Georgia Institute of Technology College of Sciences

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.

Motivated by problems in data science, we study the following questions:

(1) Given a Hilbert space V and a group G of linear isometries, does there exist a bilipschitz embedding of the quotient metric space V/G into a Hilbert space?

(2) What are necessary and sufficient conditions for such embeddings?

(3) Which embeddings minimally distort the metric?

We answer these questions in a variety of settings, and we conclude with several open problems.

Geometry arises in myriad ways within machine learning and related areas. I this talk I will focus on settings where geometry helps us understand problems in machine learning, optimization, and sampling. For instance, when sampling from densities supported on a manifold, understanding geometry and the impact of curvature are crucial; surprisingly, progress on geometric sampling theory helps us understand certain generalization properties of SGD for deep-learning! Another fascinating viewpoint afforded by geometry is in non-convex optimization: geometry can either help us make training algorithms more practical (e.g., in deep learning), it can reveal tractability despite non-convexity (e.g., via geodesically convex optimization), or it can simply help us understand existing methods better (e.g., SGD, eigenvector computation, etc.).

Ultimately, I hope to offer the audience some insights into geometric thinking and share with them some new tools that help us design, understand, and analyze models and algorithms. To make the discussion concrete I will recall a few foundational results arising from our research, provide several examples, and note some open problems.

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Bio: Suvrit Sra is a Alexander von Humboldt Professor of Artificial Intelligence at the Technical University of Munich (Germany), and and Associate Professor of EECS at MIT (USA), where he is also a member of the Laboratory for Information and Decision Systems (LIDS) and of the Institute for Data, Systems, and Society (IDSS). He obtained his PhD in Computer Science from the University of Texas at Austin. Before TUM & MIT, he was a Senior Research Scientist at the Max Planck Institute for Intelligent Systems, Tübingen, Germany. He has held visiting positions at UC Berkeley (EECS) and Carnegie Mellon University (Machine Learning Department) during 2013-2014. His research bridges mathematical topics such as differential geometry, matrix analysis, convex analysis, probability theory, and optimization with machine learning. He founded the OPT (Optimization for Machine Learning) series of workshops, held from OPT2008–2017 at the NeurIPS  conference. He has co-edited a book with the same name (MIT Press, 2011). He is also a co-founder and chief scientist of Pendulum, a global AI+logistics startup.

Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L,    I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?

The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least  (3s+1)/2 when s>0. 

We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture