HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Eric RiedlHARVARD UNIVERSITY |
Rational Curves on Hypersurfaces |

on Tuesday, September 30, 2014, at 3:00 PM in Science Center 507 | ||

Let X be a general degree d hypersurface in n-dimensional projective space, and consider the spaces of rational curves on X. Joint with David Yang, following work of Harris, Roth, Starr, Beheshti and Kumar, we prove that the space of degree e rational curves on X is irreducible and we compute its dimension for n > d+1. This resolves all but the n = d+1 case of a conjecture of Coskun, Harris and Starr. |

NUMBER THEORY SEMINAR: | Brian SmithlingJOHNS HOPKINS |
An arithmetic transfer conjecture for ramified unitary groups |

on Wednesday, October 01, 2014, at 3:00 - 4:00 PM in Science Center 507 | ||

The arithmetic fundamental lemma is a conjectural relation proposed by W. Zhang in connection with a relative trace formula approach to the Gan-Gross-Prasad conjectures for (unramified) unitary groups. It asserts a deep relation between the derivative of an orbital integral and an intersection number for cycles in a formal moduli space of p-divisible groups. I will report on an extension of the conjecture to the setting of ramified unitary groups along with, for unitary groups in three variables, its proof. This is joint work with M. Rapoport and W. Zhang. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Andy StromingerHarvard University |
Symmetry in Soft Theorem |

on Wednesday, October 01, 2014, at 4:00 PM in Science Center 507 |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Jun ZhangUniversity of Michigan, Ann Arbor |
Regularized learning in reproducing Kernal Banach Spaces |

on Wednesday, October 01, 2014, at 12:00 - 2:00 PM in Science Center 232 | ||

Regularized learning is the contemporary framework for learning to generalize from finite samples (classification, regression, clustering, etc). Here the problem is to learn an input-output mapping f: X->Y given finite samples {(xi, yi), i=1,…,N}. With minimal structural assumptions on X, the class of functions under consideration is assumed to fall under a Banach space of functions B. The learning-from-data problem is then formulated as an optimization problem in such a function space, with the desired mapping as an optimizer to be sought, where the objective function consists of a loss term L(f) capturing its goodness-of-fit (or the lack thereof) on given samples {(f(xi), yi), i=1,…,N}, and a penalty term R(f) capturing its complexity based on prior knowledge about the solution (smoothness, sparsity, etc). This second, regularizing term is often taken to be the norm of B, or a transformation f thereof: R(f) = f(||f||). This program has been successfully carried out for the Hilbert space of functions, resulting in the celebrated Reproducing Kernel Hilbert Space methods in machine learning. Here, we will remove the Hilbert space restriction, i.e., the existence of an inner product, and show that the key ingredients of this framework (reproducing kernel, representer theorem, feature space) remain to hold for a Banach space that is uniformly convex and uniformly Frechet differentiable. Central to our development is the use of a semi-inner product operator and duality mapping for a uniform Banach space in place of an inner-product for a Hilbert space. This opens up the possibility of unifying kernel-based methods (regularizing L2-norm) and sparsity-based methods (regularizing l1-norm), which have so far been investigated under different theoretical foundations. |

THURSDAY SEMINAR: | Mike HopkinsHARVARD UNIVERSITY |
Twisted K-theory |

on Thursday, October 02, 2014, at 4:00 - 6:00 PM in Science Center 507 |

GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Sobhan SeyfaddiniMIT |
The displaced disks problem via symplectic topology |

on Friday, October 03, 2014, at 3:30 - 4:30 PM in Science Center 507 | ||

We will show that a C^0-small area preserving homeomorphism of S^2 can not displace a disk of large area. This resolves the displaced disks problem posed by F. Béguin, S. Crovisier, and F. Le Roux. |

BASIC NOTIONS SEMINAR: | Laure Saint-RaymondÉcole Normale Supérieure |
The role of boundary layers in the global ocean circulation |

on Monday, October 06, 2014, at 3:00 PM in Science Center 507 |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Christian SchnellSTONY BROOK |
Kodaira dimension and zeros of holomorphic one-forms |

on Tuesday, October 07, 2014, at 4:00 PM in MIT E-17-122 | ||

I will talk about a joint paper with Mihnea Popa; our main result is that on a smooth complex projective variety of general type, every holomorphic one-form has a nonempty zero locus. Although this sounds like a problem in birational geometry, the proof uses results about D-modules on abelian varieties. |

DIFFERENTIAL GEOMETRY SEMINAR: | Paul SeidelMIT |
Steenrod squares and symplectic fixed points |

on Tuesday, October 07, 2014, at 5:15 PM *note change in time in Science Center 507 |

NUMBER THEORY SEMINAR: | Ronen MukamelUNIVERSITY OF CHICAGO |
Singular moduli and the primes of bad reduction of Teichmuller curves |

on Wednesday, October 08, 2014, at 3:00 - 4:00 PM in Science Center 507 |

DIFFERENTIAL GEOMETRY SEMINAR: | Tristan CollinsHARVARD UNIVERSITY |
The boundary of the Kahler cone |

on Tuesday, October 21, 2014, at 4:15 PM in Science Center 507 | ||

We will discuss a geometric characterization of classes of positive volume on the boundary of the Kahler cone of a compact Kahler manifold. As an application, we will show that finite time singularities of the Kahler-Ricci flow always form along analytic subvarieties. Time permitting, we will also discuss some applications to algebraic geometry, specifically to augmented base loci of nef and big line bundles on projective varieties. This work is joint with Valentino Tosatti. |

DIFFERENTIAL GEOMETRY SEMINAR: | Larry GuthMIT |
Homotopical effects of k-dilation |

on Tuesday, November 04, 2014, at 4:15 PM in Science Center 507 | ||

The k-dilation of a map measures how the map stretches k-dimensional areas. If Dil_k f < L, then it means that for any k-dimensional submanifold S in the domain, Vol_k (f(S)) is at most L Vol_k(S). We discuss how the k-dilation restricts the homotopy type of a map. Our main theorem concerns maps between unit spheres, from S^{m} to S^{m-1}. If k > (m+1)/2, then there are homotopically non-trivial maps S^m to S^{m-1} with arbitrarily small k-dilation. I find this somewhat counterintuitive. The construction has a similar flavor to constructions in the h-principle literature with lots of wrinkling. On the other hand, if k is at most (m+1)/2, there every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0. So there is a transition at k=(m+1)/2 between flexible behavior and rigid behavior. The first interesting case of the rigid behavior is k=3 and m=2. It was proven by Gromov in the 70's. The higher-dimensional cases are new. The main difficulty here is to connect the topology and the geometry. To detect that a map S^m to S^{m-1} is homotopically non-trivial requires tools from algebraic topology such as Steenrod squares. We have to connect the Steenrod squares with k-dimensional volumes of k-dimensional surfaces. (For the talk, I won't assume familiarity with Steenrod squares.) |