Monday, March 29, 2021
Time  Items 

All day 

1pm 
03/29/2021  1:00pm Abstract: We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of maxaffine spline operators (MASOs), which provide a powerful portal through which to view and analyze their inner workings. For instance, conditioned on the input signal, the output of a MASO DN can be written as a simple affine transformation of the input. This implies that a DN constructs a set of signaldependent, classspecific templates against which the signal is compared via a simple inner product; we explore the links to the classical theory of optimal classification via matched filters and the effects of data memorization. The spline partition of the input signal space that is implicitly induced by a MASO directly links DNs to the theory of vector quantization (VQ) and Kmeans clustering, which opens up new geometric avenue to study how DNs organize signals in a hierarchical and multiscale fashion. email tatianna.curtis@yale.edu for info. Location:
Zoom Meeting ID: 97670014308

4pm 
03/29/2021  4:00pm Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff’s spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. Location:
Zoom
03/29/2021  4:30pm In this talk we discuss the problem of coupling quantum field Location:
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)
