Proposed Working Groups
Algorithms for Groups and Their Applications
Algorithms for finite groups and their associated Cayley graphs have a wide range of applications in mathematics and computer science. These applications range from problems related to the classification of finite simple groups to the mixing rate of Markov chains, the design of interconnection networks for large interacting arrays of CPUs, the graph isomorphism problem (of relevance to chemical documentation), and quantum mechanics.
The primary focus of the working group is the design, analysis, and implementation of algorithms for handling finite permutation and matrix groups. Recent algorithms [1-2] set up a framework to reduce computational problems in arbitrary permutation and matrix groups to problems about finite simple groups. In turn, the algorithmic questions about finite simple groups lead to theoretical problems in these groups, mostly about statistical properties and about the structure of subgroups generated by random elements satisfying certain restrictions. The classification theorem for finite simple groups [3] asserts that the nonabelian finite simple groups fall into three infinite classes (alternating groups, classical groups of Lie type, and exceptional groups of Lie type) plus a set of 26 additional ``sporadic'' simple groups. A major project now is to develop a theory of simple groups of exceptional Lie type and of sporadic simple groups which can be exploited in the design of efficient algorithms, similar to the algorithmic handling of classical groups of Lie type [1] and alternating groups [2].
The currently most active area of computational group theory is the design and implementation of algorithms dealing with matrix groups over finite fields. There are a lot of subproblems, suitable for students at all levels (advanced undergraduate, beginning graduate, advanced graduate). Examples for each level:
- Given a long list of group elements such that one of them is in a proper normal subgroup (but we do not know which one), construct one group element in the proper normal subgroup. An efficient practical algorithm for this problem would speed up the reduction of the handling of arbitrary matrix groups to the handling of simple groups.
- Investigate how the quality of the random element generator influences randomized recognition algorithms for simple groups. In particular, investigate the reliability of the random generator ``product replacement algorithm'' in this setting.
- Design and implement algorithms for the decomposition of rank one exceptional groups of Lie type as a short product of cyclic groups. This problem is relevant for the constructive recognition of these simple groups. Implementations and experiments will be carried out in the GAP computer algebra system.
A secondary focus of the working group is the use of symmetry in the sciences. An interesting example is provided by spiral waves which appear in abundance in physics and chemistry. One famous experiment where rigidly-rotating spiral waves have been observed is the Belousov-Zhabotinsky reaction (a chemical reaction that involves the bromination and oxidation of malonic acid by catalysts). In certain parameter regimes, the spiral waves begin to meander or to drift, tracing out flower-like patterns. The package EZSPIRAL (written by Dwight Barkley) will allow us to compute and study meandering spirals on the computer. The observed phenomena, as well as other related ones, can be understood by combining elementary group theory with dynamical systems techniques. First, we will learn about the Euclidean symmetry group of the plane. Next, we consider planar differential equations and learn about Hopf bifurcations by exploring and experimenting with numerical tools. Combining this knowledge will help us to understand meandering and drifting spirals. These projects are accessible to both advanced undergraduate and beginning graduate students. Other projects will involve exploring the impact of boundary conditions on the behavior of spiral waves: this is an interesting topic related to the role that arteries play for sustaining cardiac arrhythmias.
This working group is especially well suited for the involvement of undergraduate and graduate students. Participating in the implementation of the algorithms, and the study of spiral waves, can start without much theoretical background, and it may lead to accessible problems in group theory and in the theory of computing.
| [1] | W.M. Kantor and A. Seress. Black box classical groups. Mem. AMS 149 (2001). |
| [2] | R. Beals, C. Leedham-Green, A. Niemeyer, C. Praeger and A. Seress. A melange of black box algorithms for recognising finite symmetric and alternating groups I. (Submitted). |
| [3] | D. Gorenstein, R. Lyons and R. Solomon. The classification of the finite simple groups #1. Amer. Math. Soc. Surveys and Monographs 40 (1995). |
| [4] | B. Sandstede, A. Scheel and C. Wulff. Dynamical behavior of patterns with Euclidean symmetry. In Pattern formation in continuous and coupled systems (M. Golubitsky, D. Luss, and S. Strogatz, eds.), Springer, IMA Vol. Math. Appl. 115 (1999) 249-264. |
Around Symmetry
``Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection'' (H. Weyl, Symmetry). This working group will rely on the intuitive and at the same time pervasive nature of symmetry to introduce the students to a variety of mathematical topics, starting from traditional subjects and evolving towards topics of current research.
Geometric Representation Theory: Problems which have elementary statements of instant geometric appeal can be reformulated and solved in the framework of representation theory, thus illustrating its power and effectiveness. For instance, the assertion according to which ``a convex centrally symmetric body in R^n which casts a shadow of constant area when illuminated by parallel rays from any direction is a sphere'', and more generally the problem of reconstructing a geometric body from its projections on hyperplanes, can be solved by means of elementary representation theory for orthogonal groups, viz. ``spherical harmonics''. A more sophisticated statement of a similar nature, concerning translation invariant valuations of convex bodies in R^n (P. McMullen's Conjecture) has recently been solved by using principal series representations of GL(n,R).
Automorphic Representations: The recent solution of the Fermat Last Theorem is one of the crowning achievements of 20th century mathematics. One of the critical tools for this is the use of automorphic representations for the group GL(n). Specifically the idea of Langlands of functorial correspondence (relating automorphic representation on different groups) is critical. One such example is the ``base change'' correspondence (relating groups under field extensions). The ideas used here can be illustrated very well in the finite field case where it is elementary, using just basic group theory, to establish character identities between various groups. Basic problems in number theory can be interpreted and sometimes solved using automorphic representations. Specifically how does a number in the ring of integers of a field extension of the rationals factorize into a product of prime ideals (generalizing Euclid's factorization of an integer into a product of primes). In the specific case of quadratic extensions of the rational this is an old problem called quadratic reciprocity. Here very elementary ideas from harmonic analysis (finite Fourier transforms in Heisenberg groups) can be used to solve this problem. Automorphic representation theory provides a framework on how the problem can be solved for other cases.
Fourier Analysis and Crystallography: Many of the materials we meet in everyday life are in crystalline state, a state which is characterized mathematically by a type of symmetry known as the crystalline symmetry. The determination of (the atomic structure of) crystals is done using elaborated techniques (X-ray crystallography, electron microscopy etc.), which are based on Fourier analysis. The interplay between analysis and geometry imposed by the crystalline symmetry involves rather sophisticated mathematics, which the students can learn with practical motivations in mind.
Asymptotic Symmetry in Geometry: Many of the objects and phenomena in nature satisfy mathematical structural hypotheses not exactly but with ``arbitrary accuracy short from exact''. As recently observed, ``arbitrary accuracy short from exact'' can reconcile mathematical structures which are incompatible if the hypotheses would have to be exactly verified. For example, the standard 2-dimensional Penrose tiling manifests with arbitrary accuracy both 2-dimensional crystalline symmetry and five fold symmetry. These two type of ``exact'' symmetry are mathematically incompatible. Objects which display a pattern satisfied with arbitrary accuracy but short from exact became available in geometry only very recently. The most striking and novel example of such pattern in connection with symmetry are the Penrose tilings. In addition to their interest in connection with quasi crystals,the Penrose tilings provide the student as well as the researcher with new mathematical challenges. The asymptotic symmetry is one of the many topics of interest for Asymptotic Geometry, whose attention expands considerably the class of traditional geometric objects.
Noncommutative Geometry: The stretching of geometric thinking required to treat naturally occurring ``spaces'' whose coordinates however may not commute led a noncommutative approach to geometry, whose affiliated symmetry is inherently of a ``quantum'' nature. Although the theory is not primary in any sense, its spirit and methods can be illustrated by relatively simple examples. Connes' charming proof of one of the most surprising theorems in synthetic geometry, Morley's theorem (that says that ``the trisectors of the angles of any triangle meet at the vertices of an equilateral triangle''), is such an example. When cleverly restated, as a statement involving three elements of the group of affine motions of the line, it can be verified by a direct computation and, moreover, makes sense over an arbitrary field. Another example involves Kreimer's Hopf algebra of rooted trees, which was shown to encode the combinatorics of Feynman graphs in perturbative renormalization in QFT.
Blowup of Solutions to Partial Differential Equations
Solutions of nonlinear evolution equations with given initial data either exist for all time or else ``blow up'' in finite time. There is a large body of literature dealing with the case of a single parabolic equation such as the heat equation with a nonlinear forcing term. There are many interesting results not only regarding sharp conditions for blow up, but also regarding the profile of the solution at the time of blow up. A few of these results have been extended to systems of parabolic equations. There are also blow up theorems for hyperbolic equations. Some of the results ensure that blow up will not occur for a ``very long time'', and others which deal with the profile of the solution when it blows up. An interesting area of application arises in mathematical biology for models of the movement and rearrangement of biological organisms, such as bacteria or cells in the human body, under the influence of chemical species. Such models typically consist of a mixture of several diffusion equations and ordinary differential equations, all nonlinear and coupled. The literature is quite sparse, and there are a lot of interesting questions. What makes this topic particularly appropriate for second-year students is that students do not have to know too much in order to start working on some of the models, and one may pose both rather straightforward as well as more difficult problems.
Sometimes the blowup phenomenon for systems can be somewhat surprising: e.g., Mizoguchi, Ninomiya, Yanagida and Weinberger recently constructed certain semilinear parabolic systems of two equations such that its solution may blow up for some initial data, while the corresponding kinetic system has a globally stable equilibrium. Therefore it seems to be of interest to understand connections/differences between blowup in ODEs and in PDEs. For second-year students, this kind of problem will be a good opportunity to learn some basic techniques of differential equations and possibly go further to work on hard problems, e.g., blowup profiles of solutions.
Complex Geometry and Analysis
There is a strong interplay between geometry and analysis in several complex variables. For instance: the curvature of the boundary of a domain prescribes the singularities of the canonical kernels associated to the domain (Bergman, Szegö, etc.), the order of contact of an analytic variety with the boundary of a domain dictates the strength of estimates on solutions to natural systems of partial differential equations inside the domain, and the resolution graph of an analytic variety determines the asymptotics of integrals over the variety. While there are many such connections that are fairly well-known, there are others which are only emerging and nothing like a general program exists which predicts function theoretic consequences of geometrical hypotheses (or vice-versa). In all instances where connections have been established, the connections themselves have suggested other related interactions and greatly stimulated both the analytic and geometric activity around this problem.
This working group will focus on exploring these connections by examining concrete problems related to the organizers' research programs. The goal with any particular problem will be to strip both the geometric information (often a curvature condition or assumption on the complexity of a variety) and the desired analytic output (often an inequality) down to their simplest, essential form. Projects will include explicit curvature computations on manifolds, explicit resolution of singularities of polynomial varieties, calculation of reproducing kernels by summation techniques, and direct evaluation (or estimation) of integrals defined on varieties. All of these projects have aspects which are suitable for the undergraduate level (usually by assuming a high degree of symmetry at the onset), but also range to important problems on the research frontier.
An organizing principle of this working group will be to emphasize computation and working out of special cases. Current research in complex analysis and geometry often uses techniques from algebraic geometry, partial differential equations, and differential geometry and this technical background presents a major obstacle for beginners. By focusing on special cases, we expect that students, without much theoretical background, will still have the possibility of discovering interesting relationships between analysis and geometry. Of course, we also expect that the careful understanding of special cases will clarify our view of known, general results and suggest tractable problems for further research.
| [1] | D. Cox, J. Little, and D. O'Shea. Using algebraic geometry. Springer, New York, 1998. |
| [2] | F. Zheng. Complex differential geometry. AMS, International Press, Boston, 2000. |
| [3] | S. Krantz. Function theory in several complex variables. Wadsworth & Brooks, Pacific Grove, 1992. |
Computational Neuroscience
Complex spatio-temporal patterns of neuronal activity arise throughout the central nervous system, and play a role in sensory processing, motor behavior, and learning. In particular, synchronous, oscillatory dynamics have been implicated in the generation of sleep rhythms, epilepsy, and Parkinsonian tremor [1]. Fundamental questions concern how these activity patterns are generated, how they are modulated, and what their behavioral correlates are. The projects of this working group involve learning and developing mathematical and computational tools for the modelling and analysis of biological systems. The primary focus will be in neuroscience. However, the group is open to exploring mathematical applications in other areas of biology [2], such as the cell cycle, genomics, and interacting species.
New members of the working group will become acquainted with basic facts concerning the central nervous system, including neurons, synapses, and the general organization of the brain, as well as the anatomy and physiology of a few specific neural structures or pathways [3]. Mathematical techniques from dynamical systems, such as phase plane analyses, singular perturbation theory, bifurcations, and stability theory, will be studied. Integration of the biology and mathematics will occur through the development of models to describe the activity and interactions of neurons in particular systems. Numerical software will be used to investigate the behavior of the models, and students will have the opportunity to design computer programs for their efficient simulation. We shall then analyze the spatiotemporal patterns that emerge from the models, find out the mechanisms that underlie their generation and modulation, and discuss their biological significance.
Although a typical project will concentrate on one specific system (e.g., the thalamus [4], basal ganglia [5], visual cortex, or case study in a non-neural context), the computational and mathematical analysis can be done at several levels. For example, members of the working group may work on analyzing
- intrinsic properties of single nerve cells and their responses to applied current,
- conditions under which coupled neuronal oscillators exhibit synchrony,
- how complex activity patterns and rhythms are generated in networks consisting of different types of neurons wired together,
- chaotic behavior in neuronal systems, or
- how instabilities of certain rhythms arise and what other kinds of activity patterns may develop.
| [1] | D. Terman and J. Rubin. Geometric singular perturbation analysis of neuronal dynamics, In Handbook of Dynamical Systems II, (B. Fiedler, ed.), Elsevier, Amsterdam, 2002. |
| [2] | J.D. Murray. Mathematical Biology. Springer-Verlag. |
| [3] | E.R. Kandel, J.H. Schwartz and T.M. Jessell (eds.). Principles of Neural Science. McGraw-Hill. |
| [4] | A. Bose, N. Kopell, and D. Terman. Functional reorganization in thalamocortical networks: Transition between spindling and delta sleep rhythms. P.N.A.S. 93 (1996) 15417-15422. |
| [5] | J. Rubin, D. Terman, A.C. Yew and C. Wilson. Activity patterns in a model for the subthalmopallidal network of the basal ganglia. Journal of Neuroscience 22(7) (2002) 2963-2976. |
Distribution of Extrema
One of the fundamental problems in applied probability is the computation of limit laws for extrema. These extrema are often complicated functionals of a large collection of random variables and represent some physically interesting quantity. An example of this would be the shortest route between two points on a randomly weighted graph, which is defined to be the minimum over paths connecting the two points of the sum of the random weights along the path. Classical limit theory tells us how to do these computations when the functional is relatively simple (think of finding the limiting behavior of the maximum of N independent random variables or the sum of N independent random variables). The analysis becomes more difficult when the functional is sufficiently complicated that it is not clear on an algorithmic level how to find the sample extremum.
The interplay between probabilistic and asymptotic analysis on one hand, and computational understanding of the functional on the other hand, makes these problems of extremal distributions very sensitive to the details of particular models. The focus of this working group will be to improve our algorithmic understanding of some of these problems so as ultimately to harness available probabilistic techniques. We list a couple of examples of active research areas in this vein.
DNA Folding: Given a random string of symbols from the alphabet ACGT, what is the minimal number of deletions required before the string maybe be perfectly folded so as to match pairs A-T and C-G? Depending on the definition of a perfect folding, and depending on the probability model for the random sequence, this yields several distinct extremal problems. In each case, one must first determine how to compute the minimal number of deletions for a sample string.
Spin-glass Ground States: Given random interactions between each pair of particles, what is the configuration yielding the lowest energy state? Again, there are several problems depending on the common distribution of the pairwise interactions. The first step is to understand how to efficiently compute the minimal energy for a given collection of interaction strengths.
Often the exact computation of sample extrema turns out to be provably difficult. It remains likely, however, that there are good, undiscovered ways of computing the extrema to within a factor of 1+epsilon, and/or computing the extremum correctly with probability at least 1-epsilon. These algorithmic questions provide fertile ground for student exploration. For example, in the max-cut problem (divide the vertices of a weighted graph into two halves so as to maximize the sum of the weights of edges going from one half to the other), there are conjectured algorithms to find the near-optimum with high probability, but no rigorous results. It is valuable to find algorithms that work well with trial data, even in absence of provable results about the speed or accuracy of the algorithm. Thus useful projects range from undergraduate level tasks such as the writing of code to find extrema and the collection of data based on this code, to rigorous analysis of algorithms, and ultimately to probabilistic analysis and determination of limiting distributions.
Geometric Group Theory
This area lies at the crossroads of geometry, topology and group theory. There are many open problems, of a combinatorial nature, which could be attacked by beginning graduate students at the same time they are learning the basics of these subjects.
Spaces of Nonpositive Curvature: There are many examples of polyhedra and manifolds with singular metrics (e.g., polyhedral metrics) of nonpositive curvature, [1]. Many more classes of examples remain to be discovered. Such examples arise naturally in different areas, e.g., as the configuration space of n points on a finite graph or as a compactification of the moduli space of n points on the circle. The main fact about such a nonpositively curved space is that it is the classifying space for its fundamental group. A large class of such examples come from tilings of manifolds by polytopes. Even in dimensions two and three, there are many open problems involving the classification of tilings which could be attacked by both undergraduate and graduate students.
Coxeter Groups and Artin Groups: Coxeter groups are abstract reflection groups. They play an important role in an amazing number of different areas of mathematics. In geometric group theory they provide a large class of examples of the type discussed in the previous paragraph. Associated to any Coxeter group there is an Artin group (or ``generalized braid group''). Here the issues of nonpositive curvature and the existence of a finite classifying space have not yet been resolved. After the appearance of [2], there has been a great deal of work on rigidity questions for Coxeter groups and Artin groups. The basic problem is to decide when a Coxeter diagram is determined by the underlying group. Sometimes it is and sometimes it is not. It is clear that progress on this question could be made with only elementary combinatorial arguments.
Finite group actions on graphs and surfaces: The study of finite group actions on graphs and surfaces is a topic which is accessible to beginning graduate students. A well-known conjecture in this area is that the Cayley graph of the presntation of a finite group always admits a Hamilton cycle. The study of such finite group actions is also closely tied to the problem of computing the cohomology of mapping class groups and of automorphism groups of free groups.
| [1] | M. Bridson and A. Haefliger. Metric Spaces of Non-positive Curvature. Springer, Berlin and New York, 1999. |
| [2] | R. Charney and M. Davis. When is a Coxeter system determined by its Coxeter group? J. London Math. Soc. 61 (2000) 441-461. |
Graph and Matroid Structure Theory
A graph G is a combinatorial object consisting of a family E(G) of unordered pairs of elements from a set V(G). The elements are called vertices and the pairs are called edges, a usage taken from the edges and vertices of convex polytopes. This research deals with graph structure theory, studied under minor containment. Minor containment allows contraction of edges to vertices, as well as taking subgraphs. Allowing edge-contractions gives it a topological nature. Properties such as a graph being embeddable on a fixed surface, admitting a 3-space embedding without knotted circuits, or having a tree-like structure over a given minor-closed subclass of graphs are preserved under minor inclusion. Normally, V(G) and E(G) are assumed finite, as the work in these areas has a heavy algorithmical content.
The fact that every property of finite graphs is finitely based--only finitely many graphs are minor minimal with respect to possessing the property--is a basic result proved during 1981/85, and originating from Ohio State. The general theory is called well-quasi-ordering (as a relaxation of the well-ordering principle) and has connections with Logic. Topological structure enters the picture by a theorem that all graphs in a given minor-closed class can be expressed as a finitely restricted tree-structure over a subclass of graphs "almost" embeddable in a surface of bounded genus. Applying the techniques of well-quasi-ordering allows reduction of the finiteness question from the minor-closed class to the subclass, from the "almost" embeddable to the embeddable, and (by repetition) the genus down to where the surface disappears, leaving bounded size labelled set-families, the elementary base case.
The particular open problems of most current interest involve surface embeddings, the study of tree-structure and coloring theory in graphs. The most recent advances have been in coloring theory and in graph-related matroid theory.
Matroid theory abstracts the properties of linear dependence on finite families of vectors, usually presented as the columns of a matrix. A graph G, given an arbitrary orientation, has an incidence matrix over a field F, indexed by V(G)xE(G), with entries from {-1,0,1} determined by the single negative and positive vertex incidence for each edge (loops are given all-0 columns). Then minimal dependent subsets of edges correspond to simple circuits in the graph, and independent sets to forests. It is well understood when a matrix is row-equivalent to such a graphic matrix, and graphs present the same underlying matroid independent of the orientation and the field F. Matroids representable over any field are called regular, and are also well-understood through work of Tutte in the 1950's and Seymour in 1978/79. They can be decomposed into graphic matroids, the duals of graphic matroids and a single exceptional 5x10 matrix R, by certain 1-sums, 2-sums and 3-sums. The minor relation for matrices is determined by deleting columns, removing all-0 rows and deleting a row-column pair with a common non-0 entry, after column-clearing. These operations generalize to general matroids.
The two main conjectures in matroid minors concern the case where F is finite. These are (1) F-matrices are well-quasi-ordered, and (2) (Rota) there are only finitely many minor-minimal matroids not expressible by an F-matrix. Recent work has made some progress on these conjectures, along the lines of the graph minor theorem, and this provides direction for further research in this field. Many much simpler questions need to be answered. Those which appear most assessible to this working group derive from the algorithmical nature of the context. Minor-closed classes of matroids close to the F-matrix matroids can serve as input to polynomial-time or NP-algorithms. We expect that many such classes will need to be constructed and represented effectively (in polynomial-space), and that many classical algorithmic questions will need to be answered in so doing.
Jacobi's ``Fundamenta Nova'' and Sums of Squares
This working group will initially consist of Steve Milne, his graduate student Eric Conrad, and the two Ross Assistant Professors Soon-Yi Kang and Jaebum Sohn. Milne and Conrad will soon complete an edited translation with extensive commentary of Jacobi's Fundamenta Nova from Latin into English. Conrad, whose thesis involves Jacobi elliptic functions, knows Latin and is doing the initial translation. The commentary will include a significant amount of Milne's sums of squares work as well as his recent new results on Ramanujan's tau function.
This material will emphasize explicit computational techniques from elliptic functions, continued fractions, and the combinatorics that holds all of this together. Examples range from classical sums of squares formulas of Jacobi to Milne's recent infinite families of explicit exact formulas. This will provide a sample of results from such diverse areas as Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, and Schur functions. There is ample room for the history and motivation here that involves Jacobi, Glaisher, Mordell, Ramanujan, Hardy, and Andrews.
In the working group, they together with other undergraduate and graduate students will study the above material on Jacobi's Fundamenta Nova and Milne's recent research on sums of squares. They will also consider the very recent work of Don Zagier and Ken Ono that provides a modular-forms setting for some of Milne's results. Initially they will focus on the Conrad-Milne translation of Jacobi's Fundamenta Nova and consequences and applications of Milne's sums of squares research.
Number Theory
Number theory provides many opportunities for vertical integration, since it can be entered at many different levels of background and experience. Many parts of it can be approached at a level quite accessible to advanced undergraduates and beginning graduate students and yet lead into areas suitable for doctoral thesis work and research. Below we give several examples of such topics and some references that approach these topics at various levels of sophistication.
p-adic Analysis: At its beginning levels, p-adic analysis provides a useful recapitulation of fundamental notions of real analysis in a novel setting. Direct connections to number theory can be made with the theory of p-adic L-functions [4], the beginning of which is quite elementary, and yet points to Iwasawa theory and ``Main Conjectures'' relating analytically defined p-adic L-functions with arithmetically defined characteristic polynomials [7].
Cryptology: The basics of cryptology, including the RSA and other public-key encryption methods, provide a suitable entry point [3]. At a deeper level, one can study (and analyze the expected running times of) the various algorithms for factoring integers and for primality testing [1]: some of these algorithms can be understood with very little background, while others (including the best algorithms now known) involve a deeper and more sophisticated knowledge of algebraic number theory: cyclotomic fields [7] and elliptic curves [6]. This is currently an active research area, and one in which new ideas may arise from cleverness, rather than extensive knowledge.
Coding Theory: As a complement to the secure transmission of data (cryptology) is the reliable transmission of data, the subject matter of coding theory. Again the beginnings of the subject are quite accessible, while the problem of finding efficient error-correcting codes leads directly into deeper parts of number theory: the theory of integral quadratic forms, with relations to lattices and sphere packing [2]; the theory of algebraic curves, with connections to Goppa codes [5].
| [1] | H. Cohen. A course in computational algebraic number theory. Springer, Berlin, 1996. |
| [2] | J.H. Conway and N.J.A. Sloane. Sphere packing, lattices, and groups. Springer, New York, 1998. |
| [3] | N. Koblitz. A course in number theory and cryptography. Springer, New York, 1995. |
| [4] | N. Koblitz. P-adic numbers, p-adic analysis, and zeta-functions. Springer, New York, 1996. |
| [5] | C. Moreno. Algebraic curves over finite fields. Cambridge University Press, 1991. |
| [6] | J.H. Silverman. The arithmetic of elliptic curves. Springer, New York, 1986. |
| [7] | L. Washington. Introduction to cyclotomic fields. Springer, New York, 1996. |
Reading Classics
The idea is to conduct intellectually stimulating and challenging seminars, aimed at undergraduate and graduate students, that are devoted to the study and discussion of classical mathematical texts. One of the goals will be to link the initial impetus set by works such as Euclid's elements with the impact they eventually had in modern mathematics. Doing so introduces the participients to a variety of different areas in modern mathematics and helps them to understand how mathematics developed over the centuries.
In particular, undergraduate students would be exposed, some of them for the first time, to a wealth of influential great ideas; this would show them that mathematics is more than calculus and would, hopefully, make them curious to learn more about the further development of these ideas. Graduate students would have the opportunity to deepen their knowledge, to broaden their way of thinking, and to get experience in communicating important ideas to general audiences (in this case, to the attending undergraduate students). Lastly, postdocs and faculty would moderate the discussions.
The seminar will also gain from tapping into the active visitor program of the Mathematics Research Institute (MRI) by inviting distinguished visitors to give colloquium-style presentations and panel discussions in the framework of this working group. This seminar provides also an excellent opportunity to involve foreign students by asking them to translate and explain texts written in their native language. Several of our graduate students already engaged in such activities.
Sample texts for reading:
| [1] | Euclid: The Elements. |
| [2] | Diophantus: Arithmetica. |
| [3] | Archimedes: Measurment of a Circle, On Spirals, The Sand Reckoner. |
| [4] | Euler: Introduction to Analysis of the Infinite, Elements of Algebra. |
| [5] | Newton: The Principia. |
| [6] | Gauss: Disquisitiones Arithmeticae. |
Suggested secondary literature:
| [1] | T.L. Heath. A history of Greek mathematics. Clarendon Press, Oxford, 1921. |
| [2] | B. Artman. Euclid -- The creation of mathematics. Springer, New York, 1999. |
| [3] | R. Hartschorn. Companion to Euclid. AMS, Providence, 1997. |
| [4] | I.G. Bashmakova. Diophantus and Diophantine equations. MAA, Washington, 1997. |
| [5] | S. Stein. Archimedes: What did he do besides cry Eureka? MAA, Washington, 1999. |
| [6] | W. Dunham. Euler: The master of us all. MAA, Washington, 1999. |
| [7] | V.I. Arnold. Huygens & Barrow, Newton & Hooke. Birkhäuser, Basel, 1990. |
| [8] | D.E. Smith. A source book in mathematics. McGraw-Hill Book Company, New York, 1929. |
| [9] | H. Eves. Great moments in mathematics. MAA, Washington, 1983. |
| [10] | R.S. Callinger (ed.). Classics of mathematics. Moore Pub. Co., Oak Park, 1982. |
| [11] | D.J. Struik (ed.). A source book in mathematics. Harvard University Press, Cambridge, 1969. |
Symbolic Dynamics and Applications
Symbolic dynamics arises naturally in areas as diverse as applied mathematics, number theory, combinatorics, probability, ergodic theory, coding and information theory. A plethora of motivating examples and numerous connections to other fields make symbolic dynamics an especially attractive and suitable medium for involving students at both the undergraduate and graduate level, thereby providing an excellent opportunity for achieving vertical integration and broadening education.
The term symbolic dynamics refers to dynamical systems on spaces of sequences where the dynamics is generated by a particular map, namely, the shift. These systems are readily introduced (any shift-invariant set of sequences from a fixed alphabet is a symbolical dynamical system!) and can be used to motivate, illustrate and study various key concepts in dynamical systems such as ergodicity, entropy, invariant measures and many others. They also contribute new tools to other areas. The following, by far not exhaustive, list of applications of symbolic dynamics amply demonstrates its ubiquity and potential:
- Hyperbolic diffeomorphisms, geodesic flows and billiards [2,9]
- Data storage and transmission in coding and information theory [6]
- Perron-Frobenius theory of nonnegative matrices [1]
- Ramsey theory and additive number theory via ergodic theory [3]
- Markov chains and probability [5]
- Mathematical linguistics [7]
- Chaos, Smale's horseshoe, Sharkovskii's theorem [8]
- Homoclinic and heteroclinic cycles in ODEs [4]
- Applications to rotating convection rolls in Rayleigh-Benard convection and forced oscillations of elastic beams in magnetic fields [4].
| [1] | M. Boyle and D. Handelman. The spectra of nonnegative matrices via symbolic dynamics. Ann. Math. 133 (1991) 249-316. |
| [2] | L.A. Bunimovich and Y.G. Sinai. Markov partitions for dispersed billiards. Comm. Math. Phys. 73 (1980) 247-280. |
| [3] | H. Furstenberg. Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, 1981. |
| [4] | J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, 1983. |
| [5] | B. Kitchens. Symbolic Dynamics. Springer, 1998. |
| [6] | D. Lind and B. Marcus. Symbolic Dynamics and Coding. Cambridge University Press, 1995. |
| [7] | M. Lothaire. Combinatorics on Words. Cambridge University Press, 1983. |
| [8] | J. Palis and F. Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Cambridge University Press, 1993. |
| [9] | C. Series. Symbolic dynamics for geodesic flows. Acta Math. 146 (1981) 103-128. |