In this talk we will begin by discussing some of the main concepts of machine learning at a high level. Then we will take a closer look at neural networks. And finally we will discuss an interesting application of free probability to deep learning observed by Pennington-Schoenholz-Ganguli.  The goal of this talk is to introduce mathematicians without much background in statistics to the fundamentals of machine learning while highlighting an interesting application of noncommutative probability.  Graduate students are highly encouraged to attend.

For critical percolation on the 2D triangular lattice, consider the probability that three points lie in the same cluster. The Delfino-Viti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to the imaginary DOZZ formula from conformal field theory. We prove the Delfino-Viti conjecture, and more generally, obtain the cluster connectivity three-point function of the conformal loop ensemble. Our arguments depend on the coupling between Liouville quantum gravity and the conformal loop ensemble. 
Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu.

Evolution has predisposed developing organisms to rely heavily on the repeated application of select biomechanical motifs to achieve robust growth. Within vertebrates, one prevalent architecture is the dermal condensate: these dense bundles of mesenchymal cells that as organizing centers for developing organs such as teeth, hair follicles, glands, and limbs. Though long recognized as significant, the study of these proto-organs has historically focused on biochemical regulation. This talk, however, will emphasize the mechanical role of these structures, illustrating how they can be viewed as self-organized actuators that drive tissue deformation. This discussion will be split into two parts: first, we will highlight some recently published results based on early feather morphogenesis, demonstrating how dermal condensates coordinate tissue-scale flows by localizing active contractile stresses. Second, we will use a model of the mammalian gliding membrane to examine the formation of condensates more closely, with an eye on clarifying the role of cell-matrix interactions and better understanding where condensate mechanics fits among competing hypotheses of limb growth.

Graduate school can feel like trekking through the jungle... and every adventurer needs a good survival guide! In this talk, we'll give you a few life hacks that will help along your journey through the rainforest. Older grad students who wish to attend are invited to bring their own survival strategies. Maybe we'll even see a toucan! Or a poisonous spider!