Three talks in three weeks

 
I had a talk this week about the construction of white noise test functions. It was nice actually. I was forced to articulate what I know and present it in a more coherent way. Concepts that were fuzzy before has become clearer now. I still have remaining objectives to answer but the process is the reward.

Next week, I will be presenting my results in a colloquium. I will frame the results around a central question which is a variation of a certain Lie algebra of white noise operators. This is done by replacing the number operator by a generalized number operator, called conservation operator corresponding to an operator S. Doing this changes the dimension of the base Lie algebra depending on the properties of S. One result that I have is that the linear independence of the orbits of S characterizes the dimension of the Lie algebra. The question then becomes: which operators have a linearly independent orbit? One example is the the unilateral shift. It has a very simple structure and is isomorphic to an infinite dimensional filiform Lie algebra. I found this isomorphism by grading the Lie algebra. On the other hand, an operator with linearly dependent orbit corresponds to a finite dimensional Lie algebra. As to how exactly the orbit becomes linearly dependent depends on the operator S or the initial vector zeta. This is a 15-minute talk that will be prerecorded.

The week after that I will be giving a talk on the canonical commutation relations and quantum white noise derivatives. I chose this topic because this is the algebra part of my study. I didn't think it's appropriate to talk mostly about analysis in an algebra seminar. And so the topic was chosen. I'm around 50% done with the slides since I've already given a talk on white noise distribution theory. This time I need to focus more on quantum white noise derivatives. Give a background on why it's called such, give applications in differential equations and implementation problem, and so on.

It's going to be a busy three weeks. Wish me luck. Because of my preparation for these talks, I haven't been able to focus on writing my dissertation. I will resume writing after August 10th.