Category Archives: Colloquium

Some archived events from 2007–2010

Distinguished Lecture Series

Click here to see an Online QuickTime Video Presentation of Prof Shadwick's Lectures
GENERAL AUDIENCE LECTURE The Right Answers to The Wrong Questions. A Brief History of Mathematics in Finance
February 2010 is the 50th anniversary of Eugene Wigner's On the Unreasonable Effectiveness of Mathematics in the Natural Sciences. In that paper, Wigner speculated that the benefits that mathematics had provided to physics might in future spread to 'wide branches of learning'. In this talk I will survey the history of attempts to apply mathematics in economics and finance. This is a story of missed opportunities in which the right answers to the wrong questions have had a large impact. While this lecture is about mathematics, it is for a general audience and assumes no specialist knowledge of mathematics or finance.
Risk (Mis)management and the Financial Crisis. The Impact of the All Too Probable
Extreme Value Theory is a branch of statistics which is over 80 years old. Expected Shortfall is the statistical term for the average loss beyond a given threshold. Using Extreme Value Theory to estimate Expected Shortfall is a common risk management practice in the insurance industry. In the Finance Industry risk 'management' has relied instead on normal distributions and Value at Risk. The failure to predict the losses that rocked markets in 2007, 2008 and 2009 was not a failure of markets or an example of the futility of attempting to predict market behavior with statistics. In this talk, I'll provide a brief introduction to the statistics of extremes and show that, if the correct tools had been used, the recent financial crisis (in common with earlier crashes) would have been seen to be all too probable. A cursory knowledge of probability and statistics is the only prerequisite for this material.
The Geometry of Probability Distributions. A New Source of Statistics
In studying probability distributions, a natural specialization is to consider distributions which have a finite mean. A recently discovered method of describing the probability distributions with this property leads naturally to an affine equivalence problem. The affine geometry of probability distributions reveals remarkable structure including a natural measure of dispersion about the mean, improvements on the Markov and Chebychev inequalities, a new affine invariant and a new central limit theorem. The talk is intended to be accessible to graduate students in Mathematics.

Colloquia

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Invariant measures on countable models
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.
On some conjectures of Ron Brown and Alexander Grothendieck
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
Interactions of randomness and computability
Global Weyl modules, BGG Duality and the Catalan numbers
Infinitesimals in Probability
Riemann integrals and random reals
From Zeta to L to A: Some number theory using the Riemann zeta-function, L-functions, and automorphic forms
About a Problem Arising in Radiative Heat Transfer
New Operator-Difference Schemes in Hilbert Space
Facilitating Active Learning in Calculus
Fubini's theorem using determinacy
Marine Mammals and Math
Variational and Geometric Methods in Image Processing and Analysis
Lattice embeddings into the computably enumerable Turing degrees
Quasi-homomorphism Rigidity with Noncommutative Targets
Everything you need to know to do research in lattice theory (but were afraid to ask)
Democracy is the best form of randomness extraction
Diagonal Compressions of Matrices and Numerical Shadows
Binary Trees in Financial Calculations
Calibrating the Complexity of Mathematical Proofs and Constructions
SUPER-News and updates about SUPER-M
On the Classification of Graph C*-Algebras
Buffon Needle and Singular Integrals
Computable Symbolic Dynamics
Uniform Randomness Tests
Uniform Properties
Metric Graphs: The Poor Mathematician's Riemann Surface *or* You're in Good Company if Someone Calls You One-Dimensional
Regular idempotents in beta G
Reaction-Diffusion Equations as Models for Pattern Formation in Biological Systems
A Greedy Sorting Algorithm
Theory and Applications of Optimal Control Problems with Delays
The Hierarchy of Definability
Looking back, a Descartes Sample and some Sacred Mathematics
Stable Ramsey's theorem and measure
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)
Do Dogs Know Calculus?
Some Algebras have Nonfinitely Axiomatizable Equational Theories
Observations about Perfect Lattices
Schedule of Talks
Closed range composition operators
Schur Norms: Basic Methods and Diverse Applications
A Very Large Integer
How Composition Operators (could) Solve the Invariant Subspace Problem
Multiplication Operators on the Bergman Space via Analytic Continuation
GENERAL AUDIENCE LECTURE UNEARTHING THE VISIONS OF A MASTER: THE WEB OF RAMANUJAN'S MOCK THETA FUNCTIONS IN NUMBER THEORY
FREEMAN DYSON'S CHALLENGE FOR THE FUTURE: THE MOCK THETA FUNCTIONS I
FREEMAN DYSON'S CHALLENGE FOR THE FUTURE: THE MOCK THETA FUNCTIONS II
Inverting the Math Crisis in Hawaii
On Finitely Presented Expansions of Algebras
Character Varieties
An Afternoon of Beautiful Mathematics for Girls and Their Families
Diophantine quadruples
Translation Flows and Vershik's Automorphisms
Women in Mathematics
Algebra and analysis of the generalized Routh-Hurwitz problem.
Profinite actions: graphs, groups and dynamics
Describing the Tame Geometry and the Tame Topology of Algebraic Varieties and Their Projections
On Extremal Quasi-Modular Forms for GL_2(F_q[T]) following Kaneko, Koike, and Zagier
The Euler Officer Problem
Non-existence Theorem Without In-phase and Out-of-phase Solutions in the Coupled Van der Pol Equation System
Randomness, computability, and effective descriptive set theory
The Lagrangian in Symplectic Mechanics
Global Singularity Theory in Differential Topology
Transference on Bilinear Multipliers
Complementarity in Quantum Cryptography and Error Correction
Automatic continuity of nonstandard measures: Part II
Conjugate and Cut Loci for Riemannian Metrics in 2 Dimension Sphere of Revolution with Applications to Orbital Transfer and Quantum Control
On the Heegaard Genus of Knot Exteriors
Sofic groups and dynamical systems
Interlaced Eigenvalues and Quantum Information Theory
Randomness for Continuous Measures
Butterflies: A New Representation of Links
Multiple Solutions for a Nonlinear Dirichlet Problem via Morse Index
Beautiful Mathematics Everywhere
Some Local Measures of Dependence Between Two Random Variables
Optimal Control for Systems with State Space Constraints and Applications to Semiconductors
Mathematical Models of Novel Cancer Therapies as Optimal Control Problems
Computable Structure Theory
The role of mathematical models in mankind's shift to sustainability
Determining the support of distributions from their Fourier transforms
Interpolation Sets Past and Present --- and Future?
Lecture-demonstration on On-Line Library Resources for Mathematics
Order-Bounded Operator
Card tricks, hats, false pearls, lottery, and coding theory
On Extremal Quasi-Modular Forms for GL_2(F_q[T])
following Kaneko, Koike, and Zagier

Dissertation defenses

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Multiplication and Integral Operators on Banach Spaces of Analytic Functions (Abstract)
p-adic Analysis and Mock Modular Forms
On Semifree Symplectic Circle Actions

MA defenses

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Polynomial-clone reducibility
Fourier Coefficients of Weak Harmonic Maass Forms of Integer Weight: Calculation, Rationality, and Congruences
Satellite Orbital Control
Almost Global Feedback Control of Autonomous Underwater Vehicles
Linear Differential Operators and the Distribution of Zeros of Polynomials
TIME OPTIMAL CONTROL OF A RIGHT INVARIANT SYSTEM ON A COMPACT LIE GROUP
Solving the Dirichlet Problem via Brownian Motion
Symmetry Group Solutions to Differential Equations: A Historical Perspective
Construction and Properties of Brownian Motion
The Law of the Iterated Logarithm (How far is one likely to go on a random walk?)

Undergraduate seminars

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How to prepare to be an actuary
Nonlinear algebra
Generatingfunctionology (the mothership connection)
Reaction-Diffusion Equations as Models for Pattern Formation in Biological Systems
Gerrymandering, Convexity, and Shape Compactness
Lagrange's Identity & Applications to Probability Sampling
Is a + b = c really a simple equation?
On the Markoff Equation: X2 + Y2 + Z2 = 3XYZ
Math Students Wanted: The New Master in Financial Engineering Program at UH
Some other events from 2009-2010
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Welcome Graduate Students - Lecture I:

Wed., August 18, 2010, 3:30, Keller 401

Prof. Erik Guentner

UHM Math Dept

Welcome Graduate Students - Lecture II:

Thurs., August 19, 2010, 3:30, Keller 401

Prof. Les Wilson

UHM Math Dept

International Week of Mathematics

October 11-15, 2010

SUPER-M Project

UHM Math Dept.

Week of Mathematics Hawai'i-Geneva 2010
UHM News Release

Abstract: An event co-organized by la Commission genevoise de l'enseignement des mathématiques (comprised of the mathematics department at the University of Geneva, Switzerland and the primary and secondary school system) and the SUPER-M project of the mathematics department at the University of Hawai‘i, USA.

For one week this initiative will bring together the primary/secondary educational systems and university faculty and students. All volunteer classrooms work around a common theme. This year's theme is folding. Folding usually suggests origami or other traditional folding but it is not on this aspect that these lessons are primarily developed: there is often little mathematics in origami, especially when the purpose is to follow a pre-designed model, or the mathematics are so complex that they are accessible only to the higher grade levels. The emphasis here is to use folding activities to generate mathematical questions in relation to different levels of mathematics.

SUPER-M Workshop Flyer

SUPER-M Workshop - Day 1

Fri., August 20, 2010, 9:00 a.m. - 4:00 p.m., Keller 313

SUPER-M Workshop Picnic

Fri., August 20, 2010, 6:00 p.m., Magic Island

SUPER-M Workshop - Day 2

Sat., August 21, 2010, 9:00 a.m. - 1:30 p.m., Keller 313

NSF Sponsored Workshop:
Schedule of Talks and Events

Wed., November 11, 2009, 9:00 a.m. - 3:00 p.m., Lokelani Intermediate School on Maui

UH Department of Mathematics

SUPER-M: School and University Partnership for
Educational Renewal in Mathematics

NSF Sponsored Workshop:
Schedule of Talks and Events

Sat., November 7, 2009, 9:00 a.m. - 3:00 p.m., Pauoa Elementary School in Honolulu

UH Department of Mathematics

SUPER-M: School and University Partnership for
Educational Renewal in Mathematics

NSF Sponsored Workshop:
Schedule of Talks and Events

Fri - Sat, August 21-22, 2009, 9:00 a.m. - 2:00 p.m., Keller 303

UH Department of Mathematics

SUPER-M: School and University Partnership for
Educational Renewal in Mathematics

Honors seminars for NSF scholars, and some Master's degree plan B presentations (XML format)