Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

Artificial neural networks have gained widespread adoption as a powerful tool for various machine learning tasks in recent years. Training a neural network to approximate a target function involves solving an inherently non-convex problem. In practice, this is done using stochastic gradient descent with random initialization. For the approximation problem with neural networks error rate guarantees are established for different classes of functions however these rates are not always achieved in practice due to many  local minima of the resulting optimization problem. 

The challenge we address in this work is the following. We want to find small size shallow neural networks that can be trained algorithmically and which achieve guaranteed approximation speed and precision. To maintain the small size we apply penalties on the weights of the network. We show that under minimal requirements, all local minima of the resulting problem are well behaved and possess a desirable small size without sacrificing precision. We adopt the integral neural network framework and use techniques from optimization theory and harmonic analysis to prove our results. In this talk, we will discuss our existing work and possible future promising areas of interest where this approach can potentially be adopted. 

We will describe several appearances of Milnor’s invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.

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An infinite-type surface is a surface whose fundamental group is not finitely generated. These surfaces are “big,” having either infinite genus or infinitely many punctures. Recently, it was shown that mapping class groups of these infinite-type surfaces have a wealth of subgroups; for example, there are infinitely many surfaces whose mapping class group contains every countable group as a subgroup. By extending a theorem for finite-type surfaces to the infinite-type case — the Nielsen realization problem — we give topological obstructions to continuous embeddings of topological groups, with a few interesting examples.