# Some archived events from 2007–2010

#### Distinguished Lecture Series

###### GENERAL AUDIENCE LECTUREThe 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.

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#### MA defenses

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##### 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.

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 - 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

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

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