Michael Hartz, FernUniversität in Hagen
"Dilations in finite dimensions and matrix convexity"
Pawel Sarkowicz, Department of Pure Mathematics, University of Waterloo
This week we will define elementary substructures and prove the Downward (and possibly Upward) Löwenheim-Skolem Theorem(s). To that end, we will introduce the notion of separable languages.
MC 5403
Samuel Harris, University of Waterloo
Speaker 1: Christopher Lang, Department of Pure Mathematics, University of Waterloo
"Using Group Actions to Simplify Nahm Data"
The Nahm equations are a system of differential equations for $u(k)$-valued functions on $(a,b)\subset\mathbb{R}$. Solutions of the Nahm equations are called Nahm data. By imposing certain conditions on the Nahm data, the ADHM-Nahm procedure gives rise to monopoles in $\mathbb{R}^3$. Elaborating on [1], we examine how the actions of $\mathbb{R}^3$, $u(k)$, and $\mathrm{SU}(2)$ simplify the Nahm data.
Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
"Zariski Topology 101"
We'll (at least partially) answer the following questions: when is Spec(R) compact? Hausdorff? connected? irreducible? noetherian? Also, the basic open sets that Jeff described last time can be interpreted as localisations --- we will talk about that if time permits.
MC 5479
Mohammad Mahmoud, Department of Pure Mathematics, University of Waterloo
"Degrees of Categoricity of Trees"
Greg Patchell, Department of Pure Mathematics, University of Waterloo
"Model Theory of von Neumann Algebras II"
Sam Harris, Department of Pure Mathematics, University of Waterloo
"Quantum XOR games and Connes' embedding problem"
Speaker 1: Shubham Dwivedi, Department of Pure Mathematics, University of Waterloo
"Differential Harnack estimates"
We will discuss differential harnack estimates including Hamilton’s matrix harnack estimate for solutions of the heat equation and the Li-Yau inequality. If time permits, we will discuss harnack estimates for the Ricci flow.
Speaker 2: Spiro Kargiannis, Department of Pure Mathematics, University of Waterloo
"Bubble Tree Convergence for Harmonic Maps"
Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
"Localisation and the Nullstellensatz"
Jeff defined the basic open sets $D_f$. We'll see that in fact $D_f\simeq \mathrm{Spec}(R_f)$ where $R_f$ is a localisation. We might be able to finish the proof that $\mathrm{Spec}(R)$ is Hausdorff iff $\mathrm{Kdim}(R)=0$. Lastly, we can show that if $A$ is an affine algebra then the closed points are dense in $\mathrm{Spec}(A)$.
MC 5479