RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Carl LucibelloMIT |
The Random Perceptron Problem: thresholds, phase transitions, and geometry |

on Friday, April 20, 2018, at 2:00 - 3:00 pm in CMSA Building, 20 Garden St, G02 | ||

The perceptron is the simplest feedforward neural network model, the building block of the deep architectures used in modern machine learning practice. In this talk, I will review some old and new results, mostly focusing on the case of binary weights and random examples. Despite its simplicity, this model provides an extremely rich phenomenology: as the number of examples per synapses is increased, the system undergoes different phase transitions, which can be directly linked to solvers' performances and to information theoretic bounds. A geometrical analysis of the solution space shows how two different types of solutions, akin to wide and sharp minima, have different generalization capabilities when presented with new examples. Solutions in dense clusters generalize remarkably better, partially closing the gap with Bayesian optimal estimators. Most of the results I will present were first obtained using non-rigorous techniques from spin glass theory and many of them haven't been rigorously established yet, although some big steps forward have been taken in recent years. |

RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Yas DespandeMIT |
Phase transitions in estimating low-rank matrices |

on Friday, April 20, 2018, at 3:00 - 4:00 pm in CMSA Building, 20 Garden St, G02 | ||

Low-rank perturbations of Wigner matrices have been extensively studied in random matrix theory; much information about the corresponding spectral phase transition can be gleaned using these tools. In this talk, I will consider an alternative viewpoint based on tools from spin glass theory, and two examples that illustrate how these they yield information beyond traditional spectral tools. The first example is the stochastic block model, where we obtain a full information-theoretic picture of estimation. The second example demonstrates how side information alters the spectral threshold. It involves a new phase transition that interpolates between the Wigner and Wishart ensembles. This is based on joint works with Emmanuel Abbe, Andrea Montanari, Elchanan Mossel and Subhabrata Sen. |

CMSA MIRROR SYMMETRY SEMINAR: | Xiaomeng XuMIT |
Stokes phenomenon, quantum groups and 2d topological field theory |

on Friday, April 20, 2018, at 9:30 - 10:45 AM in Sci Center 530 | ||

This talk will include a general introduction to differential equations with singularities, and its relation with symplectic geometry and representation theory. In particular, we will focus on the Stokes phenomenon of linear systems of ordinary differential equations, and construct the braiding of Drinfeld-Jimbo quantum groups as quantum Stokes matrices. We then introduce an isomonodromic deformation of Knizhnikâ€“Zamolodchikov type equations, and relate its classical limit to the theory of Frobenius manifolds. |

CMSA MIRROR SYMMETRY SEMINAR: | Alan ThompsonCambridge |
Threefolds fibred by K3 surfaces and mirror symmetry |

on Friday, April 20, 2018, at 10:45 AM - 12:00 PM in Sci Center 530 | ||

I will present a number of results about the classification of threefolds fibred by K3 surfaces. These will start out very general but rapidly specialize, first to the case where the K3 surface fibres admit a certain class of lattice polarization, then further to the case where the threefold total space is Calabi-Yau. In the process, we will begin to notice links between this classification and the classification of Fano threefolds. This turns out not to be a coincidence, but a manifestation of mirror symmetry, relating Calabi-Yau threefolds fibred by K3 surfaces to Calabi-Yau threefolds constructed by gluing together Fanos and smoothing the result. This is joint work with C. Doran, A. Harder, R. Kooistra, and A. Novoseltsev. |

CMSA MATHEMATICAL PHYSICS SEMINAR: | Baohua FuChinese Academy of Science |
Equivariant compactifications of vector groups |

on Monday, April 23, 2018, at 12:00 - 1:00 PM in CMSA Building, 20 garden St., G02 | ||

In 1954, Hirzebruch raised the problem to classify smooth compactifications of vector spaces with second Betti number 1, which is known till now up to dim 3. In 1999, Hassett-Tschinkel considered the equivariant version of this problem and obtained the classification up to dim. 3. I'll report recent progress on this (equivariant) problem. In particular, we obtain the classification up to dimension 5. |

DIFFERENTIAL GEOMETRY SEMINAR: | Jiaping WangMinnesota |
Structure at infinity for four dimensional shrinking Ricci Solitons |

on Tuesday, April 24, 2018, at 4:15 PM in Science Center 507 | ||

Ricci solitons, as self-similar solutions to the Ricci flows, play a prominent role in the study of singularity formation of Ricci flows. In this talk, we are primarily concerned with four dimensional shrinking Ricci solitons and discuss joint work with Ovidiu Munteanu on their geometric structure at infinity. |