Math Conference

David Brown: The Galois Group of Cyclotomic Fields of Fermat Primes

I will give an explicit ruler and compass construction of a regular 65537-gon.

Sándor Dobos: Why Hungary?

The fact that a small country such as Hungary has one of the largest "per capita" production of outstanding mathematicians and physicists is a phenomenon that deserves investigation.
The key may lie in the past:

• The history of Hungarian education, in particular the wisdom of one scholar, educator and statesman, Baron Loránd Eötvös.
• Origins of special schools such as Eötvös Kollegium, Lutheran High School, Fazekas High School.

• Working with the talented students, camps, trainings. Lajos Pósa, the famous student of Erdős, and his "invisible school".
• The structure of competitions and KöMaL, The Journal.

Zoltán Füredi: Sets of few distances in highdimensional normed spaces

Michael Lang: Distance-Regular Graphs: Classifying the Almost-Bipartite Q-Polynomial Case

Distance-regular graphs, particularly those with the Q-polynomial property, figure into a number of fields, including coding theory and knot theory. We have classified those that are also almost bipartite.

Kiran Kedlaya: A little semilinear algebra

The eigenspace decomposition of a finite dimensional vector space equipped with a linear transformation is one of the highlights of linear algebra. But what happens when the transformation is only semilinear, i.e., when the transformation acts nontrivially on the base field? I'll describe an old example of this situation from number theory (the Dieudonne- Manin classification). I'll then describe some more recent variations on this theme, which have shown up in the study of p-adic differential equations and of p-adic Galois representations.

Moses Klein: How to pay off your debts, $1 at a time, even with exorbitant interest

I shall present a class of sequences of natural numbers, called Goodstein sequences, which appear to grow faster than most familiar sequences but in fact all eventually decrease to 0. Most remarkably, the fact that they reach 0 has a fairly simple proof using beginning set theory, but cannot be proven using finitary methods.

Rob Matuschek: Mathematics and the Culture of the American High School

Overland High School in Aurora, Colorado, is at both typical and unique compared to many high schools accross the United States. High school math students are being taught curriculum that ranges from pre-algebra and number sense to Calculus 3, differential equations, linear algebra, and beyond. What are some of the measurable and unmeasureble marks of a quality high school education? How much college level education is available in high school and how good is it? What does it take to get into good colleges? What is high school like these days?

Dominic Mazzoni: Support Vector Machines and Kernel Methods for Machine Learning

The word "kernel" has many meanings in mathematics. Kernel methods for machine learning refer to so-called "kernel functions" that implicitly map two points into a high-dimensional space and return their dot product in that space. This allows you to take a linear algorithm that performs some operation on a set of points, and turn it into a nonlinear algorithm, with almost no additional computational cost. The most popular use of this technique has been in Support Vector Machines, where an algorithm for finding a linear separating hyperplane between two classes of points is easily modified into an algorithm for finding an arbitrarily complex nonlinear surface separating two classes of points. In this talk, we will look at the theory of kernel methods and see how it is being in the real world for such applications as handwriting recognition, speech recognition, satellite image classification, and more.

Alexander Meadows: Singular minimal surfaces and nonlinear PDEs

This talk will be a survey of some recent work on nonlinear elliptic PDEs which are related to minimal surfaces. There will be an overview of the theory of singularities of minimal hypersurfaces and recent examples of singularities found by Leon Simon using PDE techniques, motivating a more general study "singularly nonlinear" PDEs. We will then present some recent existence results based on the idea of "tornado solutions."

Terri Moore: Zero-Sequences over Finite Abelian Groups

Let G be a finite abelian group. A zero-sequence over G is a sequence of elements of G which sum to zero in G. A zero-sequence is minimal if no proper subsequence is also a zero-sequence. The set of zero-sequences over G form a monoid, called the Block Monoid of G, where the operation between two sequences is the union of the sequences. This monoid is atomic with atoms being the minimal zero-sequences. During this talk I will introduce the block monoid for a group and discuss recent results regarding maximal length minimal zero-sequences and factorizations in the block monoid.

Amy Myers: Bad Squares on Board Games

Imagine a board game (such as Monopoly) in which the roll of two dice determines the number of squares we move forward on a given turn. A particularly "bad" square (TWO hotels on Boardwalk!) looms m squares ahead of our current square. What is the probability we skip safely over it without landing on it? In this talk we consider a variation of this problem, and extend it both to "one-sided" random walks and to compositions that avoid other compositions. We focus on compositions, and model our notion of composition avoidance after the concept of "pattern avoidance" — the study of which concerns itself with permutations that avoid other permutations. Our consideration of composition avoidance leads us to extend a 1981 result of Guibas and Odlyzko concerning forbidden substrings to the case in which the strings are weighted. It also leads us to consider permutations of a multiset that avoid a given pattern. It is well known that the number of permutations of the set {1, 2, ..., n} that avoid a pattern τ is the same for any permutation τ on three letters. The same result holds for permutations of the multiset . This result was recently proved independently by Savage and Wilf.

Péter P. Pálfy: How often does the size tell us the structure of a group?

Everybody learns in the introductory group theory course that every group of prime order is cyclic. In fact, there are other numbers with this property. Namely, every group of order n is cyclic if and only if n and φ(n) are relatively prime, where &phi is Euler's totient function. This was observed by the American group theorist George A. Miller in 1899.
In 1948 Paul Erdős proved that the number of positive integers n < x with this property is asymptotically e^-γx/log log log x , where γ denotes Euler's constant.
In this survey talk we will discuss similar results characterizing those numbers n for which every group of order n is abelian, nilpotent, solvable, etc. and giving an asymptotic for the number of positive integers n < x having these properties.
In some cases the relevant question is whether a group with some property (for example, a simple group) of a given order exists. Paul Erdős has a forgotten paper on this question. We will speculate why this paper was omitted from his list of publications. Now the classification of finite simple groups yields that there are asymptotically 3·2^1/3x^1/3log x simple group orders less than x.

Cory Palmer: General neighbour-distinguishing index of a graph

It is proved that edges of a graph G with no component K₂ can be coloured using colours so that any two adjacent vertices have distinct sets of colours of their incident edges.

József Pelikán: Remember Euclidean rings?

The problem of giving exact formulas for the number of lattice points in a convex polytope has interested mathematicians for many years. Pick's Formula (c. 1890) gives the answer in dimension two, and Ehrhart achieved interesting results in higher dimensions in the 1960's. In the past 15 years, much progress has been made using the tool of toric varieties. Recent results in this field (due to many researchers including Brion, Morelli, Khovanskii, Thomas, and the speaker) have helped us understand the lattice point question much more clearly. This talk will present a brief introduction to some of these results.

András Ringler: Geometric solution of elliptic equations

"Do not enter, if you do not know geometry!" (Platon)
The author found a surprising new method to solve elliptic equations. Using this method, one can also find the second solution of these quadratic equations, given in the form: a·x - x² = b² and a·x - x² = -b². The method for the geometric solution of the quadratic equations will be shown during the presentation with the help of a computer animation.
The author is sure that the described new method will give both students and professors much joy, and he hopes that the strict Greek masters do not "roll over in their graves", being angry with him, because he introduced the -b² value into their very nice elliptic "συμπτο&muα". The author hopes that the old Greek masters will forgive him for taking the courage to continue the thoughts they began.

András Stipsicz: Knot theory: some new developments

We examine algebraic invariants giving lower bounds for minimal corrsing number, unknotting number, genus and slice genus of a knot. "Categorifications" of the classical knot polynomials will be also discussed.

Gábor Tardos: Pattern-avoidance for matrices and permutations

A 0-1 matrix A is said to contain another such matrix P if P is a submatrix of A or P can be obtained from a submatrix of A by changing a few ones to zeros. The extremal problem of finding the maximum number ex(n,P) of 1 entries an n by n 0-1 matrix can have that does not contain the "forbidden pattern" P was first studied by Zoltan Furedi. In a 1992 paper of Furedi and Peter Hajnal conjectured that if P is a permutation matrix, then ex(n,P) is linear in n.
Around the same time Richard Stanley and Herbert Wilf formulated a conjecture regarding pattern-avoidance among permutation: For any fixed permutation p there exists a constant c=c_p such that the number of n-permutations not containing p is at most cⁿ. This conjecture became widely known and several special cases were solved. The connection between the two problems, namely that the Furedi-hajnal conjecture implied the Stanley Wilf conjecture was shown by Martin Klazar in 2000.
I met Adam Marcus in my BSM THC class in 2001. He returned to Budapest to work with me as a Fulbright scholar in 2003 and we worked on pattern-avoidance problems for matrices and menaged to find a surprisingly simple proof of the Furedi-Hajnal conjecture. In the talk I present this simple argument and how it implies the Stanley-Wilf conjecture.

Russ Woodroofe: Groups, Posets, and Shellings

I'll talk about some surprising connections between fundamental definitions in group theory and fundamental definitions in combinatorics. In particular, the existence of a shelling on the partially ordered set of various substructures of a group turns out to often have very nice implications for the structure of the group. Definitions will be defined, and few prerequisites are required.