##### Department of Mathematics,

University of California San Diego

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### Math 196/296 - Student Colloquium

## Allen Knutson

#### UCSD

## Puzzles, straightening honeycombs, and adding matrices

##### Abstract:

Given the three kinds of puzzle pieces pictured on the left, define a puzzle to be a decomposition of a triangle into puzzle pieces (such that the edges match up, like in a jigsaw puzzle). Call a puzzle ``rigid'' if there is no other puzzle with the same outer boundary. A lot can be proven about puzzles (we'll do (1) and (2) in the talk): \vskip .05in \noindent 1. The number of $0$s on one side equals the number of $0$s on each of the other two sides - see if you can prove this one before the talk! \vskip .05in \noindent 2. The lines in the puzzle pieces can all be simultaneously straightened (as in the right-hand picture) if and only if the puzzle is rigid. \vskip .05in \noindent 3. There is an easy 1:1 correspondence between rigid puzzles and inequalities on the eigenvalues of the sum of two Hermitian matrices. \vskip .05in \noindent 4. The statement ``Given four generic lines in space, there are exactly two others that touch all four,'' and others like it, can be turned into puzzle-counting statements. \vskip .1in \noindent Refreshments will be provided!

Host:

### November 17, 2005

### 11:00 AM

### AP&M 2402

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