Mathematics Colloquia and Seminars

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This talks is about how Combinatorial Geometry and Data Science can help each other very productively.   

First, I illustrate how combinatorial geometry can help data science: Statistical inference often relies on the theory of maximum likelihood estimation and one can ask How much training data is needed, as a function of the dimension of the covariates of the data, before we expect an MLE to exist with high probability? Ultimately, data clouds are combinatorial in nature thus discrete methods work well. Stochastic variations of Tverberg’s theorem play a role to answer various MLE questions for example. Similarly, Tverberg type theorems indicate ways to extract the shape of data that does not require Topological tools and for the computation of ``medians’’ inside a d-dimensional data cloud. This part is joint work with  T. Hogan, D. Oliveros, E. Jaramillo-Rodriguez and A. Torres-Hernandez.  

Second, I show how data analysis can help combinatorial geometers (and other parts of Math) advance their research: Experimentally one can often generate data consisting of mathematical objects and one can try to use data analysis methods to guess conjectures about them. What makes a conjecture meaningful or interesting? We discuss a perspective of automating the process of mathematical conjecturing through data. We applied it to experimental data in combinatorial geometry, particularly in understanding sequences derived from  simple combinatorial 3-polytopes. We illustrate the potential of an agentic approach in machine-assisted mathematical research where the machine asks questions and searches for counterexamples. This part is joint work with R. Davila, J. Eddy, J. Lu, E. Fang, Z. Zhang.