We will discuss the use of a new tool to gather midterm feedback from your students, and how to interpret (midterm and final) teaching evaluations. 

Topics of this week's seminar: Introductions, the importance of setting the tone early in discussion sections, and a panel discussion with second year graduate students.

In this talk, we shall give a mathematical setting of the Random Backpropogation  (RBP) method in unsupervised machine learning. When there is no hidden layer in the neural network, the method degenerates to the usual least square method. When there are multiple hidden layers, we can formulate the learning procedure as a system of nonlinear ODEs. We proved the short time, long time existences as well as the convergence of the system of nonlinear ODEs when there is only one hidden layer. This is joint work with Pierre Baldi in Neural Networks 33 (2012) 136-147, and with Pierre Baldi, Peter Sadowski in Neural Networks 95 (2017) 110-133 and in Artificial Intelligence 260 (2018), 1-35.

I will present some recent results on equidistributive properties of toral eigenfunctions. Only a  minimal knowledge of Fourier analysis is required to follow all the details of this talk.

The celebrated Alexandrov-Bakelman-Pucci Maximum Principle (often abbreviated as ABP estimate) is a pointwise estimate for solutions of elliptic equations, which was introduced in the 1960s. It was motivated by beautiful geometric ideas and has been a fundamental tool in the study of non-divergent PDEs. More recently, this PDE technique also pays back to geometry - the ABP estimate and its extensions can be used to prove some optimal classical geometric inequalities such as the Isoperimetric and Minkowski inequalites.