Over the last 23 years, the Annual Automorphic Forms Workshop has remained a small and friendly conference. Those attending range from students, new PhD's, and established researchers. For young researchers, the conference has provided support and encouragement. For accomplished researchers, it offers the opportunity to mentor, and it provides a forum for exchanging ideas.

The workshop has become internationally recognized for both its high-quality research talks and its supportive atmosphere for junior researchers. Participants present cutting-edge research in all areas related to automorphic forms. These include Maass wave forms, elliptic curves, Siegel and Jacobi modular forms, special values of L-functions, random matrices, quadratic forms, applications of modular forms, and many other topics.

In addition to research talks, the workshop has, in the past years, featured panel discussion sessions on the topics of grant writing, mentoring and research partnerships, REUs and outreach, and opportunities for international collaborations. Based on the success of these sessions, we plan to have similar panel sessions this year as well.

Abstract: Metric graphs --- essentially Riemannian 1-manifolds with mild singularities --- share many features in common with Riemann surfaces (e.g., divisor class groups and spectral theory), and yet the analysis becomes much simpler in dimension one. In fact, it becomes so much simpler, that one can use this theory to motivate many interesting facts from complex analysis (e.g., the maximum modulus principle and Abel-Jacobi theory).

For a number theorist, these objects arise in the study of admissible metrics on an algebraic curve. For an analyst, there is an intriguing potential theory. For a geometer, they are the simplest class of tropical varieties. For a combinatorialist, they are direct limits of weighted graphs. For an applied mathematician, they are ideal models of resistive electrical networks. For me (recently), they have provided a fascinating link to arithmetic geometry. I plan to survey a bit of each of these theories.

Abstract: The operation of a discrete group G can be naturally extended to the Stone-Cech compactification beta G of G making beta G a semigroup in which left translations by elements of G and all right translations are continuous. We take the points of beta G to be the ultrafilters on G, the principal ultrafilters being identified with the points of G.

As any compact Hausdorff right topological semigroup, beta G has idempotents. Every idempotent p\in (beta G)\G determines a left translation invariant Hausdorff maximal topology T_p on G by taking as the neighborhood filter of 1 the subsets A\cup {1} where A\in p. We say that an idempotent p\in beta G is regular if p is a uniform ultrafilter and the topology T_p is regular. We show that for every infinite group G, there exists a regular idempotent in beta G. As a consequence we obtain that for every infinite cardinal k, there exists a homogeneous regular maximal space of dispersion character k, which is the answer to an old difficult question.

Another consequence tells us that there exists a translation invariant regular maximal topology on the real line of dispersion character continuum stronger than the natural topology. 

Abstract: Intuitively, we associate the process of diffusion with a homogenizing effect that leads to uniform spatial distributions. Surprisingly, as we will see through simple mathematical modeling, diffusion of chemical substances (particularly those that activate/deactive melanin) and their cross-reactions can lead to non-uniform patterns that correlate well with certain biological systems (particularly those we can necessarily see, as in animal coats). We will analyze these Reaction-Diffusion (RD) models to determine the conditions under which certain patterns form.

Abstract: In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. However, it is not obvious that this algorithm necessarily terminates.  

We show that in fact the algorithm terminates after at most $2^{n-1}-1$ steps in the worst case, and that there are super-exponentially many permutations for which this exact bound can be achieved. The proof involves a curious symmetrical binary representation.  

This is joint work with Peter Winkler. 

Abstract: Dynamical systems with delays are encountered in growth processes in economics, chemical and biomedical engineering and various others fields of applications. In this talk, we study optimal control problems with delays in control and state variables. The control process is subject to control and state constraints. The delayed (retarded) optimal control problem can be transformed into a standard optimal control problem by augmenting the state dimension. This allows us to derive a Pontryagin type Minimum Principle for retarded optimal control problems, where the adjoint function satisfies an advanced differential equation. Regularity assumptions for the control and state constraints imply that the associated multipliers are sufficiently regular. Using suitable discretization schemes, the optimal control problem is transcribed into a large-scale optimization problem which can be solved by Interior-Point or Sequential Quadratic Programming methods. This approach also provides adjoint variables which allow for a precise verification of necessary conditions. We illustrate theory and numerics by several examples, e.g., the optimal control of a continuous stirred tank reactor (CSTR) and by the computation of optimal multi-drug protocols in a generic model of the innate immune response.

Abstract: Mathematical Logicians, especially recursion theorists and set theorists, have a well-detailed structure theory for the hierarchy of definability.  We will describe this hierarchy and then the mathematical evidence, namely theorems, that it is intrinsic and unique.  

Abstract: Ramsey's theorem is an important and in many ways surprising foundational result.  It states that any coloring of the n-tuples of integers by finitely many colors admits an infinite homogeneous (or monochromatic) set, i.e., one on whose n-tuples the coloring is constant.  In broader terms, this states that complete disorder is impossible: in any configuration or arrangement of objects, however complicated or disorganized, some amount of structure and regularity is necessary.

Understanding this regularity, and how it arises, has been the subject of much research in mathematical logic, especially in computability (recursion) theory.  This talk will begin with a survey of some of the results of this analysis, and highlight several recent advances.  We will then focus on an important variant of Ramsey's theorem, obtained by restricting to colorings of 2-tuples whose value depends only on the first coordinate when the second is sufficiently large.  This "stable" form of Ramsey's theorem has served as an important tool in the logical investigation of Ramsey's theorem proper.  We introduce a measure-theoretic approach to studying this principle that sheds light on which results about it are and are not typical.

Abstract: Many advances on the algebraic side of number theory in the last 15 years (such as the solutions of the Shimura-Taniyama conjecture, Sato-Tate conjecture and Serre's conjecture, as well as decisive progress on the Fontaine-Mazur conjecture and Main Conjectures for modular forms) have relied in an essential way on improvements in the theory of Galois representations.

The aim of the three main courses is to present an overview of many of these ideas and applications, aimed at advanced graduate students and postdocs with a strong background in number theory, Galois cohomology, and basic algebraic geometry.

Abstract: A standard calculus problem is to find the quickest path from a point on shore to a point in the lake, given that running speed is greater than swimming speed. Elvis, my Welsh Corgi, has never had a calculus course. But when we play "fetch" at Lake Michigan, he appears to choose paths close to the calculus answer. In this talk we reveal what was found when we experimentally tested this ability. We will also discuss whether Elvis solving an optimization problem or a related rates problem. In short, does he bifurcate?

Abstract: In 1954, Roger Lyndon constructed a seven-element algebra with one binary operation whose equational theory failed to be finitely axiomatizable. Eleven years later, Murskii found a three-element algebra with one binary operation whose equational theory failed to be finitely axiomatizable in a more contagious manner. Since then, infinitely many more examples of nonfinitely axiomatizable algebras have been found, many with the help of the Shift Automorphism Method. We will discuss this method and the evidence it provides toward resolving two problems open since 1976.

Abstract: We call a complete lattice perfect, if it is a sublattice of lattice of the form Sp(A), where A is an algebraic lattice and Sp(A) stands for the lattice of algebraic subsets of A (subsets closed with respect to arbitrary intersections, and joins of arbitrary non-empty chains). The problem of description of perfect lattices is motivated by the fact that every lattice of subquasivarieties of any quasivariety is perfect. The question about the description of lattices of quasivarieties is known as Birkhoff-Maltsev Problem. In our paper we describe a new class of perfect lattices that we call super-lattices.

As a corollary, we show that all lattices of the form Sub(P) (lattices of subsemilattices) and O(P)(suborder lattices of a partial order) that satisfy the weak J'onsson property are perfect. The weak J'onsson property is a slight generalization of original J'onsson property D(L)=L.

We also show the examples of lattices in these classes, for which no embedding exists into join-semidistributive and lower continuous lattice. In particular, they are not perfect.

Abstract: The Schur norm ||M||_S of a matrix M is the norm of Schur (or Hadamard, or entrywise) multiplication by M, i.e.
  ||M||_S = max {||M*X||: ||X||=1},   where (M*X)(i,j)=M(i,j)X(i,j).

Here ||X|| is the operator norm of X, but other matrix norms may also be considered. Many natural matrix operations (e.g. taking the upper-triangular part) may be studied via the appropriate Schur norm. The Schur norm is surprisingly hard to compute, but we'll examine several useful approaches to this problem. We'll also discuss a variety of applications (and some open questions), e.g. von Neumann inequalities and measures of normality.

Abstract: Investigations of the left-distributive law a(bc)=(ab)(ac) have resulted, among others, in a discovery of an integer that is larger than other large numbers that cropped up elsewhere in mathematical practice. We present some results on left-distributive algebras with one generator (due to Laver, Dougherty and others) and some open problems. ref.: Dougherty-Jech, Advances of Math. vol. 130

Abstract: In this talk, I will present recent joint work with R. Douglas and S. Sun on multiplication operators by finite Blaschke products on the Bergman space of the unit disk. Using the group-like property of local inverses of a finite Blaschke product φ, we show that the largest C^*-algebra in the commutant of the multiplication operator M_φ by φ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of φ^-1 o φ over the unit disk. If the order of the Blaschke product φ is less than or equal to eight, then every C^*-algebra contained in the commutant of M_φ is abelian and hence the number of minimal reducing subspaces of M_φ equals the number of connected components of the Riemann surface of φ^-1 o φ over the unit disk.

Abstract: The legend of Ramanujan is one of the most romantic stories in the modern history of mathematics. It is the story of an untrained mathematician, from south India, who brilliantly discovered tantalizing examples of phenomena well before their time. Indeed, the legacy of Ramanujan's work (as a whole) is well documented and includes direct connections to some of the deepest results in modern number theory such as the proof of the Weil Conjectures and the proof of Fermat's Last Theorem. However, one final problem remained, the enigma of the functions which Ramanujan discovered on his death bed. Here we tell the story of Ramanujan and this final mystery.

Abstract: In his last letter to Hardy (written on his death bed), Ramanujan gave examples of 17 functions he referred to as "mock theta functions". Without a definition and without good clues, number theorists were unable to make any real sense out of these peculiar functions. Nevertheless, these examples make important appearances in many disparate areas of mathematics, a fact which inspired Freeman Dyson to proclaim:

"Mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions." --Freeman Dyson, 1987

In this lecture I will describe the solution to this challenge and give an indication of some of the open problems which have now been solved as a result.

Abstract: An algebra is a set together with several operations defined on the set. An algebra from a given class of algebras is finitely presented if the algebra can be defined by a finite set of equality relations put on its generators. A typical example is a finitely presented group. Clearly, not every algebra is finitely presented. However, one may ask the question whether a given algebra can be finitely presented if one allows to add new operations to the algebra. We motivate and discuss the question and related results from points of view of algebra, logic, and theoretical computer science.

Abstract: SL(n)-character variety of a (discrete) group G is the space of all SL(n,C)-representations of G modulo conjugation. (SL(n,C) can be replaced here by any other classical group of matrices). Goldman proved that character varieties of surface groups have a natural symplectic structure. We prove that under a certain "small" algebraic condition, character variety of a 3-manifold M with boundary F is a Lagrangian submanifold of the character variety of F. This is joint work with C. Frohman.

Abstract: Girls in elementary and middle school, along with the families, will be treated to some short expository talks about interesting mathematics. They will then have the opportunity to explore mathematical ideas more deeply at the mentor-run discovery stations.

Abstract: A Diophantine m-tuple is a set of m-positive integers
{a₁, a₂, ..., a_m} such as the product of any two of them plus 1 is a square. For example, {1,3,8,120} is a Diophantine quadruple. There are infinitely many Diophantine quadruples. It is conjectured that there is no Diophantine quintuple but this has not been proved yet. In my talk, I will survey what is known about this subject along with variations of it with rational contents, or polynomial contents, or replacing the squares by larger powers, or by Fibonacci numbers, etc.

Abstract: Consider a Riemann surfaces endowed with a zero curvature metric (which must have singularities, if the genus of the surface is 2 or more). Translation flows on such surfaces are closely related to the geodesic flow. For almost all flat surfaces, translation flows on them are ergodic, (Howard Masur, William Veech, 1982) which implies that time averages of integrable functions converge to their space average. The main result of the talk is an asymptotics for the rate of convergence. These results extend earlier work of Anton Zorich and Giovanni Forni; the approach is close in spirit to that of Giovanni Forni.

The proof uses methods of symbolic dynamics. The main tool is a special class of automorphisms introduced by Anatoly M. Vershik in 1982 (note also the 1977 work of Shunji Ito).

Abstract: The University of Hawai'i at Manoa Center for Biographical Research Brown Bag Biography Luncheon Lecture.

Abstract: In a joint work with Mikhail Tyaglov, we revisit a number of known results and establish several new connections among the following topics:

• Stieltjes and Jacobi continued fractions
• Hankel, Toeplitz, Vandermonde and other structured matrices
• Root localization of univariate polynomials, in particular, Hurwitz stability, hyperbolicity and their generalizations

Abstract: A group is residually finite, if the intersection of its subgroups of finite index is trivial. This implies that finite images approximate the group structure. There is an interplay between the asymptotic behaviour of invariants on the subgroup lattice of a residually finite group and the dynamics of its actions on profinite spaces. One can use these connections to analyze covering towers of nice spaces, like 3-manifolds or graphs. We will present the basic notions and results in the area and outline some of the open problems.

Abstract: We discuss recent progress in describing those properties of algebraic varieties and of semialgebraic sets (and definable sets for other o-minimal structures) which are preserved by diffeomorphism or bilipschitz homeomorphism. We will describe in particular G. Valette's resolution of a 1977 conjecture of L. Siebenmann and D. Sullivan as to the countability of the metric types of germs of analytic spaces.

"Scientific progress consists in the reduction of the arbitrariness of our description of phenomena" (R. Thom).

Abstract: A quasi-modular form for SL2(Z) is an "isobaric" polynomial in E2, E4, E6, the classical Eisenstein series of weights 2, 4, 6. Essentially, a quasi-modular form is "extremal" if it vanishes a lot at infinity without being zero. In several papers, Kaneko, Koike and Zagier pointed out several peculiar properties of these forms:

• they are solutions of a certain class of differential equations

• they often belong to infinite families

• they have remarkable (conjectural) arithmetic properties  

In this talk we will review these properties, then we will introduce a similar theory in non-zero characteristic, where the group SL2(Z) is replaced with GL_2(F_q[T]). Here, new structures appear, notably Greg Anderson's "t-motives", which greatly clarify Kaneko, Koike and Zagier's approach.  

N.B. The talk will be suitable for graduate students.

Tentative List of Participants:
Scott Ahlgren, University of Illinois at Urbana-Champaigne Kathrin Bringmann, University of Minnesota - Minneapolis Jan Hendrik Bruinier, Technical University at Darmstadt, Germany YoungJu Choie Pohang, Technical University, Korea Youn-Seo Choi, Korean Institute for Advanced Studies, Seoul, Korea Amanda Folsom, University of Wisconsin at Madison Sharon Garthwaite, Bucknell University, Lewisburg Pavel Guerzhoy, University of Hawaii at Manoa Paul Jenkins, University of California at Los Angeles, Winfried Kohnen, University of Heidelberg, Germany Karl Mahlburg, Massachusets Institute of Technology Ken Ono, University of Wisconsin at Madison Robert Osburn, University College, Dublin, Ireland David Penniston, Furman University, Greenville Sander Zwegers, University College, Dublin, Ireland

Abstract: We consider the period solutions of the coupled van der Pol equation system. The fact that the single van der Pol equation has a unique limit cycle which is obitally stable is well known and proved by Poincare-Bendixson theorem. The coupled van der Pol equation system we consider constructs the four-dimensional space. Therefore we can not apply Poincare-Bendixson theorem to our system. 

In our system we have two distinctive solutions: in-phase and out-of-phase solutions. We prove that the periodic solution of our coupled van der Pol equation system is in-phase or out-of-phase solution. Also we talk about the application of this system to robotics area: generation of walking patterns.

Abstract: Traditionally algorithmic notions are used to formalize the intuitive concept of randomness for infinite sequences of bits. Recently, notions from effective descriptive set theory have been used.

The interaction also goes the other way. Concepts related to randomness enrich computability theory (and might as well be applied in effective descriptive set theory). A good example for computability is the injury-free solution of Post's problem of Kucera. A further example is the class of K-trivial sets, which forms an ideal of the Turing degrees with nice properties. The construction of a noncomputable K-trivial set provides an alternative injury-free solution.

K-triviality is equivalent to being low for random and several further naturally occurring lowness properties. Recently, highness properties have been studied, partially because of their relevance in reverse mathematics. The property of being LR-hard is equivalent to a number of other properties within the sets below 0', for instance to being uniformly a.e. dominating.

Abstract: The starting point of Lagrangian and Hamiltionian mechanics is the observation that the form of Newton's third law F=ma is not invariant under general coordinate changes when the force F is the gradient of a potential F = grad(V). Lagrangian and Hamiltonian mechanics are two equivalent but essentially different solutions to make the form of the equations invariant under general coordinate changes. However, they do not have exactly the same features. The Lagrangian point of view seems to be better adapted for (quantum) field theories, whereas the Hamiltonian point of view seems to be better adapted for non-relativistic quantum mecahnics. In this talk I will argue that most (if not all) features of Lagrangian mechanics are also present in symplectic geometry (a generalization of Hamiltonian mechanics) and that one obtains a better understanding of Lagrangian mechanics when seen this way. No prior knowledge on mechanics other than Newton's law is presupposed, nor mathematics beyond the beginning graduate level (though some knowledge of differential geometry and fiber bundles will help in understanding the last part of my talk). 

Abstract: In 1950's R.Thom and H.Whitney began to study singularity theory of differentiable maps between manifolds. One can study the theory from a local or global viewpoint, both of which are interesting and attractive independently. In 1955 Thom introduced the notion of the "Thom polynomial of singularities" in order to study the universally global behavior of a generic smooth map. This is a polynomial written by cohomology classes which is Poincare dual to the homology class represented by the closure of the singular point locus. The purpose of this talk is to discuss the Thom polynomials and other obstruction classes by referring to recent progress.

Abstract: In this talk, I'll show how two basic notions in quantum cryptography and quantum error correction are complementary to each other. Error-correcting codes for quantum channels (mathematically given by completely positive maps) are the key vehicles used to avoid noise such as decoherence induced by physical attempts to build quantum computers.

Private codes for quantum channels play a central role in the development of private quantum communication networks designed to prevent adversarial attacks by eavesdroppers. It turns out that a code is private for a channel precisely when it is correctable for a complementary channel, and there is a straightforward algebraic recipe that allows one to move between the two perspectives. Moreover, an approximate version of the relationship can be quantified in terms of diamond (or completely bounded) norms for channels.

I'll begin with an introductory look at the two notions, then formulate the main result and discuss potential cross-fertilization between the fields. This talk is based on joint work with Dennis Kretschmann (TU Braunschweig) and Robert Spekkens (Cambridge)

Abstract: Measures constructed using nonstandard analysis are automatically continuous. In part I (last November) I used this remarkable fact to prove a version of the Borel-Cantelli Lemma for finitely-additive measures. In this part I will use a similar argument to prove a pretty theorem of Banach on representations of finitely-additive measures.

Abstract: We present a general result to decide when the cut locus for Riemannian metrics on a two-sphere of revolution is a single branch and the conjugate locus has an astroid shape with 4 cusps. This is applied to space and quantum dynamics.

Abstract: A knot is an embedding of the circle into the 3-sphere. Associated with any a knot K is an integer, called the Heegaard genus of the knot exterior and denoted g(K), which measures the topological complexity of K.  Given knots K_1 and K_2, one can construct their connected sum denoted K_1 # K_2.

After defining these concepts, we will survey some of the authors' results about the behavior of Heegaard genus of knot exteriors under connected sum operation.   As our main result we will prove that given integers g_i > 1 (i=1,...,n), there exist knots K_i in S^3 so that:

1) g(E(K_i)) = g_i, and:

2)g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)).

As we will see, this proves the existence of counterexamples to Morimoto's Conjecture.

Available at:

http://arxiv.org/abs/math/0701765 and http://arxiv.org/abs/math/0701766

Abstract: The sofic property of groups generalizes amenability and residual finiteness. Roughly, a group is sofic if there exists "finite approximations" of the group. I'll exhibit new isomorphism invariants for actions of sofic groups that generalize Kolmogorov-Sinai entropy. These new invariants are used to classify Bernoulli systems over a large class groups, including all infinite linear groups.

Preprint: Isomorphism invariants for actions of sofic groups

Abstract: The rank-k numerical range of a matrix M is the set of complex z such that for some rank-k projection P we have PMP=zP. Among the many generalizations that have been proposed for the classical numerical range (which is the rank-1 numerical range), this one seems especially promising. It has, for example, applications in QIT (quantum information theory); indeed, its study was first suggested by problems in quantum error correction. It also provides a striking extension of the Toeplitz-Hausdorff theorem: numerical ranges of all ranks are convex subsets of the complex plane.

In this talk we'll survey recent developments, including the connections with certain eigenvalue interlacing phenomena that have a history stretching all the way back to Cauchy. 

Abstract: The duality between measures and the sets they ``charge'' is a central theme in modern analysis. An effective analogue of this question is: Given a real x, does there exist a (probability) measure relative to which x is effectively random in the sense of Martin-Loef (so that x is not an atom of the measure)? And if such a measure exists, can we ensure that it has certain properties (non-atomic, of a certain minimum capacity, etc)? This is an ongoing project with Theodore Slaman (Berkeley).

The answers to these questions exhibit an unexpected interplay between logical and measure theoretical complexity. The techniques used in the proofs are drawn from various areas of logic and analysis, such as Turing degrees, effective compactness, determinacy, fine structure of the constructible universe, or Hausdorff measures and potential theory.

I will describe these results in a manner that is, I hope, accessible to non-logicians, too.

Abstract: A very nice class of 3-balls (called butterflies) with faces identified by pairs, such that the identification space is S³, and the image of a preferred set of edges is a link, is defined. As motivation we give some examples. It is proved that every link can be represented in this way (butterfly representation). The butterfly number of a link is also defined and we prove that this number and the bridge number of a link coincide.

Joint work with: M. Hilden (U. of Hawaii at Honolulu, USA), J. M. Montesinos (U. Complutense de Madrid, Spain), and M. Toro (U. Nacional de Colombia).

Abstract: We establish four local measures for modeling dependence between two random variables, based on coefficients of dependence between two events, and study their properties. Their use in constructing dependent random variables and in quantitatively measuring the magnitude of existing dependence is outlined.

We suggest these measures either as an alternative, or as an addition to the copula approach, recently used in dependence modeling. In our opinion, it will fill the gap in the now existing curriculum, on measuring and studying dependence, at introductory level courses on Probability and Statistics.

Abstract: In this talk I will consider single-input optimal control problems with state space constraints that have strong geometric properties, more specifically, are given by codimension 1 integral submanifolds of an admissible control for the problem. This geometric structure is utilized to construct a local embedding of a boundary arc into a local field of extremals and prove the strong local optimality of the trajectories in this field.

The emphasis is on a general approach to sufficient conditions for optimality derived through geometric constructions (regular synthesis, method of characteristics). An important step is making the connections between stronger necessary conditions for optimality that hold because of the geometric properties of the constraints and conditions that enable the construction of a parameterized family of extremals. /p>

The results are motivated by and will be illustrated for the problem of determining a base doping profile that minimizes the transit time in homogeneous bipolar transistors in electronics.

Abstract: In the talk we will show how the tools of optimal control theory can be applied to derive optimal protocols for mathematical models of novel cancer treatments. We will focus on models for tumor anti- angiogenesis, a novel medical approach to cancer treatment that aims at preventing the development of the blood vessel network a tumor needs for growth. Some mathematical models for tumor anti- angiogenesis originally introduced by a group of researchers from Harvard School of Medicine and National Cancer Institute of NIH will be analyzed as optimal control problems with the objective of minimizing the size of the tumor at the end of therapy. The dynamics of the system describes the growth of the tumor volume and its vascularization under the effects of control functions representing the dosage of the angiogenic inhibitors with a constraint on the total amount of inhibitors given imposed.

In the talk we shall present a full theoretical solution to the problem in terms of a synthesis of optimal controls and trajectories. Using tools of geometric control theory (e.g., Lie bracket computations), analytic formulas for the theoretically optimal solutions will be given. Optimal controls are concatenations of bang- bang controls (representing therapies of full dose with rest periods) and singular controls (therapies with specific time-varying partial doses). These optimal solutions, although medically not easily realizable because of the feedback form of the singular portion, provide the benchmark to which simpler, but realizable protocols can be compared. Some examples of excellent suboptimal protocols that come within 1% of the optimal values will be given.

Another novel approach to cancer treatment includes combination therapy that augments anti-angiogenic treatment with chemotherapy. The model for that treatment then also includes a killing term on the primary tumor volume which introduces a second control into the system. Due to the multi-control aspect, even with simplified dynamical equations, this becomes a challenging problem mathematically. Initial results about the structure of optimal controls will be presented and some open problems will be formulated. 

Abstract: In computable structure theory, one examines various countably infinite structures (such as linear orderings and graphs) for their computability theoretic properties. For example, the standard theorem that any two countable dense linear orders without endpoints are isomorphic can be carried out computably, in the sense that if the two countable dense linear orders are nicely presented, then there must be a computable isomorphism between them. However, there are many examples of computable structures that are isomorphic but not computably isomorphic.

This talk will be an introduction to computable structure theory, explaining some standard examples, and indicating areas of current research.

Abstract: The talk will be about the use of mathematical models by the Club of Rome, M. King Hubbert, Albert Bartlett, and Kenneth Deffeyes to foretell our current energy crisis, and outline how mathematical models are being used in my work to help ameliorate some of these problems.

Biographical Note: Prof. Antal holds a a Ph.D. in Applied Mathematics from Harvard (1973). His recent work on "flash carbonization" of organic matter (even old tires) to more efficiently produce charcoal has been taken up by several corporations, among them the largest charcoal maker in the U.S., Kingsford Charcoal (a subsidiary of Clorox). 

Abstract: Deducing properties of an object from properties of its Fourier transform (and conversely) is one of the continuing themes of Fourier analysis. I will talk about the relationship of the support of functions, measures and distributions to their Fourier transforms. Weak* limits involving bounded approximate identities play an important role.

Some of this is work by F. J. Gonzalez Vieli and myself, jointly and separately.  

Abstract: A set E of integers is an "interpolation set" if every bounded function on E extends to an almost periodic function. Sequences growing exponentially (e.g., the powers of 3), a.k.a. "Hadamard sets", are examples, but there are interpolation sets which are not finite unions of Hadamard sets. I will review the history of interpolation sets and touch upon more recent work, including that of Hare, Ramsey, and myself (in several subsets). I will also give some open questions and possible directions for future work. Prerequisites: some analysis (measure, integration, general topology) and group theory (homomorphisms of abelian groups). 

Abstract: Hamilton Library has a large collection of online resources for obtaining books, journal articles and reviews. The talk will be a tour of these resources.

Abstract: We consider a situation where coalitions are formed to divide a resource. As in real life, the value of a payoff to a given player is allowed to depend on the payoff to other players with whom he shares a common interest. There are various notions of equilibrium for this type of game, including the core and no-treat equilibrium. These stabilities may exist or not, depending on the power structure and the rule for allocating the resource. It is shown that under certain conditions, the no-treat equilibrium can exist even though the core is empty. 

Abstract: The purpose of this paper is to investigate the theory behind the new universal performance measure (the so called Omega function), which was first introduced by Con Keating and William F. Shadwick in 2002.

In the first section, we review some rudimentary probability. We then define the Omega function, introduce some of its properties, and prove these properties without continuity assumptions.

We also define the standard dispersion, a new statistic derived from the Omega function. We prove one new theorem about the range of the standard dispersion for a finite sample. The structure of the second section on the Omega function follows closely that of a recent talk given by Ana Cascon and William Shadwick. In the last section, we demonstrate these properties with real-life data.

Abstract: This thesis contains two independent papers that summarize the author's study on the statistical properties of the returns series of the daily price index series of the Shanghai Stock Exchange (SSE) (hereunder called "returns series" ) for trading days between 01/01/1991 and 14/11/2007, and the simulation of the detrended logarithmic SSE index series (hereunder called "detrended series" ) by the ergodic, generalized hyperbolic (GH) diffusion whose invariant density is the generalized hyperbolic (GH) density.

More specifically, the first paper herein gives the right tail index estimate, the left tail index estimate, for the returns series, and the Hurst index for the absolute returns series, all of which are consistent with what most researchers obtained for stock market index series and tend to argue that extreme care should be taken when making assumptions on the underlying stochastic process that generates the stock price index series. These estimates are obtained by the simplest method -- the traditional ordinary least squares (OLS) method, to avoid unnecessary or even wrong assumptions on stationarity or dependence structures of the index series, whereas some comparative estimates via different methods are also provided.

The second paper is devoted to testing out the ergodic GH diffusion by the detrended series. It constructs for the detrended series an ergodic GH diffusion whose invariant density is proportional to the GH density and estimates the unknown parameter vector by Markov chain Monte Carlo (MCMC) methods, which induces six possible models from six independent, convergent Markov chains by the Monte Carlo standard error (MCSE) convergence diagnostic. Unfortunately the Gelman-Rubin convergence diagnostic failed to converge to one even after 350,000 iterations of the Metroplolis-Hastings algorithm, since the six chains converged to their own stationary distributions at around 50,000 iterations.

After six Monte Carlo estimates of unknown parameter vector are obtained, uniform residual test is carried out for four of the six possible models due to the computational capability of Mathematica and Matlab, and the test results statistically reject all four selected models, thus rejecting the null hypothesis that the underlying stochastic process for the detrended index is an ergodic GH diffusion.

Abstract: With the numerous discoveries published recently in the study of harmonic weak Maas forms and the closely associated weakly holomorphic modular forms, it is important to understand the arithmetic properties of these forms. This paper will focus on the holomorphic parts from the q-expansions of harmonic weak Maas forms since it has been recently discovered that they have numerous applications throughout number theory.

Although the numerical data cannot be used to completely prove the irrationality of these coefficients, it does provide strong evidence to suggest that conclusion. Once this is established, we branch out into congruences that were observed through the investigation of various harmonic weak Maas forms. A recurring pattern led to the formulation of a new theorem to describe the behavior of the coefficients. It is important to note that although the numerical examples presented here are obtained focusing on modular forms of integral weight, the same techniques can be applied to produce similar results for those of half-integral weight as well.

The reason for looking exclusively at integral weights is to simplify the calculations involved when ?finding the coefficients, thus cutting down on computation time as well as increasing the number of coefficients that can be used as numerical data. The proposed theorem summarizes the new ?findings and is based on congruences demonstrated through simulations. This new theorem is in the process of being proved by Pavel Guerzhoy and his research team.  

Abstract: In this project we will be looking at the change in control required to transfer a satellite between two elliptic Keplerian orbits. We will first derive the equations of motion for our satellite and then study the controllability properties of our system. We will introduce a simple feedback controller and prove local asymptotic stability of the target orbit. The goal of this paper is to prove stability using both geometric control theory as well as stabilization methods and thus link the work done by [3] and [8]. Our primary tool to accomplish this will be LaSalle’s invariance principle. 

Abstract: An autonomous underwater vehicle (AUV) is expected to operate in an ocean in the presence of poorly known disturbance forces and moments. Owing to such an uncertain environment, it is not possible to apply open-loop control scheme for the motion planning of the vehicle. The objective of this thesis is to develop a robust feedback trajectory tracking control scheme for an AUV that can track a prescribed trajectory amidst such disturbances.

Abstract: The purpose of this paper is fourfold: (1) to survey some classical and recent results in the theory of distribution of zeros of entire functions, (2) to demonstrate a novel proof answering a question of Raitchinov, (3) to present some new results in the theory of complex zero decreasing operators, and (4) to initiate the study of the location of zeros of complex polynomials under the action of certain linear operators.

In addition, several open problems are given. 

Abstract: In this paper we will study the pulse sequences in NMR spectroscopy and quantum computing as a time control problem. Radio frequency pulses are used to execute a unitary transfer of state. Sequences of pulses should be as short as possible to minimize decoherence. Suppose we are given a controllable right invariant system on a compact Lie group. What is the minimum time required to steer the system from an initial point to a specified final point?

Abstract: In 1988, Dusa McDuff constructed the only known example of a non-Hamiltonian symplectic circle action with fixed points. Since on a Kaehler manifold, any symplectic circle action with fixed points is Hamiltonian, McDuff's construction provides an example of a symplectic manifold which cannot support a Kaehler structure. A procedure for generalizing the McDuff construction would therefore furnish a new class of symplectic, non-Kaehler manifolds. We prove some results in this direction, focusing on semifree symplectic circle actions on 6-manifolds with fixed surfaces.

Abstract: The Dirichlet problem may be interpreted as that of finding the steady-state temperature distribution on a suitably nice heat-conducting region D, given the temperature function on the boundary of D. An exposition is given on solving the problem via Brownian motion. The solution at a point x in D is shown to be the average of the temperatures at the boundary points where Brownian particles, beginning at x, first exit D. 

Abstract: In this project we will be looking at Sophus Lie's desire (his so called Id'ee Fixe) to apply Contact Transformations (what would eventually develop into the modern idea of a Lie Algebra) in order to arrive at symmetries of differential equations, and thus certain solutions. Our goal-as well as Lie's-is to develop a more universal method for solving differential equations than the familiar cook-book methods we learn in an introductory ordinary or partial differential equations class.

We answer three questions. What was the historical underpinning of Sophus Lie's theory? How do we find the symmetry Lie algebras? How do we use the symmetry Lie algebras to find solutions to the differential equation? (In order to answer these questions we will need to fill in some background material and our answers will also result in a novel derivation of the "Fundamental Source Solution.") Our second objective will be to establish a connection between solvability in Galois Theory and in Differential Equations. We will assume a familiarity with certain algebraic concepts from Abstract Algebra.

Abstract: Intuitively, we associate the process of diffusion with a homogenizing effect that leads to uniform spatial distributions. Surprisingly, as we will see through simple mathematical modeling, diffusion of chemical substances (particularly those that activate/deactive melanin) and their cross-reactions can lead to non-uniform patterns that correlate well with certain biological systems (particularly those we can necessarily see, as in animal coats). We will analyze these Reaction-Diffusion (RD) models to determine the conditions under which certain patterns form.

Abstract: The American Heritage Dictionary defines gerrymandering as the act of “dividing a geographic area into voting districts so as to give unfair advantage to one party.”

The problem of gerrymandering has led to the development of several mathematical measures of shape compactness, some of which have been used in court cases to argue for or against the legality of congressional redistricting plans. In this talk, we will show how the notion of convexity can be used to detect irregularly shaped districts. We will explore both theoretical and empirical aspects of this convexity-based measure of shape compactness.

Biographical Note: Geoff Patterson is a new TA at UHM. As an undergraduate, Geoff did three "research experiences". This talk, which combines probability and geometry, is based on one of those projects.

Abstract: The U. S. Census Bureau conducts many nationwide probability sample surveys to estimate monthly, quarterly, and annual characteristics about the economy and people of the United States. It is known that sampling and estimated results can be made more efficient by use of observed auxiliary information (variables) that are highly correlated with yet unobserved variables of interest. Simple examples will illustrate how gains in efficiency can be realized before sampling (e.g., stratification), during sampling (e.g., probability proportional to size sampling), and after sampling (e.g., ratio estimation). Thus correlation is an important and useful concept in probability sampling.

Using an elementary but important identity associated with Lagrange, this talk presents a simple proof (Wright, 1992) that Pearson's correlation coefficient r is always between -1 and 1, and that all points (x,y) fall on a straight line if and only if the square of r is 1. Another application of the identity is given to the problem of optimal allocation when sampling from a finite universe. The last part of the talk will focus on mention of some problem areas and challenges in data collection and dissemination where correlated data can help (e.g., disclosure avoidance, small area estimation, imputation for missing data) that deserve more attention from statisticians.

Abstract: It is easy to see that there are infinitely many integer solutions to the title equation: (1, 1, 1), (1, 1, 2), (1, 2, 5), (2, 5, 29), (5, 29, 433), ... We will describe what is known about the Markoff equation and what is open. For instance, an open problem is the following conjecture:

For z > 0, there exists at most one couple (x, y) with x < y < z, such that (x, y, z) is an integer solution of X² + Y² + Z² = 3XYZ.

Abstract: I will talk about our new program, its start-up (the first class will be in the fall of 2009) and the stochastic processes we use in finance.

For more information, go to the program's web page.