## Blyth 2002

### Department of Mathematics

University of Toronto

### The ninth annual R.A. Blyth Lectures in Mathematics

### Professor Dmitry Fuchs

UC-Davis

### Will give three lectures on Geometry and Algebra

**Lecture 1**

Monday, March 25, 2002 at 4:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 2102

**Paperfolding.**

How to fold a sheet of paper along a curve? What are the spatial shapes that can be obtained as a result of such paperfolding? Reformulated in terms of developable surfaces, many such problems have solutions known since the beginning of the XX century. However some experiments with paper are still waiting for a convincing explanation.

**Lecture 2**

Tuesday, March 26, 2002, 4:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 5017A

**Knots in contact geometry.**

A Legendrian curve in (the standard context) space is a smooth curve *x = x(t), y = y(t), z = z(t)* satisfying the equation * z*_{t} = *yx*_{t}. A Closed, non-self-intersecting Legendrian curve is called a Legendrian knot. The problem of Legendrian isotopic classification of Legendrian knots can be restated as a problem in geometry of curves in the plane which looks easy but is actually quite difficult. The lecture will contain an account of major old, recent, and future works in this area.

**Lecture 3**

Wednesday, March 27, 2002, 4:10 p.m.

Sidney Smith Hall, 100 St. George Street

Room 5017A

**Lie algebras in finite characteristic.**

The definition of a Lie algebra, as given in major algebra textbooks, is valid for any ground field (or a decent ring). However, there are multiple evidences showing that in the case of finite characteristic *p,* the definition of a Lie algebra needs a revision: besides the commutator operator, a Lie algebra should be furnished by a unitary operation of "raising to the power *p*". This operation is not linear, which leads to algebraic complications: even an explicit description of this operation for the most common Lie algebras is not obvious. More serious problems arise when we speak of central extensions, deformations, or, more generally, cohomology of such Lie algebras.

The Blyth Lecture Reception will follow the Third Lecture on Wednesday, March 27, 2002, at the Faculty Club, 41 Willcocks Street.

All are invited to attend.