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.
We will continue on chapter 4 of the Pierre Simons notes. We will go further on mutually indiscernible sequences and start talking about DP-ranks.
Please note room and time.
Suppose we have an algebraic group G acting algebraicly on a variety X, ie for each g ∈ G the associated map X → X is a morphism. A quotient of X by G is defined to be a variety Y and a morphism π : X → Y satisfying
(1) π−1(π(x)) = Gx for all x ∈ X.
(2) For any variety Z and G-invariant morphism X → Z, there is a unique factorization through Y.
Seminar has been cancelled due to sickness.
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 n!1 s2(n2) s2(n)
= 0 using bounds obtained from analytical methods. In the last presentation we showed that the ratio s2(n2) s2(n) can indeed hit every positive rational number. In this presentation, we show that the same is
We will take a step aside from Chapter 4 of Simon’s note, and jump the appendix to see why the theory of algebraically closed valued fields is NIP. Also, I will talk on why the model companion of difference fields is not NIP.
After reviewing the basic theory of Hall algebras, I will present a new construction of representations using duality functors.
Suppose that two finite rank, self-adjoint matrices A and B have small commutator—that is, the norm of AB−BA is small. Are there necessarily two finite rank, self-adjoint matrices A′ and B′ that commute with each other such that A′ is sufficiently close to A and B′ is sufficiently close to B? We will prove the titular theorem (which answers this in the affirmative) as well as examining a similar problem regarding unitary matrices.
For a set of numbers $A$, let the sum-set $A+A$ denote