Abstracts

Week of March 28, 2021

March 29, 2021
Applied Mathematics Spline Insights into Deep Learning 1:00pm -
Zoom Meeting ID: 97670014308

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 max-affine 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 signal-dependent, class-specific 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 K-means 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.
Group Actions and Dynamics Arithmetic and Dynamics on Markoff-Hurwitz Varieties 4:00pm -
Zoom

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.
Time permitting, we will also discuss the joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz varieties  $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.
Geometry, Symmetry and Physics Twisted Holography & Koszul Duality 4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)

In this talk we discuss the problem of coupling quantum field
theories to topological line defects and explain that it is governed by a
central mathematical notion called Koszul duality for associative algebras. We
then propose an analogous physical definition of Koszul duality for chiral
algebras. We will explain that in the context of (a twisted version of)
AdS(3)/CFT(2), in which a chiral algebra naturally arises in the CFT(2), a
deformation of this version of Koszul duality can be used to compute algebra
OPEs. This talk is based on work in collaboration with K. Costello.
March 30, 2021
Geometry & Topology Multiplicity one of generic stable Allen-Cahn minimal hypersurfaces 4:00pm -

"Allen-Cahn minimal hypersurfaces” are obtained as limits of nodal sets of solutions to the Allen-Cahn equation. Understanding the local picture of this convergence is a fundamental problem. For instance, can we avoid the situation of a nodal set looking like a multigraph over the limit hypersurface? Examples of this phenomenon, called “multiplicity” or "interface foliation”, are known when the limit hypersurface is unstable. Together with A. Neves and F. Marques we proved that generically (and in all dimensions) stable Allen-Cahn minimal hypersurfaces can only occur with multiplicity one. We will discuss this and other topics.