Invariant measures on countable models
Logic Seminar
- Dr. Cameron Freer
(UHM Math. Dept.).
Wed., October 13, 2010, 2:30, Keller 401
The Erdos-RÉnyi random graph construction can be seen as inducing a probability measure concentrated on the Rado graph (sometimes known as the countable "random graph") that is invariant under arbitrary permutations of the underlying set of vertices. The following question arises naturally: On which countable combinatorial structures is there such an invariant measure? Up until recent work of Petrov and Vershik (2010), it was not even known if Henson's universal countable triangle-free graph admitted an invariant measure.
We provide a complete characterization of countable structures admitting invariant measures, in terms of the model-theoretic notion of definable closure. This leads to a characterization for ultrahomogeneous structures, as well as new examples of invariant measures on graphs, trees, and other combinatorial structures. Joint work with Nate Ackerman and Rehana Patel.
Welcome Graduate Students - Lecture I
- Prof. Erik Guentner
(UHM Math Dept).
Wed., August 18, 2010, 3:30, Keller 401
Welcome Graduate Students - Lecture II
- Prof. Les Wilson
(UHM Math Dept).
Thurs., August 19, 2010, 3:30, Keller 401
On some conjectures of Ron Brown and Alexander Grothendieck
Colloquium
- Greg Brumfiel
(Stanford University and UH-Manoa).
December 10, 2010, 3:30, Keller 401
Associated to a field $k$ is a lattice tower of finite Galois extension fields of $k$ and the inclusions between them, inside an algebraic closure. Associated to this tower of field extensions is the lattice tower of the finite Galois groups of the extensions and the surjections between these groups. In the late 60's and 70's, work of Neukirch, Uchida, and others led to the remarkable result that if two number fields (finite algebraic extensions of $\mathbb{Q}$ have isomorphic Galois towers (in the obvious sense of matching up the groups and surjections in the two towers) then the two number fields are actually isomorphic. In the 90's, motivated by conjectures of Grothendieck dating from the early 80's, this result was extended by Pop, Mochizuki, and others to arbitrary finitely generated fields over $\mathbb{Q}$, that is, finite algebraic extensions of the field of rational functions in $d$ variables. More precisely, Grothendieck conjectured that a certain type of comparison of the Galois towers of two such fields $K$, $L$ should correspond to inclusions $K \to L$, and this also was proved.
As an illustration of the nature of the result, if $X$ is a compact Riemann surface then its field of meromorphic functions can be expressed $E(X) = \mathbb{C}(z)[w] / (f(z, w) = 0)$, where $f(z, w)$ is a polynomial in two variables. The Galois tower of $E(X)$ depends only on the topological type of $X$, that is, its genus. But polynomial $f(z, w)$ sees only finitely many complex numbers as coefficients. We then have small subfields of the meromorphic functions, $E_K(X) = K(z)[w] / (f(z,w) = 0)$, that are finitely generated over $\mathbb{Q}$. The Galois tower of $E_K(X)$, which retains arithmetic properties of $X$, essentially determines the Riemann surface $X$ itself, inside a $6g - 6$ dimensional moduli space of Riemann surfaces of genus $g \geq 2$.But Grothendieck conjectured more. Given a Riemann surface $X$ defined, say, by a polynomial $f(z, w)$ with $\mathbb{Q}$ coefficients, Grothendieck believed all rational solutions of the equation $f(r, s) = 0$ should be explained in terms of another type of comparison of the Galois tower of the field $\mathbb{Q}$ and the tower of the arithmetic meromorphic function field $E_{\mathbb{Q}}(X)$. [For example, a problem of some interest has been to understand rational solutions of $r^n + s^n = 1, n \geq 3$. The polynomial $z^n + w^n -1$ defines a Riemann surface of genus $g = (n-1)(n-2)/2$. The story goes that in the 80's Grothendieck thought he might be able to settle such arithmetic questions by his Galois tower considerations, or at least recover Falting's results on the finiteness of the number of rational solutions when $g \geq 2$.]These `section conjectures' of Grothendieck remain unproved. I want to discuss some related conjectures about ordered fields that Ron Brown and I have been discussing since 2007. For example, is it possible to express the real numbers as a composite field $\mathbb{R} = \mathbb{Q}^r F$, where $\mathbb{Q}^r$ is the field of real algebraic numbers and $\mathbb{Q}^r \cap F = \mathbb{Q}$? Presumably the answer is `no', even though there are examples of such decompositions $ \mathbb{C} \simeq \mathbb{Q}^rF[i]$. (The usual conjugation in $\mathbb{C}$ badly scrambles $F$. Put another way, the field $\mathbb{Q}^r F$ here is a big ordered field, but quite different from $\mathbb{R}$. All the Dedekind cuts of $\mathbb{Q}$ determined by elements of the ordered field $F$ in known examples of such decompositions of $\mathbb{C}$ are rational cuts, or $\pm\infty$).More generally, if $R$ is any ordered field with $R = \mathbb{Q}^r F$ as above, and such that $R[i]$ is algebraically closed, Ron has suggested exactly what the nature of the field $F$ should be. In particular, elements of $F$ should always produce rational Dedekind cuts of $\mathbb{Q}$, or $\pm\infty$. These suggestions, if true, imply some of the Grothendieck section conjectures about rational points, and seem to provide a new way of looking at the Grothendieck conjectures.
Towards a Theory of Dynamical Complex Multiplication
Seminar
- Diane Yap
Friday Dec. 10, 2010, 1pm in Keller 402
After introducing some background in the field of arithmetic dynamics (iteration of rational maps on projective varieties), the speaker willdescribe a fundamental open problem in the field: the question of a uniform bound for rational preperiodic points. There is a natural analogy between rational maps on the projective line and multiplication-by-m maps on an elliptic curve, in which torsion points on elliptic curves correspond in a natural way to preperiodic points for a rational map. Given Mazur's and Merel's theorems on uniform bounds for torsion points on elliptic curves over number fields, the dynamical uniform boundedness conjecture seems reasonable. However, the proof techniques rely heavily on the group structure for elliptic curves, so there is no hope of generalizing them to a dynamics setting.
An elliptic curve with complex multiplication (CM) is one in which the endomorphism ring of the curve is strictly larger than the ring of integers. For these maps, there is a more elementary proof of a uniform bound for torsion points due to Olson. The speaker will briefly outline Olson's proof, and explain the ways in which she hopes to generalize his work in developing a theory of dynamical complex multiplication.
Interactions of randomness and computability
Colloquium
- Prof. AndrÉ Nies
(Dept. of Computer Science , University of Auckland, New Zealand).
Wed., December 1, 2010, 3:30, Keller 401
Randomness and complexity are closely connected. We briefly consider the meaning of the two concepts in the sciences. Thereafter, we provide mathematical counterparts of the two concepts for infinite sequences of bits. Later on in the talk, we discuss mathematical theorems showing the close relationship between the two.
For a mathematician, randomness of a sequence of bits is usually understood probability-theoretically. She may think of a random sequence as the outcomes of a sequence of independent events, such as coin tosses. Theorems about random objects hold outside some unspecified null class; for instance, a function of bounded variation is differentiable at every ''random'' real. It makes no sense to say that an individual real is random. To obtain individual sequences that are random in a formal sense, one introduces a notion of an effective null class. A sequence is random in that sense if it avoids each effective null class. For instance, Chaitin's halting probability is random in the sense of Martin-Loef, a concept central in the hierarchy of effective randomness notions. Every computable function of bounded variation is differentiable at any Martin-Loef random real; conversely, if the real is not Martin-Loef random then some computable function of bounded variation fails to be differentiable (Demuth 1975; recent work of Brattka, Miller and Nies). Effective randomness notions interact in fascinating ways with the computational complexity of sequences of bits. For instance, being far from Martin-Loef random is equivalent to being close to computable in a specific sense (Nies, Advances in Math, 2005).
Global Weyl modules, BGG Duality and the Catalan numbers
Colloquium
- Nathan Manning
(Math. Dept., University of California, Riverside).
Thurs., November 4, 2010, 2:30, Keller 401
The discovery of BGG duality for category O in the 1970s stimulated research in that area and provided homological insight and information about the blocks of the category. We present partial progress in an attempt to obtain a result of this kind for affine Lie algebras, and the problem takes a surprisingly combinatorial direction.
Infinitesimals in Probability
Colloquium
- Prof. David Ross
(UHM Math. Dept.).
Fri., September 17, 2010, 3:30, Keller 401
Nonstandard analysis makes it possible to assign positive infinitesimal probabilities to events in a mathematically sensible way. One might expect this ability to be of primary importance in the practice of applying nonstandard methods in probability theory. In fact, the philosophical literature in the foundations of mathematics often does use infinitesimals in this way. However, for mathematicians who use nonstandard methods in probability, this is not the major reason why the technology is useful.
This talk will look at some recent arguments using infinitesimals from philosophy, explain why they don't work, and illustrate why transfer and saturation, both properties familiar to mathematical logicians, are more central to nonstandard probability.
Riemann integrals and random reals
Logic, lattice theory and universal algebra seminar
- Prof. Dale Myers
(UHM Math. Depart.).
Wed., September 1, 2010, 2:30, Keller 401
From Zeta to L to A: Some number theory using the Riemann zeta-function, L-functions, and automorphic forms
Colloquium
- Prof. Ellen Eischen
(Dept. of Mathematics, Northwestern Univ.).
Fri., August 27, 2010, 3:30, Keller 401
A century before Riemann established a link between the zeta function and the distribution of prime numbers, the values of the zeta function at certain points had already been linked to important mathematical data. Notably, Euler showed that the values of the zeta function at negative integers are rational and, in fact, closely related to the Bernoulli numbers. Later, Kummer showed that special values also satisfy certain strong congruence properties modulo powers of a prime (leading to a "p-adic" theory).
The Riemann zeta-function is actually the first example of a larger class of functions called "L-functions." In analogue with the zeta-function, L-functions are closely tied to data of number-theoretic significance. Certain values of some L-functions also satisfy strong congruence properties (leading to a p-adic theory). In this talk, I will introduce L-functions, their significance in number theory, and some related topics in my research.
About a Problem Arising in Radiative Heat Transfer
Colloquium
- Prof. Oscar Lopez Pouso
(Dept. of Applied Mathematics, Universidade de Santiago de Compostela , SPAIN).
Thurs., August 26, 2010, 3:30, Keller 403
Among the three main ways of heat transfer, namely conduction, convection, and radiation, the third one becomes important and even dominant under certain circumstances, the more clear to explain is that radiation becomes dominant when temperatures are very high. That situation is referred to in the engineering bibliography as "radiative equilibrium" (see M. F. MODEST, "Radiative Heat Transfer", Elsevier, 2003). We will talk about the heat equation in radiative equilibrium when the spatial domain is the 1-D slab.
The references are the following: [1] Maurizio FALCONE and Oscar LOPEZ POUSO, Analysis and comparison of two approximation schemes for a radiative transfer system. Mathematical Models and Methods in Applied Sciences, vol. 13, number 2, pp. 159-186, February 2003. [2] Oscar LOPEZ POUSO and Rafael MUNOZ SOLA, About the solution of the even parity formulation of the transient radiative heat transfer equations. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A: Matematicas, vol. 104, number 1, pp. 129-152, 2010.
New Operator-Difference Schemes in Hilbert Space
Summer Colloquium
- Prof. Mehmet Emir Koksal
(Florida Institute of Technology).
Fri., July 23, 2010, 1:30, Keller 401
Facilitating Active Learning in Calculus
Special Seminar
- Prof. Jennifer Hyndman
(Univ. of Northern British Columbia, British Columbia, Canada).
Tues., July 13, 2010, 1:30, Keller 4024
Logic, lattice theory and universal algebra seminar
- Prof. Dale Myers
(UHM Math. Depart.).
Tues., May 4, 2010, 3:30, Keller 404
Among other things this talk deals with an alternate, easier, proof of Fubini's theorem using the Axiom of Determinacy.
Special Colloquium: THE DEPARTMENT OF MATHEMATICS AND GRADUATE WOMEN IN SCIENCE PRESENTS
- Alexis Rudd
(UH Graduate Student, Department of Zoology).
Fri., April 30, 2010, 3:30, Keller 401
Marine mammals, including dolphins and whales, spend most of their time underwater. As a result, traditional visual methods of behavioral observation can be inadequate. New methods, especially those allowing sound, allow us to gain a better understanding of the behavior and biology of these animals. I will give an overview of some of these methods, and talk about how mathematics, physics, computer programming, and engineering are becoming an indispensable part of the study of dolphins and whales in the wild.
Variational and Geometric Methods in Image Processing and Analysis
Colloquium
- Prof. Darryl D Holm,
(Imperial College London, England).
Wed., March 31, 2010, 2:30, Keller 401
I will speak about some current research at Imperial College London on variational and geometric methods in image processing and analysis. Applications include Computer Tomography images of the beating heart and Magnetic Resonance Images of brain development (cortex folds). I will concentrate on the goals, methods of approach and opportunities for mathematical analysis in our recent work on image processing and analysis. Many types of mathematics apply in this problem, including soliton theory!
Lattice embeddings into the computably enumerable Turing degrees
Colloquium
- Prof. Steffen Lempp
(University of Wisconsin at Madison).
Mon., March 29, 2010, 3:30, Keller 401
This talk will attempt to explain in lattice-theoretic terms the status quo of the problem of which finite lattices can be embedded into the computably enumerable Turing degrees, concentrating on the lattice theory rather than the computability-theoretic constructions.
In 2000, Lerman proved a characterization of the embeddable finite lattices for join-semidistributive lattices, building on joint work with Lempp in the mid-1990's. (It is generally believed that once the join-semidistributive case is solved, the full characterization is not too difficult.) Subsequent work by Lempp, Lerman and Solomon produced only minor improvements and ended in a stalemate on this problem which has been open for almost 50 years. The problem with Lerman's condition is that it is not known to be decidable (and so in particular not expressible by first-order formula in the language of lattices); rather, it requires the existence of finite sequences of sequences of lattice elements subject to fairly delicate transition rules. I will try to explain Lerman's condition in detail, motivating it somewhat by a hint of where each feature comes from in computability-theoretic terms.
Quasi-homomorphism Rigidity with Noncommutative Targets
Colloquium
- Prof. Narutaka Ozawa
(University of Tokyo, Japan).
Mon., March 22 2010, 3:30, Keller 401
As a strengthening of Kazhdan¥s property (T), property (TT) was introduced by Burger and Monod. In this talk, I will add more rigidity to (TT) and introduce property (TTT). This property is suited for the study of rigidity phenomena for quasi-homomorphisms with noncommutative targets and e-representations.
Logic, lattice theory and universal algebra seminar
- Prof. J.B. Nation
(UHM Math. Depart.).
Mon., March 15, 2010, 3:30, Keller 401
- Prof. BjØrn Kjos-Hanssen
(UHM Math. Depart.).
Mon., March 8, 2010, 3:30, Keller 401
Diagonal Compressions of Matrices and Numerical Shadows
Colloquium
- Prof. John Holbrook
(Univ. of Guelph, Ontario, Canada).
Fri., February 19, 2010, 3:30, Keller 401
A k-dimensional compression of an operator T (acting on a complex Hilbert space) is the restriction to a k-dimensional subspace L of PT, where P is orthogonal projection onto L. In terms of a matrix M representing T, such a compression is the k x k northwest corner of U*MU, where U is any unitary.
Recently the lore of numerical ranges has been enriched by the study of rank-k numerical ranges (a concept arising naturally in quantum information theory). This work may be regarded as a highly successful theory of scalar k-dimensional compressions. The higher-rank numerical ranges may also be viewed as regions of higher density within the classical numerical range. In this talk we'll first review these ideas and then extend them to the study of diagonal (i.e. normal) compressions and the determination of numerical shadows, i.e. densities or measures on the classical numerical range induced by the map u -> (Mu,u) when the unit vector (pure quantum state) u is chosen at random.
Binary Trees in Financial Calculations
Financial Math Lecture Series
- Ed Mirani
(MFE student in the Shidler School of Business).
Thurs., February 11, 2010, 12:00, Keller 413
The lecture is based on Shreve's book about binary trees used for financial calculations.
Calibrating the Complexity of Mathematical Proofs and Constructions
Colloquium
- Prof. Richard A. Shore
(Dept. of Math, Cornell University).
Mon., February 8, 2010, 3:30, Keller 401
We will discuss two related measures of complexity for mathematical theorems and constructions. One asks what proof techniques (or formally axioms) are needed to prove specific theorems. The other asks (for existence proofs) how complicated (in the sense of computability) are the objects that are asserted to exist.
For this talk we will consider some illustrative examples from Combinatorics. In particular, we will consider several theorems of matching theory such as those of Frobenius, (M. and P.) Hall and Kˆnig. While in the infinite case these theorems seem both different and yet somehow the same, an analysis of the countable case in terms of computability or provability clearly distinguishes among them and assigns precise levels to their complexity. At the most complicated level we will consider lies the Kˆnig Duality Theorem: Every bipartite graph has a matching such that one can choose a vertex from each edge of the matching so as to produce a cover, i.e. a set with an element from every edge. This theorem cannot be proven using algorithmic methods even when combined with compactness (Kˆnig's lemma for binary trees) or full Kˆnig's lemma. We will show that it requires highly nonelementary methods as typified by constructions by transfinite recursion, choice principles and, for some versions, even more. If time permits, we may also mention the calibration of some results of Ramsey theory that lie at the other (low) end of our classification scheme: Ramsey's theorem for n-tuples for different n and some consequences such as the theorems of Dilworth and Erdos-Szekeres. (Every infinite partial order has an infinite chain or antichain and every infinite linear order has an infinite ascending or descending sequence.) We will not use, or even consider, any formal systems and no knowledge of logic is presupposed. We will work instead with an intuitive notion of what it means for a function to be computable, i.e. there is a computer program that calculates it given time and space enough and no mechanical failures. We will also explain the relevant combinatorial notions.
SUPER-News and updates about SUPER-M
Colloquium
- Prof. Monique Chyba
(UHM Dept. of Math).
Fri., January 29, 2010, 3:30, Keller 401
On the Classification of Graph C*-Algebras
Colloquium
- Prof. Efren Ruiz
(Dept. of Math, UH Hilo).
Fri., January 22, 2010, 3:30, Keller 401
In recent years, there has been a great deal of interest in graph algebras associated to graphs. In turns out that graph algebras have an attractive structure theory in which algebraic properties of the algebra are related to combinatorial properties of paths is the directed graph.
In this talk, I will show how to construct a graph C*-algebra from a directed graph and introduce a K-theoretical invariant that we claim will classify all graph C*-algebras with finitely many ideals. I will give a class of graph C*-algebras that supports our conjecture.
Buffon Needle and Singular Integrals
Colloquium
- Prof. Alexander Volberg
(Dept. of Math, Michigan State University).
Fri., January 15, 2010, 3:30, Keller 401
We consider the interplay between GMT and Harmonic Analysis. The testing ground for us will be the probability of the long thin needle to land near Cantor sets. We will list several unsolved problems in this area.
NSF Focused Research Group in Algorithmic Randomness
- Prof. Steve Simpson
(Penn State Univ. Math. Depart.).
Tues., January 5, 2010, 3:30, Keller 401
NSF Focused Research Group in Algorithmic Randomness
- Prof. Laurent Bienvenu
(Univ. of Paris 7 Math. Depart.).
Thurs., January 7, 2010, 3:30, Keller 401
Logic Seminar
- Prof. David Ross
(UHM Math. Depart.).
Fri., December 4, 2009, 1:30, Keller 4th floor
Metric Graphs: The Poor Mathematician's Riemann Surface *or* You're in Good Company if Someone Calls You One-Dimensional
Colloquium
- Prof. Xander Faber
(McGill University).
Mon., November 23, 2009, 3:30, Keller 401
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.
Regular idempotents in beta G
Colloquium
- Prof. Yevhen Zelenyuk
(University of Witwatersrand, Jahannesburg, South Africa).
Fri., November 13, 2009, 3:30, Keller 401
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.
Undergraduate Colloquium
- Prof Jim Dator, Director, Hawaii Research Center for Future Studies
(UHM Political Science Dept.).
Mon., November 9, 2009, 1:30, Keller 402
Reaction-Diffusion Equations as Models for Pattern Formation in Biological Systems
Undergraduate Colloquium
- John C. Rader
(UH Math).
Tues., October 27, 2009, 12 noon, Keller 401
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.
A Greedy Sorting Algorithm
Colloquium
- Prof Sergi Elizalde
(Dept of Math., Dartmouth College).
Fri., October 16, 2009, 3:30, Keller 401
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.
Theory and Applications of Optimal Control Problems with Delays
Colloquium
- Prof Helmut Maurer
(Univ. of Muenster, Institute for Computational and Applied Mathematics).
Mon., October 12, 2009, 3:30, Keller 401
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.
The Hierarchy of Definability
Colloquium
- Prof Theodore A. Slaman
(Dept of Math., University of California, Berkeley).
Fri., October 9, 2009, 3:30, Keller 401
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.
Looking back, a Descartes Sample and some Sacred Mathematics
Graduate Student Orientation Talk
- Prof Adolf Mader
(UH Dept. of Math.).
Thurs., August 20, 2009, 3:30, Keller 401
Stable Ramsey's theorem and measure
Colloquium
- Damir Dzhafarov
(Department of Mathematics, University of Chicago).
Thurs., August 20, 2009, 2:30, Keller 401
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.
Top 10 Tips for Math Grad Students (from someone who has not been a grad student for a very long time and therefore probably doesn't know what he's talking about)
Graduate Student Orientation Talk
- Prof David Ross
(UH Dept. of Math.).
Wed., August 19, 2009, 3:30, Keller 401
Do Dogs Know Calculus?
Colloquium
- Prof Timothy Pennings
(Dept of Math., Hope College).
Wed., June 24, 2009, 1:30, Keller 401
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?
Some Algebras have Nonfinitely Axiomatizable Equational Theories
Colloquium
- Prof Kate Scott Owens
(Dept of Math., University of South Carolina).
Wed., May 6, 2009, 3:30, Keller 401
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.
Observations about Perfect Lattices
Colloquium
- Prof. Kira Adaricheva
(Stern College for Women, Yeshiva University, New York).
Wed., Apr. 15, 2009, 3:30 Keller 401
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.
Workshop on Groups, Embeddings and Applications
Thurs.-Sun., Mar. 19-22, 2009
Closed range composition operators
Special Analysis Seminar
- Prof. Pratibha Ghataga
(Cleveland State University).
Mon., Mar. 16, 2009, 3:30, Keller 401
Schur Norms: Basic Methods and Diverse Applications
Colloquium
- Prof. John Holbrook
(Dept of Math, University of Guelph, Canada).
Fri., Mar. 13, 2009, 3:30, Keller 401
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.
A Very Large Integer
Special Logic Seminar
- Prof. Thomas Jech:
(Penn. State University (Retired)).
Wed., Mar. 4, 2:30
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
How Composition Operators (could) Solve the Invariant Subspace Problem
Special Analysis Seminar
- Prof. Joel Shapiro
(Michigan State University (Retired); Portland State University (Adjunct)).
Wed., Mar. 4, 3:30, Keller 401
Special Analysis Seminar
- Prof. Dechao Zheng
(Vanderbilt University).
Mon., Mar. 2, 3:30, Keller 401
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 f, we show that the largest C*-algebra in the commutant of the multiplication operator Mf by f on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of f-1 o f over the unit disk. If the order of the Blaschke product f is less than or equal to eight, then every C*-algebra contained in the commutant of Mf is abelian and hence the number of minimal reducing subspaces of Mfequals the number of connected components of the Riemann surface of f-1 o f over the unit disk.
GENERAL AUDIENCE LECTURE UNEARTHING THE VISIONS OF A MASTER: THE WEB OF RAMANUJAN'S MOCK THETA FUNCTIONS IN NUMBER THEORY
Distinguished Lecture Series:
See Prof. Guentner's website for details.
- Prof. Ken Ono
(Solle P. and Margaret Manasse Professor of Letters and Science and The Hilldale Professor of Mathematics University of Wisconsin at Madison ).
Wed., Feb. 25, 4:30, Bilger 150
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.
FREEMAN DYSON'S CHALLENGE FOR THE FUTURE: THE MOCK THETA FUNCTIONS I
Distinguished Lecture Series:
See Prof. Guentner's website for details.
- Prof. Ken Ono
(Solle P. and Margaret Manasse Professor of Letters and Science and The Hilldale Professor of Mathematics University of Wisconsin at Madison ).
Thurs., Feb. 26, 3:30, Keller 401
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.
FREEMAN DYSON'S CHALLENGE FOR THE FUTURE: THE MOCK THETA FUNCTIONS II
Distinguished Lecture Series:
See Prof. Guentner's website for details.
- Prof. Ken Ono
(Solle P. and Margaret Manasse Professor of Letters and Science and The Hilldale Professor of Mathematics University of Wisconsin at Madison ).
Fri., Feb. 27, 3:30, Keller 401
Inverting the Math Crisis in Hawaii
Math Summit
-
Dr. Uri Treisman
(Founder and Director of the Charles A. Dana Center, Univ. of Texas at Austin).
Sat., Feb 21, 9:00-3:00, School of Architecture Auditorium
Colloquium
- Prof. Bakhadyr Khoussainov
(Department of Computer Science, University of Auckland, New Zealand).
Wed., Jan. 7, 2009, 3:30, Keller 401
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.
Character Varieties
Colloquium
- Prof. Adam Sikora
(Dept of Math, State University of New York at Buffalo ).
Wed., Dec. 10, 2008, 3:30, Keller 401
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.
An Afternoon of Beautiful Mathematics for Girls and Their Families
Special Lecture
- Professors Monique Chyba, Mirjana Jovovic, and Michelle Manes
(UHM Math Dept).
Sun., Dec. 7, 2008, 1-5, UH Campus Ballroom
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.
Diophantine quadruples
Colloquium
- Prof. Florian Luca
(Instituto de Matematicas de la UNAM, Morelia, Mexico).
Mon., Dec. 1, 2008, 3:30, Keller 401
A Diophantine m-tuple is a set of m-positive integers {a1, a2, ..., am} 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.
Translation Flows and Vershik's Automorphisms
Colloquium
- Prof. Alexander Bufetov
(Dept of Math, Rice University).
Mon., Nov. 24, 2008, 3:30, Keller 401
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).
Special Lecture
- Prof. Monique Chyba
(UHM Math Dept).
Thurs., Nov. 13, 2008, 12:00-1:15, Henke Hall 325, 1800 East-West Road
The University of Hawai'i at Manoa Center for Biographical Research Brown Bag Biography Luncheon Lecture.
Algebra and analysis of the generalized Routh-Hurwitz problem.
Colloquium
- Prof. Olga Holtz
(Dept of Math, UC Berkeley).
Mon., Nov. 10, 3:30, 2008, Keller 401
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
Profinite actions: graphs, groups and dynamics
Colloquium
- Prof. Miklos Abert
(University of Chicago).
Fri., Oct. 24, 2008, 3:30, Keller 401
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.
Describing the Tame Geometry and the Tame Topology of Algebraic Varieties and Their Projections
Colloquium
- Prof. David Trotman
(Univ. de Provence, Marseille, France).
Fri., Oct. 10, 2008, 3:30, Keller 401
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).
On Extremal Quasi-Modular Forms for GL_2(F_q[T]) following Kaneko, Koike, and Zagier
Colloquium
- Prof. Federico Pellarin
(University of Saint-Etienne, France).
Fri., August 8, 2008, 3:30, Keller 401
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.
Colloquium
- Prof. Curt Lindner
(Auburn Univ.).
Wed., July 16, 2008, 1:30, Keller 401
Non-existence Theorem Without In-phase and Out-of-phase Solutions in the Coupled Van der Pol Equation System
Colloquium
- Prof. Ben T. Niohara
(Musashi Institute of Technology, Tokyo, Japan).
Fri., May 23, 2008, 3:30, Keller 402
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.
Randomness, computability, and effective descriptive set theory
Logic Colloquium
- Prof. Andre Nies
(University of Auckland, New Zealand).
Thurs., May 29, 2008, 3:30, Keller 401
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.
The Lagrangian in Symplectic Mechanics
Colloquium
- Prof. Gijs Tuynman
(University of Lille I, France).
Fri., May 30, 2008, 3:30, Keller 401
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).
Global Singularity Theory in Differential Topology
Colloquium
- Prof. Kazuhiro Sakuma
(Kinki University, Osaka, Japan).
Mon., May 19, 2008, 3:30, Keller 401
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.
Special Analysis Seminar
- Prof. Parasar Mohanty
(Dept. of Math. and Statistics, India Institute of Tech.).
Wed., May 7, 2008, 1:30, Keller 303
Complementarity in Quantum Cryptography and Error Correction
Colloquium
- Prof. David Kribs
(University of Guelph).
Wed., Apr. 30, 2008, 3:30, Keller 401
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)
Automatic continuity of nonstandard measures: Part II
Analysis Seminar
- Prof. David Ross
(UHM Math. Dept.).
Mon., Apr. 28, 2008, 2:30, Keller 303
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.
Conjugate and Cut Loci for Riemannian Metrics in 2 Dimension Sphere of Revolution with Applications to Orbital Transfer and Quantum Control
Colloquium
- Prof. Bernard Bonnard
(University of Bourgogne, France).
Fri., Apr. 25, 2008, 3:30, Keller 401
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.
On the Heegaard Genus of Knot Exteriors
Colloquium
- Prof. Yo'av Rieck
(University of Arkansas).
Mon., Apr. 21, 2008, 3:30, Keller 401
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.
Analysis Seminar
- Prof. Lewis Bowen
(UHM Math. Dept.).
Mon., Apr. 21, 2008, 2:30, Keller 303
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.
Interlaced Eigenvalues and Quantum Information Theory
Colloquium
- Prof. John Holbrook
(UHM Math. Dept. & Univ. of Guelph (Retired)).
Fri., Apr. 18, 2008, 3:30, Keller 401
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.
Colloquium
- Prof. Jan Reimann
(University of California, Berkeley).
Fri., Apr. 11, 2008, 3:30, Keller 401
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.
Butterflies: A New Representation of Links
Colloquium
- Prof. Debora Tejada
(Universidad Nacional de Colombia at Medellin).
Thurs., Apr. 3, 2008, 3:30, Keller 401
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).
Colloquium
- Prof. Jorge Cossio
(Universidad Nacional de Colombia at Medellin).
Wed., Apr. 2, 2008, 3:30, Keller 401
Faculty Lecture Series: Sharing our Work and Knowledge
- Prof. Monique Chyba
(UHM Math. Dept.).
Wed., Mar. 19, 2008, 3:30, Hamilton Library Room 301
Some Local Measures of Dependence Between Two Random Variables
Colloquium
- Prof. Boyan Dimitrov
(Kettering University, Michigan).
Thurs., Mar. 20, 2008, 3:30, Keller 401
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.
Optimal Control for Systems with State Space Constraints and Applications to Semiconductors
Colloquium
- Prof. Heinz Schaettler
(Depart. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri).
Fri., Mar. 14, 2008, 3:30, Keller 401
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. 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.
Mathematical Models of Novel Cancer Therapies as Optimal Control Problems
Mathematical Biology Seminar
- Prof. Urszula Ledzewicz
(Southern Illinois University).
Wed., Mar. 12, 2008, 12:30, Keller 302
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.
Colloquium
- Prof. Barbara Csima
(University of Waterloo, Canada).
Feb. 8, 2008, 2008
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.
Undergraduate Mathematics Seminar
- Prof Michael J. Antal, Jr.
(Coral Industries Distinguished Professor of Renewable Energy Resources, UHM Hawaii Natural Energy Institute).
Feb. 6, 2008, 2008
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).
Analysis Seminar
- Prof. Colin C. Graham
(University of British Columbia).
Feb. 11, 2008
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.
Colloquium
- Prof. Colin C. Graham
(University of British Columbia).
Feb. 13, 2008
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).
Colloquium
- Sara Rutter
(Hamilton Library Science & Technology Librarian).
Fri., Feb. 29, 2008, 3:30, Keller 401
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.
Order-Bounded Operator
Special Analysis Seminar
- Austin Anderson
(UHM Math Dept).
Mon., Apr. 13, 2009, 2:30 p.m.
Card tricks, hats, false pearls, lottery, and coding theory
Undergraduate Analysis Seminar
- Prof. Claude Levesque
(Universite Laval in Quebec, Canada).
Wed., Mar. 5, 2007, 1:30, Keller 303
On Extremal Quasi-Modular Forms for GL_2(F_q[T])
following Kaneko, Koike, and Zagier
Colloquium
- Prof. Federico Pellarin
(University of Saint-Etienne, France).
Fri., August 8, 2008, 3:30, Keller 401
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.
