## Algebra and Geometry Seminar

The seminar is devoted to classical and modern questions of algebra and geometry. Most talks are two-hour informal talks. Comments and discussions are highly encouraged.

### Folding polynomials and cutting cakes

by Yuri Burda  (www) | University of Toronto
Time: 15:00  (Thursday, Jun. 28, 2012)
Location: BA6183, Bahen Center, 40 St George St
Abstract:
Which polynomials have the most exceptional arithmetical and dynamical properties? I will discuss one possible answer to this vague question. A complex polynomial of degree n is (almost everywhere) an n-to-1 map. If such a polynomial is defined over the integers, one can reduce it modulo any prime. Usually the resulting map from Z/pZ to itself won’t be 1-to-1. Thus a polynomial that defines invertible mappings modulo infinitely many primes has very exceptional arithmetical properties. I will discuss a way to classify all such polynomials, which uses beautiful result of Weil relating geometrical and arithmetical properties of a planar curve. It turns out that all polynomials with the exceptional arithmetical property above have (by coincidence?) exceptionally simple (integrable) dynamics. I will then discuss multidimensional analogues of these polynomials (called folding polynomials) that have equally remarkable arithmetical and dynamical properties. To make the story easier to digest, a homemade cake will be provided to everyone attending.

#### Dates in this series

· Monday, Jan. 11, 2010: On rational functions invertible in radicals  (Yuri Burda)
· Wednesday, Apr. 21, 2010: Dynamical foliations (Tanya Firsova)
· Friday, Mar. 11, 2011: NEW MINI-COURSE: Inifinitesimal Hilbert 16th problem: inside the proof (Dmitry Novikov)
· Friday, Mar. 25, 2011: Inifinitesimal Hilbert 16th problem: inside the proof - 2nd meeting (Dmitry Novikov)
· Friday, Apr. 01, 2011: Inifinitesimal Hilbert 16th problem: inside the proof II (Dmitry Novikov)
· Friday, Apr. 08, 2011: Algebraic Geometry and Convex Geometry (Askold Khovanskii)
· Tuesday, Apr. 19, 2011: Algebraic Geometry and Convex Geometry II (Askold Khovanskii)
· Tuesday, Oct. 11, 2011: On two classical invariants of singularities (Dmitry Kerner)
· Tuesday, Oct. 18, 2011: Topological approach to Klein's resolvent problem (Yura Burda)
· Thursday, Nov. 10, 2011: Folding polynomials as examples of discrete algebraically integrable systems (Yuri Burda)
· Tuesday, Nov. 15, 2011: Folding polynoimals as examples of discrete integrable systems (Yuri Burda)
· Tuesday, Jan. 17, 2012: Some parameter spaces of curves on toric surfaces, their tropicalizations, and applications (Jessie Yang)
· Tuesday, Jan. 24, 2012: Hermite's form of a quintic equation and geometry of cubic surfaces (Yuri Burda)
· Tuesday, Jan. 31, 2012: Obstructions to the local approximation of tropical curves in planes (Kristin Shaw)
· Tuesday, Feb. 07, 2012: Constructing the 17-gon with a ruler and a compass (Liudmyla Kadets)
· Tuesday, Mar. 27, 2012: Discriminant of transversal singularity type (Dmitry Kerner)
· Tuesday, Apr. 10, 2012: Discriminant of transversal singularity type (Dmitry Kerner)
· Thursday, Jun. 28, 2012: Folding polynomials and cutting cakes (Yuri Burda)
· Monday, Oct. 01, 2012: Radon and Fourier transforms for differential forms (Theodore Voronov )
· Tuesday, Oct. 09, 2012: Multiplicity in generic families of maps (Gal Binyamini)
· Tuesday, Oct. 23, 2012: Weil reciprocity law and Newton Polygons (Lucy Kadets)
· Tuesday, Oct. 30, 2012: Newton polyhedra theory and Bernstein--Kushnirenko theorem (Askold Khovanskii)
· Tuesday, Nov. 06, 2012: Universal Grobner basis of an ideal (Askold Khovanskii)
· Tuesday, Nov. 13, 2012: One more time about universal Gröbner bases (Askold Khovanskii)
· Tuesday, Nov. 20, 2012: Introduction to tropical bases (Kristin Shaw)
· Tuesday, Nov. 27, 2012: Combinatorics of affine birational maps (Ilya Karzhemanov)
· Monday, Dec. 10, 2012: Arithmetic properties of Chebyshev polynomials (Yuri Burda)
· Wednesday, Jan. 30, 2013: Geometry of numbers and finiteness of ideal class group (Kiumars Kaveh)
· Wednesday, Feb. 06, 2013: Geometry of numbers and finiteness of ideal class group (part II) (Kiumars Kaveh)
· Tuesday, Mar. 19, 2013: Stringy invariants of p-cyclic quotient singularities and weighted counts of Artin-Schreier extensions of the power series field (Takehiko Yasuda)