Events

Thai Hoang Le, University of Texas at Austin

“Intersective polynomials and Diophantine approximation”

It has been known since Vinogradov that for each k, there is an exponent θ = θ(k) such that for every positive integer N and real number α, we have min1≤n≤N ∥αnk∥ ≪ N−θ, the bound being uniform in α and N (where ∥·∥ denotes the distance to the nearest integer). More generally, there is an exponent θ = θk,l such that for any polynomials f1, . . .

Nikolai Vavilov, St. Petersburg

“Structure of Chevalley groups: the proof from the book”

Joint with Mikhail Gavrilovich, Sergei Nikolenko and Alexander Luzgarev. New generation proofs of normal subgroup structure, commutator formulae, and the like, over an ARBITRARY commutative ring are given, based on the study of minimal modules of exceptional groups.

Robert Garbary, Pure Mathematics University of Waterloo

“Rational Maps”

I am going to establish some results concerning rational maps into abelian varieties. In no particular order, these are the following results:
1) Any rational map from a smooth variety to an abelian variety is in fact a morphism 2) Any rational map from An or Pn to an abelian variety is constant 3) Birationally equivalent abelian varieties are isomorphic (as abelian varieties).

J C Saunders, Pure Mathematics Department, University of Waterloo

“Sums of Digits in q-ary expansions”

Let sq(n) denote the sum of the digits of a number n in base q. For example, s2(n) represents the number of 1s in the binary expansion of n. In 1978, Kenneth B. Stolarsky showed that
lim inf s2(n2) = 0 n→∞ s2(n)

Rahim Moosa Department of Pure Mathematics University of Waterloo

“NIP Theories XII”

We aim to complete chapter 3 of Pierre Simon’s notes.

Correction: time is 3:00 pm.

Rahim Moosa, Pure Mathematics, University of Waterloo

“Abelian varieties as complex tori”

I will discuss how the complex points on an abelian variety have the structure of a complex torus. I will also discuss the Riemann-Hilbert relations which answer the question: When is a complex torus the complex points of an abelian variety?

Tom Tucker, Rochester University

“Integral points in two-parameter orbits”

Let K be a number field, let f : P1 − − > P1 be a nonconstant rational map of degree greater than 1 that is not conjugate to a powering map, let S be a finite set of places of K, and suppose that u,winP1(K) are not preperiodic under f. We prove that the set of (m,n)inN2 such that fm(u) is S-integral relative to fn(w) is finite and effectively computable.

Shuntaro Yamagishi, Pure Mathematics, University of Waterloo

"Sidon Problem in polynomial ring over finite field"

Given a sequence of natural numbers $\omega$, we define $r_n(\omega) = | \{ (a,b) : a+b = n, a< b, \text{ and } a,b \in \omega  \}|$.  In 1954, Erdos proved that there exists a sequence $\omega$ such that $\log n \ll r_n(\omega) \ll \log n$. We consider the analogue of this question in polynomial ring over finite field.

Tyrone Ghaswala, Pure Mathematics, University of Waterloo

"The geometry of Yang-Mills fields, Part 04"

Throughout the Spring 2013 term, we will (as a group) be reading through and lecturing on "The Geometry of Yang-Mills Fields" by Sir Michael Atiyah. All are welcome to attend.