aa r X i v : . [ m a t h . HO ] J un THE ROOTS OF MATHEMATICAL THOUGHT
HARRY TAMVAKIS
Six years ago, I wrote the opinion piece [T2], titled ‘Mathematics is a quest fortruth’. My aim in this sequel is to expand upon the sentiment expressed there that‘the roots of mathematical thought lie within the deepest recesses of the humanmind, where logos and mythos come together, in search of our very nature’. I willexamine the moment when new mathematics is created, which is also a moment ofdiscovery and revelation. There is a certain mystical quality to this event, whichis to a large extent very personal. In conclusion, I will discuss what I believemathematical research can tell us about ourselves, and our role in the world.I agree with Dieudonn´e [D, II.6] and Hardy [H] that the main reason whichcompels us to do research in mathematics is intellectual curiosity, the attractionof enigmas, the desire to know the truth. Note however that neither [T2] nor thepresent paper claim that anything mathematical is actually true.To give credence to my thoughts, I have to draw upon my own experience, whichinevitably entails some self-promotion. I apologize in advance for the latter, andwill have something more to say about that at the end. I feel compelled to writebecause I have not seen this topic discussed in the same way before, despite whatI strongly suspect, that I am not alone. Eleven years ago, I solved my favoriteproblem in all of mathematics, in a manner so effortless that I did not realize thefull significance of the event until much later. The story of how that happened willaccompany the more important points to be made along the way.I was reading books by my second birthday, and exhibited curiosity about allsorts of things. Around that time, my family moved to the United States frommy native country of Greece, so I had two languages to play with. An incidentthat affected me occurred in fourth grade elementary school, in Skokie, Illinois,when our teacher taught us about area, and showed how to derive the formula πR for the area of a circle. I remember the proof to this day. He cut the circleinto two semicircles, and divided each of them using radii into a large and equalnumber of triangular wedges. The two halves of the circle unfold along these linesby straightening their outer rim, like a sliced orange, and fit together to form an(approximate) rectangle whose side lengths are πR and R , respectively.I noticed that there was one shape our teacher had mentioned without givingus a formula for its area: the ‘oval’. When I asked ‘What is the area of an oval?’,he said it requires calculus, and I would have to wait until college, or perhaps latein high school, to learn that. This led me on a quest to teach myself calculus andfind the answer, which lasted for years. The plot thickened when we moved backto Greece a few years later, as I did not know the translation of Latin terms like‘calculus’ and ‘oval’ into Greek. The few books I managed to gather on the subject Date : June 12, 2020. were not very helpful, and I never understood the precise definition of a limit untilit was taught to me in class, during my last year of high school.In fact, a proof of the formula for the area of an ellipse does not require calculus.An ellipse with semiaxes of lengths a and b with a < b can be defined as theintersection of a cylinder, whose horizontal cross section is a circle C of radius a ,with a slanted plane. The ellipse is obtained by dividing C into two halves andstretching each half in the vertical direction by a factor of b/a . The area of theellipse is thus b/a times the area πa of C , or πab . This argument using similartriangles is not beyond the comprehension of a curious fourth grader. However Iam glad that my teacher did not tell me this, because the journey I embarked onto discover the answer was much more valuable than the destination.In secondary school, I found the answers that mathematics gave to the key ques-tion of ‘why?’ more satisfying than those offered in my science classes. Althoughmathematics and science are close relatives and cross-pollinate each other, thereremains a fundamental distinction between them, in that scientific validity dependsupon agreement with experiments, whereas mathematical facts rely on logicallyrigorous proofs from abstract first principles. As such, the latter are among thevery few things that (so far) survive the test of time.This proximity to truth and beauty was a major part of the attraction of math-ematics for me. Another was my participation and success in math competitions,which exposed me to challenging problems. In downtown Athens, there were weeklyinformal training sessions for mathematical olympiads, organized by the HellenicMathematical Society, and I decided to go have a look. The very first problem Ifaced upon my arrival there was the following. A school has 1000 students. Each student has a locker, and these are numberedfrom 1 to 1000. The first student opens all the lockers. The second student closesevery second locker. The third student goes to every third locker, closing thoselockers which are open, and opening those which are closed. This continues in thesame manner, with the n th student going to every locker whose number is a multipleof n , and opening/closing those which are closed/open, for each n ≤ . Howmany lockers will be open at the end of this procedure? After overcoming my initial reaction, here is how I solved this: Consider a fixedlocker, say locker number 12. The students touching this locker are numbers 1,2, 3, 4, 6, and 12, which are the divisors of 12. The locker remains closed at theend because 12 has an even number of divisors. Observe that the divisors of anypositive integer n can be organized in pairs of the form ( d, n/d ), such as the pairs(1 , , ,
4) in our example. Therefore n will have an even number ofdivisors unless there is a divisor d of n such that d = n/d , that is, unless n = d isa square. Hence the lockers numbered 1 , , , , , . . . will remain open, and, sincethere are 31 squares less than 1000, the answer to the problem is 31.This delightful conundrum, whose solution requires only basic arithmetic, hastwo important lessons to teach us. The first is the utility of working on a specificexample, and then trying to generalize from there. The second lies deeper. Whenconfronted with the question, one’s first impression is that of a long line of lockers,some of them changing their state with every passing student. It is a picture ofdizzying complexity, similar to the famous sieve of Eratosthenes, which detects theprimes. In order to solve the problem, one needs to concentrate on a fixed locker HE ROOTS OF MATHEMATICAL THOUGHT 3 and carefully analyze what happens to it. This involves changing your point ofview, from a global/horizontal to a local/vertical one (see the figure below). ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ● ○ ● ○ ● ○ ● ○ ● ○ ● ○ ● ○ ● ○ ● ● ● ○ ○ ○ ● ● ● ○ ○ ○ ● ● ● ○ ● ● ○ ○ ○ ○ ○ ● ● ○ ● ○ ● ● ○ ○ ● ● ○ ● ○ ○ ○ ● ○ ○ ● ○ ● ○ ○ ○ ● ● ○ ● ● ○ ○ ● ○ ○ ○ ○ ● ○ ○ ○ ● ● ○ ● ● ● ○ ● ○ ○ ○ ○ ○ ○ ○ ○ ● ● ○ ● ● ● ● ● ○ ○ ○ ○ ○ ○ ● ○ ● ● ○ ● ● ● ● ○ ○ ○ ○ ○ ○ ○ ● ○ ● ● ○ ● ● ● ● ○ ● ○ ○ ○ ○ ○ ● ○ ● ● ○ ● ● ● ● ○ ● ● ○ ○ ○ ○ ● ○ ● ● ○ ● ● ● ● ○ ● ● ● ○ ○ ○ ● The importance of a new perspective shedding light in an area of mathematicswill be a theme of this narrative. The examples are myriad, at every level of thesubject, for instance the introduction of variable quantities and instantaneous ratesof change, which turned the ancient problem of the computation of areas from astatic to a dynamic one, and led to the fundamental theorem of calculus.My undergraduate study at the University of Athens provided me with a solidbackground in analysis, which impressed my teachers during my first year of grad-uate school at the University of Chicago. It came as a surprise to me that five yearslater, I was writing my thesis in arithmetic algebraic geometry. My favorite sub-ject was complex analysis, and under the direction of my first advisor Narasimhan,I went from the study of Riemann surfaces and Hodge theory to Grothendieck’stheory of schemes and intersection theory in algebraic geometry. Narasimhan thensuggested that I have a look at Gillet and Soul´e’s arithmetic intersection theory, ageneralization of Arakelov theory to higher dimensions, which added number the-ory to the mix. Since the glue that held all this together was intersection theory, itmade sense after my second topic exam to switch advisors to Fulton, who was alsoat Chicago, and whose brilliant book on the subject had kept me going.One of the main difficulties with higher dimensional Arakelov theory is a lackof examples where explicit computations are possible. Maillot had succeeded incomputing arithmetic intersection numbers on Grassmannians, and I decided to tryto extend his work to more general flag manifolds. This involved moving beyondthe case of hermitian symmetric spaces, which have a canonical theory of harmonicdifferential forms, and also learning a fair amount of combinatorics.The advice I received from Fulton was to first try doing some small examples.This is certainly the way one should begin any such project, however I found tomy frustration that I was unable to compute even one new example using theknown methods. Instead, what was required was a different perspective: my Ph.D.dissertation was one of the first works to use representation theory in Arakelovgeometry, and led the way to a new arithmetic Schubert calculus . During the long
HARRY TAMVAKIS road to its completion, I came to appreciate all that mathematics has to offer, andthe beauty found in areas that seem far from one’s initial interests.I remember the moment when I realized that I could solve my thesis problem.I was alone in my living room, sitting on the couch, when my mind turned again,as it had so many times before, to the calculation I sought to achieve. And then,in a sudden flash of inspiration, I saw how I could do it. The solution combinedin a novel way ingredients that I had developed gradually over the previous years,and proved that all the natural intersection numbers were rational numbers. Onthat occasion, I experienced the ineffable, incomparable feeling of exciting newmathematics taking shape right before (or rather, behind) my eyes.The following excerpt from a famous poem by Dionysios Solomos describes inthe best way that I know the exhilaration that the researchers of the unknownexperience at the moment of mathematical discovery. Mother, magnanimous in suffering and glory,Though your children forever live in hidden mysteryIn reflection and in dream, what grace have these eyes,These very eyes, to behold thee in the deserted forest . . .
The ‘Mother’ is the ontological mother – the mother that gives birth to us all.She is the truth – no matter how defined – that mathematicians seek in theirresearch endeavors. Her children live in hidden mystery, alone and in the dark,searching for a pathway to that truth. We have two essential tools at our disposal:deductive reasoning (reflection) and guesses or conjectures, which are rooted inhuman intuition and fantasy (dreams). Much of the work happens subconsciously,over long periods of time. And when the moment of enlightenment finally comes,one is left speechless in admiration and awe. The only feeling then is one of wonder,joy, and gratitude for the gift that was bestowed upon you, to be present there withyour mind’s eye open, at the very moment when the Mother reveals a bit more ofher light. We have here an essentially otherworldly experience, as the researcherunderstands that he or she is more the recipient or channel of knowledge, ratherthan its creator. No other area of inquiry illustrates in such a profound way theillusion of human agency: we apparently have original mathematical ideas, but theconclusions we obtain using them could not have been otherwise.There is ample evidence that new mathematics is discovered, often independentlyby different people at different times, and is a revelation to us. A striking example ofthis occurred in my first joint paper with Kresch on the arithmetic Grassmannian.For such spaces, Gillet and Soul´e formulated arithmetic analogues of Grothendieck’sdifficult standard conjectures on algebraic cycles.Kresch and I used arithmetic Schubert calculus to prove these conjectures for theGrassmannian of lines in projective space [KT]. In the course of our work, we wereled to make a conjecture of our own about a certain family of
Racah polynomials .This indirect and miraculous connection between the problem we set out to solve,the theory of hypergeometric orthogonal polynomials, and the Racah coefficientsor 6- j symbols in quantum mechanics came as a complete surprise. I am indebted to the Greek philosopher Christos Malevitsis, who loved this passage anddiscussed its significance in his writings. The translation and interpretation here are my own.
HE ROOTS OF MATHEMATICAL THOUGHT 5
To state our conjecture, let k , n , and T be integers, and define(1) R ( k, n, T ) := min( k,n ) X i =0 ( − i (cid:18) ki (cid:19)(cid:18) k + ii (cid:19)(cid:18) ni (cid:19)(cid:18) n + ii (cid:19)(cid:18) T − i (cid:19) − (cid:18) T + ii (cid:19) − . Conjecture 1.
For any integers k, n, T with ≤ k, n ≤ T − , we have − ≤ R ( k, n, T ) ≤ . We found much computer evidence in support of Conjecture 1, and proved itwhen n ≤ n = T −
1. The latter case is interesting: although one can show,using the Wilf-Zeilberger method, that in general there is no ‘closed form’ for thesum defining R ( k, n, T ), when n = T − R ( k, T − , T ) = k X i =0 ( − i TT + i (cid:18) ki (cid:19)(cid:18) k + ii (cid:19) = (1 − T )(2 − T ) · · · ( k − T )(1 + T )(2 + T ) · · · ( k + T ) . The second equality in (2) is a special case of an identity proved by Pfaff, 220 yearsago, and (independently!) by Saalsch¨utz, 130 years ago.The most vivid evidence for Conjecture 1 is the following picture, which waskindly provided to us by Wilf. In the figure, we let T := 51 and plot the values R ( k, n,
51) on the lattice points 0 ≤ k, n ≤
50, then connect the resulting dotsby line segments. In fact, the bound | R ( k, n, | ≤ k and n aretaken to be real parameters lying in [0 , integrality of k and n .
10 20 30 40 50 10 20 30 40 50-1-0.500.51 10 20 30 40
HARRY TAMVAKIS
Conjecture 1, which surprised Askey, remains open as of this writing. Fortunatelyfor Kresch and me, the arithmetic Hodge index conjecture for the GrassmannianG(2 , N ) is equivalent to the weaker inequality T − X k =1 ( − k +1 R ( k, n, T ) H k < T − X k =1 H k which was within our reach. Here H k := 1 + + · · · + k is a harmonic number .My research on the Arakelov theory of Grassmannians and flag manifolds wasfollowed by further joint papers on their quantum cohomology rings. These exoticintersection theories are both deformations of the classical Schubert calculus. Thelatter is the study of the usual cohomology ring of the same spaces, and an old andrich subject. The work of twentieth century mathematicians provided algorithmsto do computations in Schubert calculus, but I realized that there remained fun-damental gaps in our understanding. A perplexing mystery beyond the Lie type Awas the problem of representing polynomials , to which we turn next.We begin with the example of the Grassmannian G( m, n ), consisting of all m -dimensional C -linear subspaces of C n . This space has a natural decomposition(3) G( m, n ) = [ I X I into Schubert cells X I , one for each subset I := { i , · · · , i m } of { , . . . , n } with | I | = m . Every subspace V in G( m, n ) can be represented uniquely by an m × n matrix A in reduced row echelon form with row space V . We have V ∈ X I if andonly if the pivot ‘1’s in A lie in columns i , . . . , i m . The Schubert class [ X I ] is thecohomology class of the closure of X I . The cell decomposition (3) implies that theSchubert classes form an additive basis for the cohomology group of G( m, n ):H ∗ (G( m, n ) , Z ) = M I Z [ X I ] . To understand the ring structure of H ∗ (G( m, n ) , Z ), it is better to parametrizethe Schubert classes by integer partitions λ = ( λ ≥ · · · ≥ λ m ) with λ ≤ n − m rather than by subsets I . The formula λ r := n − m + r − i r for 1 ≤ r ≤ m gives abijection between these two parameter spaces. If I corresponds to λ , we denote theSchubert class [ X I ] by σ λ . For every integer p ∈ [1 , n − m ], the class c p := σ ( p, ,..., is known as a special Schubert class . Giambelli [G] was able to express a generalSchubert class σ λ as a polynomial in the special classes c , . . . , c n − m :(4) σ λ = det( c λ i + j − i ) ≤ i,j ≤ m . In the Giambelli formula (4), and in later equations, we set c := 1 and c p := 0if p / ∈ [0 , n − m ]. The polynomials on the right hand side of (4) are algebraicrepresentatives for the Schubert classes, and lead naturally to an algebraic modelfor the cohomology ring of Grassmannians.In fact, the determinant in equation (4) had appeared in the work of Jacobi andTrudi, 60 years earlier. The symmetric functions studied by these authors wereeventually named Schur polynomials , in honor of Schur’s work relating them tothe characters of the general linear group. Although this remarkable connectionbetween the cohomology of Grassmannians and representation theory was (and
HE ROOTS OF MATHEMATICAL THOUGHT 7 continues to be) influential, later research showed that it breaks down when onelooks at more general homogeneous spaces.In the twentieth century, the study of the Grassmannian and related symmet-ric spaces was enriched by writing them as quotients of Lie groups. The generallinear group GL n = GL n ( C ) acts transitively on G( m, n ), and the stabilizer of thesubspace h e , . . . , e m i under this action is the maximal parabolic subgroup P ofinvertible matrices in the block form (cid:18) ∗ ∗ ∗ (cid:19) where the lower left block is an ( n − m ) × m zero matrix. It follows that G( m, n ) =GL n /P . The Weyl group of GL n is the symmetric group S n , and if B ⊂ P denotesthe Borel subgroup of upper triangular matrices, then we have a decompositionGL n /P = [ w ∈ S Pn BwP/P, where S Pn is the set of permutations w such that w ( i ) < w ( i + 1) for all i = m .This agrees with the Schubert cell decomposition (3) of G( m, n ) given earlier.The above picture generalizes to the case when G is a classical complex Lie groupand P is any parabolic subgroup of G , so that the quotient space G/P is a compactmanifold parametrizing (isotropic) partial flags of subspaces in C N . If B ⊂ P is aBorel subgroup, then we have a cell decomposition G/P = [ w ∈ W P BwP/P and a corresponding direct sum decomposition(5) H ∗ ( G/P, Z ) = M w ∈ W P Z σ w of the cohomology group of G/P , where W P is a certain subset of the Weyl groupof G . The Giambelli problem is to determine the analogue of formula (4), that is,to find canonical polynomial representatives for the Schubert classes σ w on G/P .Once again, the answer required a change in perspective. The most challengingand original part of this was joint work with Buch and Kresch, which examined thecase of symplectic Grassmannians. Here we equip C n with a non-degenerate skew-symmetric bilinear form ( , ), and say that a linear subspace V of C n is isotropic if the restriction of ( , ) to V vanishes identically. We let IG = IG( n − k, n ) =Sp n /P denote the symplectic Grassmannian consisting of all isotropic subspacesof dimension n − k , for some fixed k ≥
0. The Schubert classes σ λ on IG can beindexed by k -strict partitions λ . The condition ‘ k -strict’ means that all parts λ i of λ with λ i > k are distinct, and reflects the fact that the rows of the matrices inreduced row echelon form which represent V must be pairwise orthogonal.Using a Pieri rule for the products c p · σ λ of a special Schubert class c p := σ ( p, ,..., with a general one, we proved that the c p for p ∈ [1 , n + k ] generate thering H ∗ (IG , Z ), and could access the Schubert calculus there. With the help of acomputer, we observed that known determinantal and Pfaffian formulas represented σ λ in extreme cases, when all the parts of λ were at most k , or, respectively, greaterthan k . However, a search of the extensive literature for an operation on matriceswhich interpolates naturally between a determinant and a Pfaffian in the sense HARRY TAMVAKIS required proved fruitless. To make matters worse, there was no a priori reason whythere should be any nice formula for σ λ , and, given the presence of relations amongthe c p , there were plenty of reasons to be pessimistic. A situation all too familiarto those researching the unknown: we were groping in the dark.Instead of the old language of determinants and Pfaffians, the answer we foundto the Giambelli problem for IG employed Young’s raising operators . Given aninteger sequence α := ( α , α , . . . ) with finite support and indices i < j , define R ij ( α ) := ( α , . . . , α i + 1 , . . . , α j − , . . . ) . A raising operator R is any monomial in the R ij ’s. Moreover, let c α := c α c α · · · ,and for any raising operator R , let R c α := c R ( α ) . The main theorem of [BKT]states that for any k -strict partition λ , we have(6) σ λ = Y i G/P variesin an algebraic family, to obtain formulas for the cohomology classes of degeneracyloci of vector bundles . The main theorems gave unique, combinatorially explicit,and intrinsic Chern class formulas for all the degeneracy loci involved.I finished writing [T1] and stared in disbelief at the end result. Although it didrequire some new ingredients, the paper seemed almost trivial, with most proofsconsisting of just a few lines. The article [T1] introduced a new, intrinsic point ofview in Schubert calculus, one that came entirely naturally to me, the culminationof an understanding of the subject that had developed over many years.We are not always so fortunate that the answers to our hardest questions turnout to be so simple. Nevertheless, I believe that simplicity is the hallmark of truth.I also began to appreciate, quoting E. Artin, that our difficulty is not in the proofs,but in learning what to prove. Indeed, the ability to ask the right questions iscritical, and something that can only be taught by experience.To the casual observer looking at the sequence of papers leading up to [T1],it appears as though the author had it all planned out, assembling the necessarycomponents over time, until the final synthesis. Of course, this is completely false,as a detailed examination of the record shows. Many of the pieces were foundwhile solving problems in Arakelov theory and quantum cohomology, not directlyrelated to [T1]. Moreover, a large part of the work was intuitive, with some sectionsincluded in papers not because they were needed there, but because they were toobeautiful to omit, and trusting there would be an application someday.As I began my education, I was drawn to mathematics because I love truth, andsought it there. I soon realized that although mathematics is seemingly humanity’smost credible attempt at finding permanence and truth, it has not reached thatgoal, not by a long shot. Nor does mathematics have answers yet to our deepestand most pressing existential questions. Since aquaintance with truth is the basis ofall knowledge, I conclude with Socrates that in essence, I know nothing . Everythingwritten below should therefore be taken with that caveat in mind.Because I can’t honestly claim to possess knowledge with any certainty, what Iam left with are beliefs. Like any scientist, my beliefs are supported by evidenceculled from my life experiences. The aforementioned ones are the most relevant forthe purposes of this brief exposition, but many more remain unsaid.I believe that the apparent existence and consistency of the mental structurewe call mathematics is a miracle, as wondrous as the universe around us. Thebeautiful equations such as (1) and (6) which emerge out of our collective mind areas unforseeable as any revelatory event. These miracles are unlike those that dependon religious faith: once seen, they can be reproduced, shared, and admired togetherwith other human beings. However, they are far too surprising and otherworldlyto be explained away as our inventions, or a product of solely rational thought.I am under no illusion that I knew what to expect, or even that I was in the dri-ver’s seat, when [T1] was written. Like most of my research efforts, the work beganwith a pen, paper, and wishful thinking. Although the article soon materialized onmy desk, I have no satisfactory explanation as to why all the different ingredients required were there just when they were needed, and fit together like a charm, sothat the proofs turned out to be so easy.To paraphrase Gibran, it is not up to you to direct the course of mathematics,for mathematics, if it finds you worthy, directs your course . I therefore cannot ingood faith take credit for [T1], or by extension any of my papers. This is not justbecause of the debt that [T1] owes, like all scientific research, to the many otherworks by various authors that preceded it. More importantly, I am convinced thatthe fact that [T1] was going to be written was determined in advance .This is what my career has helped to teach me about the human condition: thatthe choices we make, consciously or not, are predetermined. Science supports thenotion that our sense of self and freedom of the will are both illusory. Mathemati-cians are in a unique position to understand this, because their success depends oncircumstances which are figments of their imagination, and yet assuredly beyondtheir control. Moreover, in both the research process and its fruits, mathematiciansexperience and perceive miracles. At the same time, as a poster in my daughter’selementary school points out, math is everywhere.Assuming that human society can reconcile with determinism and overcome itsanthropocentric posture without destroying itself in the process, what is left for thehuman race to strive for? Recall the importance of adopting the right perspective,and also the Mother, magnanimous in suffering and glory. In my opinion, whatremains is to collectively move closer to Her point of view. This will require ahumble reckoning with the truth about our endeavors, an admission of ignorance,and above all, bestowing this attitude towards life to our children, and looking tothem – for forgiveness first, and, ultimately, for guidance. References [BKT] A. S. Buch, A. Kresch, and H. Tamvakis : A Giambelli formula for isotropic Grassman-nians , Selecta Math. (N.S.) (2017), 869–914.[D] J. Dieudonn´e : Pour l’honneur de l’esprit humain , Les math´ematiques aujourd’hui, His-toire et Philosophie des Sciences, Librairie Hachette, Paris, 1987.[G] G. Z. Giambelli : Risoluzione del problema degli spazi secanti , Mem. R. Accad. Sci. Torino(2) (1902), 171–211.[H] G. H. Hardy : A Mathematician’s Apology , Cambridge University Press, Cambridge, Eng-land; Macmillan Company, New York, 1940.[KT] A. Kresch and H. Tamvakis : Standard conjectures for the arithmetic GrassmannianG(2,N) and Racah polynomials , Duke Math. J. (2001), 359–376.[M] I. G. Macdonald : Symmetric functions and Hall polynomials , Second edition, The Claren-don Press, Oxford University Press, New York, 1995.[T1] H. Tamvakis : A Giambelli formula for classical G/P spaces , J. Algebraic Geom. (2014), 245–278.[T2] H. Tamvakis : Mathematics is a quest for truth , Notices Amer. Math. Soc. (2014), p.703.[T3] H. Tamvakis : Theta and eta polynomials in geometry, Lie theory, and combinatorics ,First Congress of Greek Mathematicians, 243-284, De Gruyter Proc. Math., De Gruyter,Berlin, 2020. University of Maryland, Department of Mathematics, William E. Kirwan Hall, 4176Campus Drive, College Park, MD 20742, USA E-mail address ::