# 2005 Academy Colloquium Series

## Impossibility theorems on integration in elementary terms

### Brian Conrad (University of Michigan)

It has been said that integration is an art, whereas differentiation is merely a skill. Even beginning calculus students quickly recognize that integration is a fundamentally much harder problem than differentiation. There are certain simple kinds of indefinite integrals of "elementary functions" that seem to be resistant to attack by the methods of calculus, or in fact any methods that can ever be devised: it was proved by Liouville in the 19th century that some rather simple integration problems cannot be solved "in elementary terms". (Of course, one has to give a reasonable and precise definition of what "elementary" means.) Such results are to be considered as an analogue of the beautiful Galois theory of polynomials, for which the analogue of solving for an integral in elementary terms is solving for the roots of a polynomial "in radicals" (and so Liouville's results are analogous to Galois' theorem on the impossibility of solving the general quintic in radicals). This leads to a powerful analogy between the theories of solving for roots of polynomials and solving certain kinds of differential equations.

The theory of differential fields provides an elegant framework for understanding how to actually prove impossibility results of the sort discovered by Liouville. We will present some basic aspects of the theory of differential fields and use this theory to explain how one can apply considerations with simple differential equations to prove that specific integration problems really cannot be solved in "elementary terms." In particular, we will see why exp(-x^{2}) cannot be integrated in elementary terms. Proceedings article

## The Multiple Facetes of the Associahedron

### Jean-Louis Loday (CNRS, Strasburg, France)

The triangle admits higher analogues called the tetrahedron and then the standard simplex. The square admits higher analogues called the cube and then the hypercube. Did you know that the pentagon also admits higher analogues ? Its called the associahedron (or sometimes the Stasheff polytope) and has many nice properties: geometric, combinatorial, topological and algebraic. Lecture notes Proceedings article

## E6, E7, E8

### Nigel Hitchin (Oxford University, UK)

These three symbols represent to a modern mathematician three Dynkin diagrams, but in a sense they have been known to mathematicians for hundreds of years. When the Greeks classified the regular solids, they were really showing they knew about E6, E7 and E8 and when the 19th century geometers got excited about the 27 lines on a cubic surface or the 28 bitangents to a quartic curve they were really studying E6 and E7 in their own terms. Today string theorists search for a theory of the universe by looking at E8 X E8. The talk will aim to show how E6, E7 and E8 lie at the heart of some very concrete mathematics. Lecture notes

## Bott Periodicity in Topological, Algebraic and Hermitian K-Theory (Part I)

### Max Karoubi (University of Paris)

In this first lecture about Bott periodicity, we summarize the history of the proof which leads naturally to topological K-theory of Banach algebras. Another vision is in terms of iterated loop spaces of the general linear group, leading to a striking list of homotopy equivalences between homogeneous spaces which might be interpreted in terms of Clifford algebras. Lecture notes Proceedings article (Includes parts I & II)

## Bott Periodicity in Topological, Algebraic and Hermitian K-Theory (Part II)

### Max Karoubi (University of Paris)

In this second lecture of the series, we introduce Quillen's K-theory (for discrete rings) and ask ourselves how far we can go for a periodicity theorem in this "algebraic" K-theory. We state the Lichtenbaum-Quillen conjecture which is the true analog of Bott periodicity in this context (through a "Galois descent"). Finally, we conclude the lecture with remarks about Hermitian K-theory where a "real" periodicity can be stated and proved, using topological ideas. Lecture notes Proceedings article (Includes parts I & II)

## Sizes and Scales in the Subatomic World

### Gerard ’t Hooft (ITP, Utrecht)

The tiniest objects that physicists can detect, are studied using very large particle accelerators and detection devices. We can observe structures 1000 times smaller than an atomic nucleus. What Nature may be like at even smaller scales can be deduced only from theoretical extrapolation and speculation. The Laws of Physics appear to admit "scaling", which means that smaller structures are very similar to the larger ones. Eventually, however, a limit is reached, where our theoretical imagination simply comes to an end.

What we found so-far can be used to figure out how the Universe may have come into being. According to one theory called "inflation", the Universe inflated from nearly pointlike to about one cm, in a time interval shorter than what is needed by light to traverse an atomic nucleus. Extraordinary as this may sound, such theories can be put to a test ...

## Prime Numbers

### Peter Sarnak (Princeton University)

The topic of distribution of prime numbers subject to various constraints is an old and fascinating one. There have been some interesting recent developments. We discuss some of the basic tools that have proven effective in this study. Lecture notes

## Quivers in Algebra, Geometry, and Representation Theory

### Victor Ginzburg (University of Chicago)

A quiver is an oriented graph; a representation of a quiver is an assignment of a vector space to each vertex and of a linear map to each oriented edge of the quiver. For quivers corresponding to Dynkin graphs of A-D-E types one has a complete classification of finite dimensional representations. On the contrary, for a general quiver, the classification of finite dimensional representations is known to be a wild problem, that is, no such classification could possibly exist. The most interesting, intermediate 'tame' case, is the one of 'extended Dynkin graphs' obtained by adjoining one extra vertex to a Dynkin graph (these extended graphs are Dynkin diagrams for Kac-Moody affine Lie algebras). We shall discuss the famous McKay correspondence. It provides a bijection between extended Dynkin graphs and finite subgroups G in the group SL(2) of complex two-by-two matrices with determinant 1. The space of G-orbits in the 2-plane is a singular variety that has a very nice smooth resolution. It turns out that the topology and algebraic geometry of this resolution is encoded in representation theory of the corresponding extended Dynkin quiver. This geometry plays a key role in explaining a mysterious connection between finite subgroups in SL(2) and simple Lie algebras. Associated naturally with any quiver is its Hall algebra, an associative algebra depending on a parameter 'q'. Multiplication table for the Hall algebra is comprised of polynomials in 'q' with nonnegative integral coefficients. When q is a prime power, the values of these polynomials at q count the number of certain representations of the given quiver in vector spaces over a finite field with q elements. We will explain how the Hall algebras are related to Symmetric functions, representation theory of Symmetric groups and, eventually, to Quantum groups. Lecture notes