451
Current Church Position:Webelos Den Leader
If you have moved within six months, please list the previousWard/Branch and Bishop/Branch President:
Mission?Yes
Where and when?Michigan DetroitMission, July 2002 -July 2004
Supplemental QuestionsWHAT IT MEANS TO BE A BYU FACULTY MEMBER: Faculty are an important part of BYU's uniqueness. They combinepractical and academic expertise with a commitment to the University, its students, and the gospel of Jesus Christ. Thisuniqueness is characterized by faculty who observe the behavioral standards of the University and refrain from any behavior orexpression that seriously and adversely affects the University or its sponsoring institution, The Church of Jesus Christ of Latter-day Saints. Faculty who are members of The Church of Jesus Christ of Latter-day Saints accept as a condition of employment thestandards of conduct consistent with qualifying for temple privileges. LDS and non-LDS faculty are expected to be role models fora life that combines the quest for intellectual rigor with spiritual values and personal integrity including adherence to the HonorCode and Dress & Grooming Standards. I HAVE READ AND UNDERSTAND THAT A CONDITION OF EMPLOYMENT AT BYUINCLUDES COMPLIANCE WITH THESE REQUIREMENTS.YES Honor Code and Dress & Grooming Standards: As a matter of personal commitment, all faculty are to demonstrate in daily livingon and off-campus those moral virtues encompassed in the gospel of Jesus Christ, and will be honest, live a chaste and virtuouslife, obey the law and all campus policies, use clean language, respect others, abstain from alcoholic beverages, tobacco, tea,coffee, and substance abuse, participate regularly in church services, observe the Dress & Grooming Standards, and encourageothers in their commitment to comply with the Honor Code. I WILL ABIDE BY THE REQUIRED BRIGHAM YOUNG UNIVERSITYHONOR CODE AND DRESS & GROOMING STANDARDS.Yes EEO Statement: Brigham Young University, an equal opportunity employer, does not discriminate on the basis of race, color,gender, age, national origin, veteran status, or against qualified individuals with disabilities. All faculty are required to abide by theUniversity's Honor Code and Dress & Grooming Standards. Preference is given to qualified candidates who are members in goodstanding of the affiliated church, The Church of Jesus Christ of Latter-day Saints. Successful candidates are expected to supportand contribute to the academic and religious missions of the University within the context of the principles and doctrine of theaffiliated church. I HAVE READ AND UNDERSTAND THE BRIGHAM YOUNG UNIVERSITY EEO STATEMENT.Yes
AgreementPLEASE READ THE FOLLOWING CAREFULLY
Brigham Young University, an equal opportunity employer, is a private university with unique goals and aspirations that arise fromthe mission of its sponsoring institution, The Church of Jesus Christ of Latter-day Saints. Preference is given to LDS applicants. Itis a condition of employment that faculty members observe the behavior standards of the University, including the ChurchEducational System Honor Code with its Dress and Grooming Standards, and refrain from behavior or expression that seriouslyand adversely affects the University mission or the Church. Faculty who are members of the Church also accept as a condition ofemployment the standards of conduct consistent with qualifying for temple privileges. All faculty are expected to be role modelsfor a life that combines the quest for intellectual rigor with spiritual values and personal integrity. (For more information regarding
the Honor Code, please click here.)
Recognizing the influence of faculty members on students, the BYU Board of Trustees, under the guidance of the FirstPresidency, has asked us to contact your ecclesiastical leaders to determine your eligibility for employment at BYU.
To the best of my knowledge, the information included in this application is accurate. BYU has my permission to contact myecclesiastical leaders and other references and I request that my ecclesiastical leaders and other references provide BYUinformation concerning my eligibility for employment at BYU. I waive any privilege I have to the information, and I release fromany liability all persons supplying or receiving information pursuant to this request.
BY SIGNING BELOW or SUBMITTING THIS FORM ELECTRONICALLY, I certify that I have read and agree with thesestatements.
Alan Reid ParryApplicant's Name Applicant's Signature Date
Search CommitteeDepartment of Mathematics275 TMCBBrigham Young UniversityProvo, UT 84602-6539
Dear Committee Members:
I am writing to apply for either a tenure-track assistant professorship or a visiting assistant pro-fessorship in your department. I am currently a graduate student at Duke University under thesupervision of Hubert L. Bray and will be graduating with a Ph.D. in May 2013.
My research interests are in differential geometry, geometric analysis, and general relativity includingnumerical relativity. As such, I could potentially work closely with Michael Dorff in your departmentas well as across disciplines with Eric Hirschmann and David Neilson in the physics departmentat BYU, and I have overlapping research interests with several other people in your departmentincluding Jessica Purcell, Jeffrey Humpherys, and Vianey Villamizar. My current research concernsa possible geometric explanation of dark matter called wave dark matter, which is an excitingand relatively new direction of research at the intersection of geometric analysis, general relativity,computational physics, and astrophysics. I have given invited talks on this topic at Duke University,the University of Miami, and Lafayette College.
I have considerable teaching experience as a graduate student instructor having independently taught12 different semester long courses ranging from algebra to calculus. I have been recognized for myteaching ability as a recipient of the L.P. and Barbara Smith Award for Teaching Excellence. I havealso served as a teaching assistant at the recent summer graduate workshop on mathematical generalrelativity at MSRI.
I am a member in good standing of the Church of Jesus Christ of Latter-day Saints and would loveto work in a department, like the BYU mathematics department, which encourages an environmentcompatible with the standards of the Church.
I have arranged for letters of recommendation from Hubert L. Bray (Duke), Pengzi Miao (Miami),and Justin Corvino (Lafayette) concerning my research and from Jack Bookman (Duke) concerningmy teaching to be sent via mathjobs.org. My remaining materials can also be found there. I havealso made the requisite application at yjobs.byu.edu. My papers and more information on me canbe found at my webpage http://fds.duke.edu/db/aas/math/grad/alrparry. If you have anyquestions, please contact me via the email address or phone number below.
I appreciate all the time and consideration you give to my application and look forward to hearingfrom you.
Department of Mathematics Phone: 435-760-8361Duke University, Box 90320 Email: [email protected], NC 27708-0320 USA Webpage: http://fds.duke.edu/db/aas/math/grad/alrparry
Education
PhD Mathematics Duke University May 2013 (Expected)Advisor: Hubert L. BrayThesis Title: Wave Dark Matter
and Dwarf Spheroidal Galaxies
MA Mathematics Duke University December 2008
MS Mathematics Utah State University August 2007Advisor: Ian M. Anderson
BS Mathematics Utah State University May 2006Magna Cum Laude
Research Interests
General Relativity, Geometric Analysis, Mathematical Physics
Professional Experience/Employment History
Duke University Department of MathematicsGraduate Student Instructor/Teaching Assistant/Research AssistantAugust 2007 - presentResponsibilities included conducting research and either independently teaching a section of an intro-ductory level course in mathematics or heading a lab for one such section as often as required by myprogram. Labs and courses taught are listed below.
Courses Taught Math 31L - Laboratory Calculus I Spring 2011Math 41L - One Variable Calculus Fall 2010Math 25L - Laboratory Calculus and Functions I Summer 2010Math 25L - Laboratory Calculus and Functions I Spring 2010Math 25L - Laboratory Calculus and Functions I Fall 2009Math 25L - Laboratory Calculus and Functions I Summer 2009Math 25L - Laboratory Calculus and Functions I Fall 2008
Labs Taught Math 32L - Laboratory Calculus II Fall 2007
1
Utah State University Department of Mathematics and StatisticsGraduate Student Instructor/Teaching AssistantJanuary 2006 - August 2007Responsibilities included independently teaching a section of an introductory level course in mathe-matics each semester (summers optional) and attending annual teacher workshops. Courses taught arelisted below.
Courses Taught Math 1210 - Single Variable Calculus I Summer 2007Math 1050 - College Algebra Spring 2007Math 1050 - College Algebra Fall 2006Math 1010 - Intermediate Algebra Summer 2006Math 1010 - Intermediate Algebra Spring 2006
Professional Service
Duke Mathematics Department Calculus Committee, January 2010 - present
Awards, Honors, and Distinctions
L.P. and Barbara Smith Award for Teaching ExcellenceDuke University Mathematics Department, September 2010
L.P. and Barbara Smith Award for Beginning TeachersDuke University Mathematics Department, February 2009
University Club ScholarshipUtah State University, 2001
Robert C. Byrd ScholarshipUtah State Board of Education, 2001
Conferences Attended
MSRI Summer Graduate Workshop on Mathematical General Relativity, July 9-20, 2012, MSRI, Role:Teaching Assistant
27th Annual Geometry Festival, April 27-29, 2012, Duke University, Role: Attendee/Technology Lia-son
41st Barrett Memorial Lectures in Mathematical Relativity, May 11-14, 2011, University of Tennessee,Knoxville, Role: Attendee
2
Invited Talks
Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies, Math-Physics Colloquium, Lafayette Col-lege, October 3, 2012
A Pictorial Introduction to General Relativity, Mathematical Adventures and Diversions Talk, LafayetteCollege, October 3, 2012
Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies, Geometry and Physics Seminar, Universityof Miami, September 26, 2012
Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies, Geometry/Topology Seminar, Duke Uni-versity, September 4, 2012
Crazy Cosmological Correlations and How General Relativity Has and Might Continue to ExplainThem, Graduate/Faculty Seminar, Duke University, April 20, 2012
Dark Matter Matters, First Annual Math Slam, Duke University, March 23, 2012
Publications
Alan R. Parry, Wave Dark Matter and Dwarf Spheroidal Galaxies, PhD Thesis, Duke University,Durham, NC (In Preparation, Spring 2013)
Alan R. Parry and Hubert L. Bray, Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies I,(In Preparation, Fall 2012)
Alan R. Parry, A Survey of Spherically Symmetric Spacetimes, (Preprint, October 2012)[arXiv:1210.5269 [gr-qc]] http://arxiv.org/abs/1210.5269
Alan R. Parry, A Classification of Real Indecomposable Solvable Lie Algebras of Small Dimension withCodimension One Nilradicals, Master Thesis, Utah State University, Logan, UT (August 2007)http://gradworks.umi.com/14/48/1448067.html
My research is in differential geometry and geometric analysis. In particular, I am interested in oneof the most natural applications of these topics, the theory of general relativity. General relativity isEinstein’s theory of gravity and currently the dominant theory to explain the large scale structure ofthe universe making it an exciting field full of big questions about some of the most fascinating objectsin the cosmos. Furthermore, general relativity is described using the language of semi-Riemanniangeometry, a beautiful subset of differential geometry that naturally extends the ideas of calculus tomanifolds that are not necessarily flat and succinctly describes their curvature. Thus general relativitylies at one of the most innate and appealing intersections between mathematics and physics.
My current main field of research, and the topic of my PhD thesis, concerns the possibility ofdescribing dark matter, which makes up nearly one-fourth of the energy density of our universe, froma geometrical point of view. This idea is relatively new and has been most recently championed bymy advisor Hubert Bray. To discuss my research and set up the specific problems on which I plan towork, a brief description of some background material about the theory of general relativity and theidea of dark matter as well as some recent work by mathematicians and physicists alike concerningdark matter and galaxies is required. I devote the next three sections to discussing such a background.Section 5 will describe the problem in which I am interested, as well as my previous work and plannedfuture work on the topic.
2 Basic Ideas of General Relativity
As a precursor to general relativity, Einstein published his theory of special relativity in 1905, whichoverturned many of the natural assumptions about the universe that Newton had made in his theoryof gravitation and relativity. Chief among these differences was that special relativity asserted thattime and space were two parts of the same thing and that the speed of light was constant, two ideascompletely foreign to the Newtonian model. This theory was given an elegant mathematical setting byHermann Minkowski in 1906, who described the special relativity spacetime as a differentiable manifold,N , of three spatial dimensions and one time dimension, coupled with a semi-Riemannian metric, g,whose line element is of the form
ds2 = −dt2 + dx2 + dy2 + dz2 (1)
where I have used geometrized units to set the speed of light c = 1. In Minkowski space time, thechange in time of two events is not invariant in all inertial frames as it was assumed to be by Newton,but in fact, the invariant interval is the space time interval ds defined by the line element above. Fromthis fact, that ds is invariant in all inertial frames, one can obtain, among other things, that the speedof light is constant. This metric also splits the set of vectors in the tangent space to N at p, TpN ,into three sets, timelike, spacelike, or null vectors depending on the sign of the dot product of thevector with itself with respect to the inner product induced on each tangent space by the metric in
1
Alan R. Parry
Figure 1: The earth traveling along a geodesic in the curved spacetime generated by the sun. Imagecredit: http://einstein.stanford.edu/.
(1). Finally, this mathematical description of special relativity gave an elegant interpretation of inertialobservers in the spacetime as being those who follow the geodesics, or non-accelerating curves, whosevelocity vectors are timelike.
These ideas were carried over into Einstein’s theory of general relativity, first published in 1916,which generalizes special relativity to apply to non-inertial reference frames as well. In this theory,Einstein removed the requirement that the metric be the Minkowski one in (1). By removing thisrequirement, the set of allowable spacetimes and metrics dramatically increases to include spacetimeswhich are intrinsically curved. This intrinsic curvature was interpreted by Einstein as the presence ofenergy density, whether it be made up of matter, radiation, etc. Thus we obtain a beautiful equationwhich identifies a purely mathematical object describing the curvature of the spacetime with a physicalobject that describes where the energy density lies in the spacetime. This equation is called the Einsteinequation and is given by
G = 8πT (2)
where G = Ric−(R/2)g is the Einstein curvature tensor, consisting of a formula involving the Riccicurvature tensor, Ric, the scalar curvature, R, and the metric, g, and T is the classical stress energytensor from physics. Note that G, Ric, and R are all objects that contain information about the intrinsiccurvature of the spacetime. This equating of energy density and curvature is completed by the conceptthat test particles which follow timelike geodesics are in free fall, that is, they are only acting under theinfluence of gravity. This takes advantage of the natural covariant derivative induced by the Levi-Civitaconnection, whose corresponding non-accelerating curves (i.e. geodesics) are not necessarily straightlines. From a physical point of view, it fundamentally changed the way we understood gravity. Gravitywas no longer a force. Instead, gravity is the phenomenon that free falling objects follow the curvedgeodesics of a curved spacetime. For example, the earth orbits the sun not due to some imaginary cordtethering it to the sun, but instead orbits because it is following a geodesic of the curved spacetimecreated by the sun, that is, the earth is trying to follow a straight line, but since the spacetime aroundit is curved it gets stuck in the dimple caused by the sun and orbits. Figure 1 illustrates this.
This notion of equating energy density with spacetime curvature has been incredibly successfulat predicting and explaining observed phenomena, including many of which were inconsistent with aNewtonian view of gravity. We have already mentioned that it is consistent with the observation thatthe speed of light is constant, but general relativity also explains the observations of (1) time dilationand length contraction for moving reference frames and observers near massive bodies, a concept usedin practice today to sync the clock of a GPS on the ground with that of the clock on the GPS satellite,
2
Alan R. Parry
(2) relativistic precession, which explained the discrepancy in the observed precession of the perihelionof Mercury with the prediction of Newtonian gravity, (3) black holes, (4) the big bang, and mostrecently (5) dark energy, which is responsible for the accelerating expansion of the universe and can bedescribed by a simple modification of the Einstein equation with a cosmological constant[18].
This physical interpretation of the geometry of spacetime has also led to many connections withgeometric analysis. For example, the positive mass theorem, first proved in certain dimensions bySchoen and Yau [20, 21], and the Riemannian Penrose inequality in the case of a single black hole byHuisken and Ilmanen[9] and then in the case of any number of black holes by Bray[3], just to name afew.
The next topic I present is a problem in astrophysics that, while it has been known for severaldecades, it is still not very well understood. That is the problem of dark matter.
3 Dark Matter
While general relativity has been extremely successful at explaining a lot of physical phenomena inthe universe, there does exist a small number of observations which, at first glace, are in apparentcontradiction with the predictions of general relativity. One of these is concerning how fast stars, gas,and dust are orbiting in spiral galaxies. There are two methods of determining the rotational speeds ofthese objects in a spiral galaxy if the galaxy is at the appropriate angle that objects at different radiican be resolved but also tipped enough that objects on the left and right of the center of the galaxyfrom our vantage point are either moving toward us or away from us. Under these conditions, red andblue shift can be used to determine the rotational velocities of objects at different radii. Moreover, dueto the fact that the luminosity at all wavelengths of the galaxy is proportional to the amount of regularmass in a galaxy, one can obtain an approximate mass profile of the regular mass and use Newtonianmechanics, which is what general relativity reduces to on a galactic scale, to compute the rotationalspeed at each radii. We call a plot of the rotational speed at each radii the rotation curve.
These two methods have been performed on many spiral galaxies, see [1] amongst other referenceson this matter. What has been found in every spiral galaxy is that the stars, gas, and dust at distantradii are moving much faster than predicted and that instead of the rotation curve dropping off quicklyat large radii, it tends to remain flat. One of the frames in Figure 2 shows the two rotation curves forthe Andromeda galaxy overlayed on a picture of the galaxy itself.
There are only two possible explanations for such a discrepancy between computation and correctdata. Either the law of gravity used is incorrect on at least the galactic scale and requires an overhaulsimilar to how general relativity overhauled Newtonian mechanics, or there is more matter present inthe galaxy than can be accounted for by the luminosity alone. This kind of matter is called dark mattersince it interacts gravitationally but does not give off any kind of light or interact in any other observedway (e.g. it is collisionless and hence frictionless). Both approaches to this problem have been and arestill being considered, but due to another observation, most astrophysicists favor the solution of darkmatter.
This observation is one of the bullet cluster, an image of which is presented in the remaining framein Figure 2. The bullet cluster is actually two clusters of galaxies (a cluster of galaxies being a groupof many galaxies gravitationally bound together) that have recently collided with one another. In thiscollision, the vast majority of the regular luminous mass, that part which is comprised of gas and dust,was slowed down due to friction in the colliding gas clouds. However, the stars and planets, which aretoo far apart to collide, as well as the dark matter passed through the collision without slowing down.The two separate components have been resolved using gravitational lensing. The pink part cloud isthe gas and dust, while the blue is everything else. The blue portion, however, must be almost entirelycomprised of dark matter since the gravitational lensing effects are far too great to be accounted for by
3
Alan R. Parry
Figure 2: Left: The rotation curves, both observed and calculated, for the Andromeda galaxy.Credit: Queens University. Right: The Bullet Cluster. The pink clouds are where the gasand dust from the galaxy clusters is located, while the blue represents the dark matter. Credit:NASA/CXC/CfA/M.Markevitch et al.; Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.;Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.
the leftover stars and planets that made it out of the collision. Thus the bullet cluster represents anevent where dark matter has literally been stripped from galaxy clusters. This is not something thatwould be expected if the rotational curves could be explained by simply correcting the law of gravityon the galactic scale.
So dark matter appears to exist and, in fact, makes up a significant portion of the energy densityin the universe. Specifically, the energy density of the universe seems to be currently made up of threeconstituents, dark energy, dark matter, and regular or baryonic matter. Dark energy accounts for about72% of the energy density of the universe, while dark matter accounts for 23% and baryonic matteraccounts for only about 4.6% [7]. Baryonic matter is described extremely well in the realm of quantummechanics, while dark energy, on the other hand, only seems to be described adequately using generalrelativity. But no one is quite sure of how to describe dark matter. As such, there is active research todescribe dark matter from both the quantum mechanical and general relativistic perspectives.
There is ample research into considering dark matter as a particle and exploring the consequencesof this idea in the realm of quantum mechanics, see [8, 17] for review articles. Particularly interestingto my research is the attempts to obtain an energy density profile for the dark matter halo around agalaxy that agrees well with observation. Two profiles, one from Navarro, Frenk, and White[13] andanother from Burkert[5], are two such profiles that have been studied well and that I have found useful.
However, another avenue of approach to finding a way to describe dark matter may lie in the fieldof general relativity. In the next section, I introduce some important ideas presented by my advisorHubert Bray in a recent paper on the axioms of general relativity that suggest such an approach.
4 Axioms of General Relativity
In another example of mathematics lending ideas to physics, Hubert Bray recently developed a set ofaxioms upon which one could define general relativity[4]. The axioms also suggest the most naturalgeneralization of vacuum general relativity, which is vacuum general relativity with a cosmologicalconstant. I reprint these axioms here because they are particularly relevant to my work.
Axiom 0 Let N be a smooth spacetime manifold with a smooth metric g of signature (−+ ++) and
4
Alan R. Parry
a smooth connection ∇. In addition, given a fixed coordinate chart, let ∂i, for 0 ≤ i ≤ 3,be the standard basis vector fields in this coordinate chart and define gij = g(∂i, ∂j) andΓijk = g(∇∂i∂j , ∂k). Moreover, let
M = gij and C = Γijk and M ′ = gij,k and C ′ = Γijk,` (3)
be the components of the metric and connection in the coordinate chart and all of their firstderivatives.
Axiom 1 For all coordinate charts Φ : Ω ⊂ N → R4 and open sets U whose closure is compact andin the interior of Ω, (g,∇) is a critical point of the functional
FΦ,U (g,∇) =
∫Φ(U)
QuadM (M ′ ∪M ∪ C ′ ∪ C) dVR4 (4)
with respect to smooth variations of the metric and connection compactly supported in U , forsome fixed quadratic function QuadM with coefficients in M . Note that a quadratic function,QuadY (xα), has the form
QuadY (xα) =∑α,β
Fαβ(Y )xαxβ (5)
for some functions Fαβ of Y .
Fixing the connection as the Levi-Civita connection, vacuum general relativity is recovered fromthese axioms because the Einstein-Hilbert action
∫R dV , where R is again the scalar curvature of the
spacetime, satisfies the requirements of the axioms and because metrics which are critical points of thisaction, with respect to the aforementioned variations, satisfy
G = 0. (6)
Additionally, vacuum general relativity with a cosmological constant, which describes dark energy, isrecovered in the same way from the action
∫R − 2Λ dV , which also satisfies the requirements of the
axioms. That is, metrics which are critical points of this action satisfy
G+ Λg = 0. (7)
Both vacuum general relativity and dark energy can be obtained from these axioms without re-moving the usual assumption that the connection is the Levi-Civita one. Moreover, it is clear thatmany major advances in the theory of gravity have been made by removing assumptions. From New-tonian gravity to special relativity, the assumptions that time intervals were invariant among inertialreference frames and that there could be no maximum velocity were removed. From special relativityto general relativity, the assumption that the metric was flat was removed. The metric is the mostfundamental object of the spacetime. The second most fundamental object is the connection, thoughit is often overlooked because the Levi-Civita connection, which is induced by the metric, is normallyassumed. Thus it seems natural to be curious of what results if the assumption that the connection isthe Levi-Civita connection is removed.
This is exactly one of the questions treated in Bray’s paper, in particular detail in one of theappendices[4]. A connection that is not the Levi-Civita connection is not necessarily metric compatibleor torsion free. While I leave the details to Bray’s paper, ultimately the fully antisymmetric part ofthe torsion tensor can be related to a scalar field f : N → R. In order for the connection and metric
5
Alan R. Parry
to be critical points of the corresponding functional of the type in 4, the scalar field and metric mustsatisfy the following set of equations
G+ Λg = 8πµ0
(2
Υ2df ⊗ df −
(|df |2
Υ2+ f2
)g
)(8)
2gf = Υ2f (9)
where 2g is the Laplacian operator induced by the metric g, µ0 is some constant which can be absorbedinto f if desired, and Υ is a fundamental constant of the system. This system is called the Einstein-Klein-Gordon system of equations and can also be directly obtained from an action, though not strictlyspeaking an action of the type in 4. Specifically this action is
FΦ,U (g, f) =
∫Φ(U)
R− 2Λ− 16πµ0
(|df |2
Υ2+ |f |2
)dV (10)
That is, if g and f are critical points of this functional under compactly supported variations, then gand f satisfy (8) and (9). In my current research, I consider a complex scalar field, f : N → C andthis same action. In this case, the Einstein-Klein-Gordon equations are
G+ Λg = 8πµ0
(df ⊗ df + df ⊗ df
Υ2−
(|df |2
Υ2+ |f |2
)g
)(11)
2gf = Υ2f (12)
which reduces to (8) and (9) if f is real.Vacuum general relativity and vacuum general relativity with a cosmological constant are the most
natural general relativity theories that lie within the scope of the axioms. These two theories describetwo of the most important cases of the universe, that of vacuum and that of dark energy. The nextsimplest theory that is consistent with the axioms is that of a single scalar field as described above. Soit is natural to ask if this could describe something physical as well. In particular, dark matter is thenext most fundamental portion of the universe. Could this scalar field theory describe dark matter?From here on, we will call this theory of dark matter, wave dark matter, due to equation (12) being awave-like equation.
Currently, my research is dominated by this question of wave dark matter. How is it describedgeometrically? What are generic solutions to the Einstein-Klein-Gordon equations in various settings?What are the physical predictions associated with assuming that dark matter is described via a scalarfield? Are these predictions consistent with observation? All of these are big picture questions on whichI hope to shed light in my research career, particularly in the next several years. The next sectiondiscusses some of the known properties of wave dark matter, including some of my recent research onthese questions, as well as the directions I plan to continue to follow in my research.
5 Wave Dark Matter
It is certainly plausible that wave dark matter could describe the geometrical and gravitational effectsof dark matter and in fact scalar fields have been shown to have many of the same properties that areattributed to dark matter particularly on the cosmological scale[12]. On the galactic scale, there aremany questions, but also many hints.
For example, the spiral structure of spiral galaxies like the one depicted in Figure 3 is most oftenattributed to the existence of density waves that rotate through the galaxy pulling gas and dust together,
6
Alan R. Parry
Figure 3: Left: M51-The Whirlpool galaxy. Density waves are thought to be the cause of grand spiralstructures like those found in this photo. Photo Credit: Nasa/ESA. Right: The elliptical galaxy whosegalactic center is the quasar MC2-1635+119. Notice the interleaved waves of in the luminosity. OriginalPhoto Credit: NASA, ESA, and G. Canalizo (University of California, Riverside).
which then forms large stars, which are vibrant but short lived, creating the beautiful spiral arms inits wake[2]. What causes these density waves? It was already known that elliptic potentials createdensity waves and spiral structure and Bray showed that such elliptic potentials can be generatedin the framework of wave dark matter and performed simulations that created spiral galaxies withflat rotation curves[4]. Some examples of his simulations are displayed in Figure 4. More work stillneeds to be done, however, to determine whether or not wave dark matter is consistent with all of theobservations of spiral galaxies.
As shown in Figure 3, elliptical galaxies also exhibit wave-like behavior in their energy density. Theinterleaved waves of luminosity offer a promising possible consistency with wave dark matter, thoughthis is still an open problem being studied by Bray. As with spiral galaxies and density waves, there arealready ideas that might describe these interleaved waves in elliptical galaxies, but none are conclusive.Thus it makes sense to continue to consider every possibility, which, as shown by Bray[4], includeswave dark matter.
Another type of galaxy that is particularly related to my current work is a subset of ellipticalgalaxies called dwarf spheroidal galaxies. These are ideal test beds to start testing a dark mattertheory because they are the smallest galaxies known with a significant amount of dark matter, theyare almost entirely dominated by their dark matter component (more than 99% dark matter in somecases), and they are roughly spherically symmetric. This greatly simplifies the mathematics requiredto describe them. My most recent and current research is in the vein of describing the predictionsof wave dark matter in the setting of a dwarf spheroidal galaxy. In particular, in a joint work withHubert Bray, we were able to use these dwarf spheroidal galaxies to obtain an estimate on what thefundamental constant Υ that appears in the Einstein-Klein-Gordon equations must be in order forwave dark matter to agree with observations.
However, the first part of my research was to discuss how best to describe spherically symmetricdark matter halos, which led me to some results that prove useful in the general setting of any kind ofspherically symmetric spacetime.
5.1 Spherically Symmetric Spacetimes
In a paper I recently placed on the arXiv[15], I presented several different metrics and coordinatesystems that can be used to describe a spherically symmetric spacetime. I also rederived many well
7
Alan R. Parry
Figure 4: Top Row: Photos of spiral galaxies NGC 1300, NGC 4314, NGC 3310, and NGC 488 [14].Bottom Row: Hubert Bray’s wave dark matter simulations on the bottom row (showing gas, dust, andstars, not dark matter)[4].
known facts about spherically symmetric spacetimes by utilizing a set of coordinates and choice of met-ric functions that made such results readily apparent and strikingly simple. In particular, I considereda general spherically symmetric metric in polar-areal coordinates, which can always be written in theform
g = −e2V (t,r) dt2 +
(1− 2M(t, r)
r
)−1
dr2 + r2 dσ2 (13)
where V (t, r) and M(t, r) are smooth functions on the spacetime N , and dσ2 is the standard metric onthe unit sphere. I showed that the function M is the Hawking mass in this spacetime and gave a proofthat the Hawking mass is monotonic in spherical symmetry, a well known result, but one that is madeparticularly simple in these coordinates and the above form of the metric. Next, let νt be the unitvector field in the ∂t direction, then, given the Einstein equation G = 8πT , define the energy densityas µ(t, r) = T (νt, νt). I also showed that the Einstein equation implies that M is the flat integral ofthe energy density. That is,
M(t, r) =
∫Et,r
µ(t, r) dVR3 (14)
where Et,r is the ball of radius r at time t. This suggests that M can be interpreted as the mass of thesystem. Furthermore, let νr be the unit vector field in the ∂r direction and define the pressure of thesystem as P (t, r) = T (νr, νr). In this case, I showed that, in the Newtonian limit where M r andP ≈ 0, the Einstein equation implies that
Vr =M
r2and ∆R3V = 4πµ (15)
which are the defining equations of the Newtonian potential. This shows that in the low field limitthe function V can be interpreted as the gravitational potential function of the spacetime. More
8
Alan R. Parry
importantly, I showed that if the stress energy tensor T is known for all t and r, then solving theequations
Mr = 4πr2µ (16)
Vr =
(1− 2M
r
)−1(Mr2
+ 4πrP
)(17)
at each time t solves the Einstein equation G = 8πT on this spacetime. This fact is remarkable sincethese equations are first order equation, but solve the Einstein equation, which is second order inthe metric. This is due to the fact that while G is generally second order in the metric, in sphericalsymmetry, it is not second order in these components. These results were made particularly useful indescribing wave dark matter in spherical symmetry.
5.2 Spherically Symmetric Wave Dark Matter
Define the stress energy tensor of a spherically symmetric spacetime in the presence of a complex scalarfield, f(t, r), via equation (11) (on the galactic scale we can ignore the cosmological constant Λ). Definethe energy density and pressure functions as in the previous section. In addition, let
p(t, r) = fte−V(
1− 2M
r
)−1/2
(18)
Then equations (11), (12), (16), and (17) and the results above show that solving the system
Mr = 4πr2µ0
(|f |2 +
(1− 2M
r
)|fr|2 + |p|2
Υ2
)(19)
Vr =
(1− 2M
r
)−1(M
r2− 4πrµ0
(|f |2 −
(1− 2M
r
)|fr|2 + |p|2
Υ2
))(20)
ft = peV√
1− 2M
r(21)
pt = eV
(−Υ2f
(1− 2M
r
)−1/2
+2frr
√1− 2M
r
)+ ∂r
(eV fr
√1− 2M
r
)(22)
solves the Einstein-Klein-Gordon equations in spherical symmetry. One major aspect of my futureresearch is concerned with evolving given initial conditions f(0, r) and p(0, r) via the above systemin order to understand the solutions to such a system. In particular, what physical predictions aboutwave dark matter do these solutions make? Do such predictions match observations? To this end, Ihave written Matlab code which solves this system numerically.
However, there is a more natural setting to begin investigating solutions to the above system: astatic spacetime. Very recently, Salucci et al. computed Burkert mass profiles whose predicted velocitydispersion profiles are best fits to the observed velocity dispersion profiles of the eight classical dwarfspheroidal galaxies orbiting the Milky Way[19]. In an upcoming paper that is a joint work betweenmyself and Hubert Bray[16], we compare the mass profiles corresponding to solutions of equations(19)-(22) that yield static metrics, namely those of the form f(t, r) = eiωtF (r) for ω ∈ R and F realvalued, to these best fit Burkert mass profiles to get an estimate of Υ. Solutions of this type are calledstatic states and can be classified by the number of zeros each F has, no zeros corresponding to a groundstate, one zero corresponding to a first excited state, etc. A few examples are these scalar field staticstates are in Figure 5. We found an upper bound for the constant Υ under certain conditions which
9
Alan R. Parry
Figure 5: Plots of static scalar fields (specifically the function F(r)) in the ground state and first,second, and third excited states. Note the number of nodes(zeros) of each function.
seem reasonable on the scale of dwarf spheroidal galaxies, specifically, we found that Υ < 1000 yr−1.Hubert Bray and I have planned another paper on these static spherically symmetric spacetimes forwhich we will attempt to acquire a lower bound on the parameter Υ.
5.3 Future Directions
One immediate observation of our above analysis is that comparing static spacetimes to dwarf spheroidalgalaxies is restrictive since the metric of a dwarf spheroidal galaxy, while in dynamical equilibrium,does not need to be static. Thus, after obtaining lower bounds for Υ by the above procedure, the nextproject in this vein I plan to undertake is to consider various scalar fields, f , which consist of sums of thedifferent static excited states mentioned above as initial conditions and evolve such initial conditionsvia equations (19)-(22). The resulting solutions will not necessarily be static, so the first question toask is what do these solutions look like as they evolve? Furthermore, under what conditions are thesesolutions stable and in what sense? Can regular matter distributions stabilize any of the unstablesolutions? If so, is there a favored ratio of dark matter to regular matter that has the most stabilizingeffects? How does such a ratio compare to the observed ratio of dark matter and matter in ellipticalgalaxies given by the elliptical galaxy version of the Tully-Fisher relation? Note that this version of theTully-Fisher relation relates the luminosity of the elliptical galaxy, which is controlled by the baryonicmatter, to the circular speeds of orbiting objects, which is controlled by the total amount of matter(both dark and baryonic), in the galaxy and is consistent across different elliptical galaxies[6, 10].
Another experiment concerning these sums of static states that I would like to conduct, is to placea set of stars following the matter distribution profile necessary to stabilize the wave dark matter.Requiring these stars to follow orbiting geodesics and evolving the entire spacetime in time will simulatea galaxy with a wave dark matter halo on the scale of a dwarf spheroidal galaxy. With a simulationlike this, we can compute directly the velocity dispersion of the simulated galaxy and compare thequalitative behavior of this predicted velocity dispersion profile to the observed dispersion profilesof dwarf spheroidal galaxies. It may even be possible to parameterize these sums of static states andcompute best fit velocity dispersion profiles of solutions of this type to the observed velocity dispersion,which may yield a better estimate for the value of Υ.
In order to generalize this work on wave dark matter further, we will need to remove the assumptionof spherical symmetry and instead consider axially symmetric and triaxial spacetimes. This is becauseit has been observed that the dark matter halos for spiral galaxies, in particular our own, are notspherically symmetric but triaxial[11]. This presents some difficulty since numerically solving theaxially symmetric or triaxial Einstein-Klein-Gordon equations is nontrivial. This presents an excellentopportunity for collaboration with other mathematicians or physicists who have strong backgroundsin numerical analysis or numerical relativity, as these are fields in which I would like to gain moreexperience.
10
Alan R. Parry
In addition to the research interests I have outlined above, there are several other areas in which Ihave a great deal of interest. I have spent a lot of time recently working with computational techniquesto solve PDEs numerically while constructing the Matlab program I mentioned above that evolves thespherically symmetric Einstein equations. I have found these numerical techniques and how they areapplied to general relativity particularly useful and exciting. In the future, I will continue to work inthis direction to expand my knowledge of numerical relativity techniques, especially in my continuedresearch into wave dark matter. I have also studied fundamental geometric relativity theorems such asthe positive mass theorem and the Penrose inequality for black holes. I find these results in geometricanalysis very interesting and plan to explore these types of topics even more in the future.
Thus, I see my study of wave dark matter, and the corresponding analysis of the Einstein-Klein-Gordon equations, both from a theoretical and computational point of view, as my first big problem atthe intersection of geometric analysis and general relativity. With my advisor’s help, I have successfullybegun my own research program on this topic. However, I also look forward to studying many otherproblems in geometric analysis and general relativity as well.
References
[1] K. G. Begeman, HI Rotation Curves of Spiral Galaxies, Astronomy and Astrophysics 223 (1989), 47–60.
[2] G. Bertin and C.C. Lin, Spiral Structure in Galaxies: A Density Wave Theory, MIT Press, 1996.
[3] Hubert L. Bray, Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem, Journal of DifferentialGeometry 59 (2001), 177–267.
[4] , On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity, arXiv:1004.4016 [astro-ph.GA](2010).
[5] A. Burkert, The Structure of Dark Matter Halos in Dwarf Galaxies, The Astrophysical Journal Letters 447 (1995),no. 1, L25.
[6] Ortwin Gerhard, Andi Kronawitter, R. P. Saglia, and Ralf Bender, Dynamical Family Properties and Dark HaloScaling Relations of Giant Elliptical Galaxies, The Astronomical Journal 121 (2001), no. 4, 1936.
[7] G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N.Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer,G. S. Tucker, and E. L. Wright, Five-Year Wilkinson Microwave Anisotropy Probe Observations: Data Processing,Sky Maps, and Basic Results, The Astrophysical Journal Supplement Series 180 (2009), no. 2, 225.
[8] Dan Hooper and Edward A. Baltz, Strategies for Determining the Nature of Dark Matter, Ann.Rev.Nucl.Part.Sci.58 (2008), 293–314.
[9] G. Huisken and T. Ilmanen, The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality, Journal ofDifferential Geometry 59 (2001), 353–437.
[10] Andi Kronawitter, R. P. Saglia, Ortwin Gerhard, and Ralf Bender, Orbital Structure and Mass Distribution inElliptical Galaxies, Astronomy and Astrophysics Supplement Series 144 (2000), 53–84.
[11] David R. Law, Steven R. Majewski, and Kathryn V. Johnston, Evidence for a Triaxial Milky Way Dark Matter Halofrom the Sagittarius Stellar Tidal Stream, [arXiv:submit/0028100 [astro-ph.GA]] (2009).
[12] Tonatiuh Matos, Alberto Vzquez-Gonzlez, and Juan Magaa, ϕ2 as dark matter, Monthly Notices of the Royal As-tronomical Society 393 (2009), no. 4, 1359–1369.
[13] J. F. Navarro, C. S. Frenk, and S. D. M. White, The Structure of Cold Dark Matter Halos, The Astrophysical Journal462 (1996may), 563, available at arXiv:astro-ph/9508025.
[14] NGC1300 photo credit: Hillary Mathis/NOAO/AURA/NSF. Date: December 24, 2000. Telescope: Kitt Peak NationalObservatorys 2.1-meter telescope. Image created from fifteen images taken in the BVR pass-bands. NGC4314 photocredit: G. Fritz Benedict, Andrew Howell, Inger Jorgensen, David Chapell (University of Texas), Jeffery Kenney (YaleUniversity), and Beverly J. Smith (CASA, University of Colorado), and NASA. Date: February 1996. Telescope:30 inch telescope Prime Focus Camera, McDonald Observatory. NGC3310 photo credit: NASA and The HubbleHeritage Team (STScI/AURA). Acknowledgment: G.R. Meurer and T.M. Heckman (JHU), C. Leitherer, J. Harrisand D. Calzetti (STScI), and M. Sirianni (JHU). Dates: March 1997 and September 2000. Telescope: Hubble WideField Planetary Camera 2. NGC488 photo credit: Johan Knapen and Nik Szymanek. Telescope: Jacobus KapteynTelescope. B, I, and H-alpha bands.
[15] Alan R. Parry, A Survey of Spherically Symmetric Spacetimes, arXiv:1210.5269 [gr-qc] (2012).
[16] Alan R. Parry and Hubert L. Bray, Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies I, in preparation(2012).
[17] J. R. Primack, D. Seckel, and B. Sadoulet, Detection of Cosmic Dark Matter, Ann.Rev.Nucl.Part.Sci. 38 (1988),751–807.
[18] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan,S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J.Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, Observational Evidence from Supernovae for an AcceleratingUniverse and a Cosmological Constant, The Astronomical Journal 116 (September 1998), 1009–1038, available atarXiv:astro-ph/9805201.
[19] Paolo Salucci, Mark I. Wilkinson, Matthew G. Walker, Gerard F. Gilmore, Eva K. Grebel, Andreas Koch, ChristianeFrigerio Martins, and Rosemary F. G. Wyse, Dwarf Spheroidal Galaxy Kinematics and Spiral Galaxy Scaling Laws,Monthly Notices of the Royal Astronomical Society 420 (2012), no. 3, 2034–2041.
[20] Richard Schoen and Shing-Tung Yau, On the Positive Mass Conjecture in General Relativity, Commun. Math. Phys.65 (1979), no. 1, 45–76.
[21] , Proof of the Positive Mass Theorem. II, Commun. Math. Phys. 79 (1981), no. 2, 231–260.
I have found teaching to be one of the most rewarding experiences I have had and I have takenevery opportunity in my professional career to teach and even sought more opportunities than isnormal for someone in my stage of professional development. These opportunities include, tutoringas an undergraduate, being a lab instructor, and independently teaching a total of 12 courses as agraduate student during my time at both Duke University and Utah State University, which rangedover the topics of intermediate and college algebra and several types of first year calculus courses. Ihave also been a member of the Duke University mathematics department calculus committee, whichdiscusses and makes changes to the first year calculus curriculum at Duke, since January 2010.
I have been very successful as a teacher and have been recognized by my peers for my work.In 2009, I received the L.P. and Barbara Smith Award for Beginning Teachers and in the followingyear, I received the L.P. and Barbara Smith Award for Teaching Excellence, which is a prestigiousaward given annually usually to one or two graduate students in the mathematics department at Duke,see http://www.math.duke.edu/first_year/lpsmith.html for more information. Additionally, Igenerally receive excellent student evaluations from the courses I teach.
In the Fall of 2010, I was also selected to teach a first year calculus course at Duke while itwas still being developed, which was designed for stronger students and incorporated the use of acomputer algebra system to discuss problems more deeply. I participated on a team consisting ofthe three instructors of the different sections, two graduate students and one senior faculty member,whose purpose was to figure out how best to incorporate the computer algebra system and evaluatewhat aspects of the original design of the course worked and what aspects could be modified. Severalsections of the course is now being taught regularly in the fall semesters in the mathematics departmentat Duke and has become a success.
The remainder of this statement will discuss the specifics of my teaching philosophy, which hashelped me become a successful teacher.
1 Teaching Philosophy
My teaching philosophy is centered on three key ideas which I believe a teacher must implement inorder to be successful. First, he must be chiefly motivated by genuine care and concern for his students’welfare and success in the course. Second, he must foster an enjoyable atmosphere to engender studentinvestment in the course. Finally, he must acquire and utilize certain skills and methods that takeadvantage of that student investment and motivation in order to facilitate the student achieving thedesired course goals. I will discuss each of these aspects separately.
The motivation of the teacher is key to effective teaching. There are many things that can motivatea teacher, but if the motivation to teach is centered on the wrong reasons, the instructional experiencecan be significantly poorer. For example, if a teacher’s only reason to be in the classroom is becausehis department requires him to do so every so often, he will be less likely to invest a lot of time andeffort in the course, inevitably creating a sub-par experience for him and his students. On the otherhand, if a teacher has genuine concern for his students and their success, he will be more willing and
more likely to put forth the extra time to help his students outside of the class room, to look for moreinnovative ways to teach the material, and to think reflectively on how he might improve each time heteaches. Students can tell when a teacher invests in them and it, in turn, makes them want to investin that teacher’s course.
I have experienced this myself. There was a semester during my graduate studies where I became sofocused on completing my research goals at the time, that I was more motivated by my time constraintsthan about helping my students. My students complained about the course more and did more poorlythan my students had in the past. The following semester, though I still had the same research demandson my time, I thought more about my students and worked hard to make myself more available forthem and took advantage of other time-saving resources so that I could invest more in the class overallbut still have time for research. My students performed much better this time around and were muchmore pleased with my instruction.
In the same way that a teacher’s level of investment is determined by what kind of motivation hehas, a student’s investment level is also governed by his motivation. In introductory math courses, thisis a particular problem especially since math is not most students’ idea of fun. And in my experience,the class a student enjoys the most is the one into which he invests the most time and energy. As such,I strive in my classroom to foster an enjoyable environment where students can have fun and at thesame time learn something.
In order to create such an atmosphere, I try to be humorous and tell an occasional related story.Humor throughout a lecture is an attention inducer and helps the students enjoy and be themselves.This lessens the pressure on the student and allows the student to perform better. Other ways I createan enjoyable atmosphere is to use examples that draw on the experiences of the students, currentevents, or even my current research, and also to engage the class in a discussion of how to proceedfrom one step to another in the math lecture. This participation increases the students’ interest in thematerial as well as helps them learn how to tackle many different problems.
Once the teacher has such an atmosphere, student motivation begins to increase automatically andthey tend to invest more in class because they are happy to be there. The students gain a connectionwith the teacher that makes it easier to facilitate learning. When I have a student come up to me andtell me that my class (their math class) is their favorite class that semester, I know I am doing my jobto motivate them.
Finally, once both the instructor and student are properly motivated, then they are ready to tacklethe course goals. In college level math courses, the general goal of any course is for students to becomeproficient at not just recalling, understanding, and performing mathematical methods, but also to beable to analyze a more complex problem and break it down into smaller problems they can solve andput it back together. Especially in teaching introductory college math courses, one of the most usefulmethods to achieve this goal is to simply be capable of presenting topics in a variety of ways or withdifferent examples especially since the ways that students think can be extremely varied. I have beenworking on how to do this since I was an undergraduate tutor. Whenever I prepare to teach, I spendsome time determining different ways to discuss a topic and also borrow new ideas from colleagues. Thisalso naturally carries over to what kinds of assignments and practice I assign. Many of my colleagueshave had a lot of good ideas and by utilizing them, I have been able to reach a wider variety of studentsthan if I restricted myself to teaching only the way I learned or approached the material.
To conclude, I quote Parker Palmer, “The connections made by good teachers are held not in theirmethods but in their hearts” (Palmer, The Courage to Teach, 2007, p. 11). Methods are good andhelpful, but if the correct motivations of both teacher and student are not present, learning is muchmore difficult. The teacher must care for the student and create an atmosphere that would engendera student to care to be a part of the class. Once that connection is made, the learning part is mucheasier and much more natural.
2
Alan R. Parry
2 Student Evaluations of Teaching
The purpose of this part of this document is simply to highlight some of the responses I have receivedon my student evaluations of teaching forms (at Duke, these are called teacher course evaluations). Ialso give some statistical information at the end. For reference, Math 25L, Math 31L, and Math 41Lare all different versions of first year calculus courses at Duke University.
2.1 Student Evaluation Responses
Spring 2010 - Math 25L Course “Parry is one of the best profs I have had at Duke. He is verywilling the help anytime, answers emails quickly, and always encourages hard work. I have neverhad a prof so willing to help whether periodically through the semester or right before an exam.... Parry was always available. He stayed after class many times to help with extra problems andto answer questions.”
Spring 2011 - Math 31L Course “The instructor was one of the best math teachers I have had sofar in my schooling and I come from a strong math/science Governor’s school (Magnet school).Wonderful job at teaching. ... Prof. Parry will be recommended by me to others.”
Spring 2010 - Math 25L Course “The instructor is very enthusiastic and want you to do well. Hereally teaches [and] explains the subject well. I am very glad I had him as a teacher for math.He is an asset to the math department.”
Spring 2011 - Math 31L Course “He makes the concepts easy to understand and is available forhelp and questions.”
Spring 2010 - Math 25L Course “The instructor was really great at explaining concepts so thatthe class really understood everything.”
Fall 2009 - Math 25L Course “The student/instructor interaction was very comfortable, in classparticipation was encouraged.”
Fall 2009 - Math 25L Course “Great instructor, couldn’t have been more accessible or helpful. ...GREAT TEACHER”
Summer 2010 - Math 25L Course “Instructor was very enthusiastic and helpful.”
3
Alan R. Parry
2.2 Student Evaluation Statistics
The following is a collection of some of the questions asked on our teacher course evaluation formsat Duke and the average score I received in each class. All scores are on a scale of 1 (very poor) to5 (excellent). The number of students refers to the number of evaluations collected. The columns areabbreviated as follows
QI - Quality of instruction
E - Instructor was enthusiastic about the course.
A - Instructor was accessible outside of class.
IP - Participation in class discussion was encouraged.
Course # of Students QI E A IP
Fall 2008 - Math 25L 22 3.91 4.73 3.90 4.10Summer 2009 - Math 25L 9 4.25 4.89 4.56 4.00Fall 2009 - Math 25L 12 4.25 4.75 4.42 4.33Spring 2010 - Math 25L 11 4.45 4.91 3.88 4.09Summer 2010 - Math 25L 6 4.33 4.83 4.17 4.50Fall 2010 - Math 41L 18 3.39 4.50 3.22 3.44Spring 2011 - Math 31L 18 4.41 4.76 4.41 4.35
The purpose of this letter is to very highly recommend Alan Parry for a postdoctoral positionin your department. Alan will receive his Ph.D. from Duke University under my supervisionin 2013. Alan’s work is very exciting and is at the intersection of
• geometric analysis
• general relativity
• computational relativity
• applications to astrophysics
What makes Alan very exciting and a bit unusual is that in addition to proving theoremsin geometric analysis and relativity, he is also at the forefront of connecting ideas in math-ematical relativity to actual observations about the universe. In other words, he knows themathematics and theory of the universe on the large scale (geometric analysis and generalrelativity) and is making connections to the observations and physics of the universe on thelarge scale (astronomy and astrophysics).
Alan Parry’s thesis projects are, in large part, motivated by my paper on the arXiv athttp://arxiv.org/abs/1004.4016 entitled “On Dark Matter, Spiral Galaxies, and the Axiomsof General Relativity.” The paper begins by using a geometric analytic motivation topropose axioms which imply a generalization of classical general relativity which results intwo matter fields: The first is the cosmological constant (aka dark energy). The second is ascalar matter field satisfying the Einstein-Klein-Gordon equations. Given that dark energyand dark matter are the two main mysterious matter fields in the universe and the stronggeometric analytic motivation for this second scalar matter field, it then seems natural toask, “Could this scalar field represent dark matter?” Since this matter field satisfies a waveequation, we have come to call this scalar field dark matter model “wave dark matter.” Thepaper then studies the predictions of this wave dark matter model and points out how itcould cause spiral patterns in galaxies. Galaxies are good tests for dark matter theoriesbecause most of the mass of galaxies is dark matter. Furthermore, as seen in Figure 1,the simulations of spiral galaxies based on this geometric dark matter theory are strikinglysimilar to photos taken of actual galaxies. While this does not prove that this is the correctmodel of dark matter, this model of dark matter is consistent with observations, has some
interesting successes, and hence is a viable theory of dark matter that needs to be studiedthoroughly. Alan Parry is at the forefront of this study.
Figure 1: The similarity of the photos of spiral galaxies on the top row to the simulationsof wave dark matter on the bottom row (showing gas, dust, and stars, not dark matter)motivate studying this model of dark matter further, as Alan Parry is doing.
Alan Parry has been my “right hand man” for the last three years and has been instrumentalin helping me study the partial differential equations at the foundation of this wave darkmatter theory, namely the Einstein-Klein-Gordon equations with a cosmological constantand regular baryonic matter, shown below:
G+ Λg = 8π
df ⊗ df + df ⊗ df
Υ2−
(|df |2
Υ2+ |f |2
)g
+ 8πTB (1)
f = Υ2f (2)
In the above equations, the speed of light and the universal gravitational constant havebeen set to one, G is the Einstein curvature tensor, is the wave operator (the divergenceof the gradient) of the Lorentzian metric g, f is the scalar field which we suggest causesthe curvature of spacetime attributed to dark matter via the right hand side of equation1, TB is the contribution to the stress-energy tensor due to regular baryonic matter (theperiodic table of elements) which is mostly visible, Λ is the cosmological constant which isthe standard model for the observed accelerating expansion of the universe (but may beignored on the scale of galaxies), and Υ is a new constant of nature which is fundamentalto the dynamics of f .
Here are some of Alan Parry’s accomplishments to date:
2
1. Alan has become an expert on spherically symmetric 3+1 dimensional spacetimesand wrote a beautiful paper surveying this subject which is available on the arXivat http://arxiv.org/abs/1210.5269.
2. Alan has written programs to compute spherically symmetric static spacetime solutionsto the Einstein-Klein-Gordon equations.
3. Alan has created a simulator to study the dynamics of spherically symmetric solutionsof the Einstein-Klein-Gordon equations (the spacetime metric and the scalar field arefunctions of t and r).
4. Alan has become knowledgeable about dwarf spheroidal galaxies, the smallest knownclumps of dark matter in the universe.
5. Alan has used his computed solutions to the Einstein-Klein-Gordon equations to findbest fits of these models to dwarf spheroidal galaxies.
6. In doing so, Alan has made some of the first estimates of this new fundamental constantof nature Υ predicted by the wave dark matter model, to appear on the arXiv in thenear future as a joint paper with me.
Important questions on which Alan is currently working fall into the general categories of:
1. Find special solutions to the Einstein-Klein-Gordon equations which model observa-tions as well as possible.
2. Understand the stability characteristics of these solutions to make sure they are phys-ically relevant.
Stability is a very important question. For example, a pencil standing vertically on its tipis a solution of Newtonian physics, but any perturbations of its angle from being perfectlyvertical, no matter how small, will cause the solution to change dramatically (the pencilwill fall). Similarly, solutions to the Einstein-Klein-Gordon equations which are not stableare not physically relevant since the odds they would ever occur are zero.
These stability questions are very subtle and highly mathematical. In addition, they willrequire modeling the regular matter correctly since the regular matter also affects solu-tions and generally will be a stabilizing factor. Alan’s spacetime simulator was written inlarge part to study these stability questions. More generally, a combination of simulationsand proofs using geometric p.d.e. techniques are needed to study solutions to the Einstein-Klein-Gordon equations. The simulations are needed to “see what happens” typically sinceunexpected phenomena could occur that are hard to anticipate. Once we know what hap-pens typically, and once it is clear what is special about these solutions (such as being staticor stationary, etc.), then one can approach the problem of proving theorems that rigorouslycharacterize the special properties of these solutions. This is Alan’s (and my) overall gameplan.
Alan has already presented his work to other expects in geometric relativity, Pengzi Miaoat the University of Miami and Justin Corvino at Lafayette College, both of whom are also
3
writing letters about Alan’s work. At Lafayette, I understand that Alan gave two lectures,one on his research and one a colloquium for a general audience with a good attendanceand questions from undergraduates. Alan’s presentation at the University of Miami wasalso a success and had at least one astrophysicist in the audience, who, after hearing histalk, invited Alan to participate in the “Miami 2012” conference on elementary particles,astrophysics, and cosmology sponsored by the Department of Physics at the University ofMiami. Hence, by the end of the year Alan will have given invited presentations to 2 outsidemath audiences and 1 outside astrophysics audience. This demonstrates the appeal thatAlan’s work has across disciplines.
Alan is just as passionate about his work as I am about mine. His thesis projects are muchharder than what I would typically give to a graduate student and he has learned moreas well. This is all a result of Alan’s passion and desire to be challenged with worthwhileproblems that have a chance to change the way we understand the world. Among mycurrent 4 graduate students, all of whom are very good, he is the leader who inspires theothers to work hard and to strive to achieve their maximum potential.
In addition to his exciting research, Alan is also a great teacher. He has won two (!) teachingawards at Duke, the first student that I have known to do that. I also take teaching seriouslyand generally get good evaluations, but when I talk about teaching with Alan I always learnnew ideas. In fact, as exciting as Alan’s research is, his first love is teaching, and it shows.
In summary, Alan would be a great catch for any department looking for a postdoc who willmake an immediate impact with his wide range of interests, exciting research, hard workingattitude, and great teaching. He is much broader than the typical postdoc, which makeshim different, but in a good way. Inviting Alan to join your department will allow others inyour department to learn more about geometric analysis, general relativity, computationalrelativity, and astrophysics, and even more interestingly, how these fields connect. Plus,Alan is a superb teacher. Add it all up and you will be glad you hired him.
Please contact me if there is any more information that I may provide.
Sincerely,
Hubert L. BrayProfessor of Mathematics and PhysicsDuke University
4
October 31, 2012 To the Hiring Committee: I am writing this letter in support of Alan Parry, who is applying for a job in your department. Alan has done and continues to do very interesting work involving differential geometry and geometric analysis as applied to general relativity and astrophysics. He cares deeply about his research, about communicating his research, and in being a careful, dedicated teacher. Alan would be an ideal postdoctoral or tenure-‐track candidate in mathematics, applied mathematics, or physics. Last summer I co-‐organized a Summer Graduate Workshop on Mathematical Relativity at the Mathematical Sciences Research Institute in Berkeley, CA. I asked Alan’s Ph.D. advisor Hugh Bray if he thought Alan would be a suitable TA for the program. Hugh responded enthusiastically! Alan proved to be very helpful to many students, some of whom needed extra help with some background material. In the second week of the program, Alan presented some of his work and that of his advisor on the axioms of general relativity, and the presentation generated requests for the slides of his talk. Alan got along really well with many of the students in the program, providing the organizers a link to how the students were finding the pace and motivation of the topics presented. We were very happy to have had Alan as a TA. Alan’s advisor will speak more in detail about Alan’s work. Bray’s formulation of the axioms for general relativity, leading to a study of the Einstein-‐Klein-‐Gordon equation and modeling of spiral galaxies, is an intriguing starting point for Alan’s investigations. Alan’s work involves analysis and numerical solution of the differential equations which result under a spherical symmetry ansatz, and comparing the results with physical data. The results are interesting first steps in trying to model dark matter and determine what would be a new fundamental constant, and at this point, there are more interesting questions than answers, so Alan has plenty of work ahead of him in the next few years. We invited Alan to Lafayette College to give a special joint Mathematics-‐Physics Colloquium. He also volunteered to give a student-‐focused lunchtime talk on general relativity in the Mathematics Department. In almost a decade at Lafayette, I think Alan’s lunch talk was the most well attended of any I can recall in the series. The positive comments after the talk, from students to faculty, indicated that he had hit just the right level in introducing a very tough subject. Alan spent lunch with physics faculty and students, and then spent some time conversing with
Easton, Pennsylvania 18042-1773 TEL 610-330-5267 FAX 610-330-5721 www.lafayette.edu
Department of Mathematics
an astrophysicist at Lafayette, who is trying to measure gravitational waves using pulsars. The afternoon colloquium was also well attended, and he presented some of his research. Alan got some very good questions from the audience (mostly physicists and physics students), did a nice job answering them, and followed up to the questioners about whether he had satisfactorily addressed their concerns. Alan was very comfortable discussing his work with both mathematicians and with physicists, which will be crucial in his career-‐-‐-‐he will have to communicate with both camps in order to get his work published in the right journals, and to have his work funded. Alan is off to a great start so far. In summary, I recommend Alan Parry for a postdoctoral or tenure-‐track position. I spoke at some length with him in Berkeley and Lafayette, and I know he is a serious researcher and a dedicated teacher. I look forward to seeing the progress he will make on the very interesting mathematical models he is studying. Sincerely, Justin Corvino Associate Professor of Mathematics
