POSTDOC SEMINAR: | Emily RiehlHARVARD UNIVERSITY |
On the duality between "free" and "forgetful" constructions |

on Friday, October 31, 2014, at 2:30 -3:30 PM in Science Center 530 | ||

Groups, rings, and compact Hausdorff spaces have underlying sets and admit "free" constructions. Moreover, each type of object is completely characterized by the shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding provides formulas for direct and inverse limits. After we describe a typical encounter with adjunctions, monads, and their algebras, we introduce a new "homotopy coherent" version of this adjoint duality together with a graphical calculus that is used to define a homotopy coherent algebra. |

GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Nate BottmanMASSACHUSETTS INSTITUTE OF TECHNOLOGY |
Singular quilts and a proposed A-infinity 2-category |

on Friday, October 31, 2014, at 3:30 PM in Science Center 507 | ||

I will describe work-in-progress with Katrin Wehrheim to construct the "symplectic A-infinity 2-category `Symp' ", whose objects are symplectic manifolds and where hom(M,N) is the immersed Fukaya category Fuk(M- x N). The structure maps will be defined by counting moduli spaces of pseudoholomorphic quilts with a figure eight singularity. A formal consequence of Symp is a symplectic analogue of Fourier--Mukai functors. After describing the blueprint for Symp, I will present two analytic results about singular quilts: a removal of singularity for the figure eight singularity, and a Gromov compactness theorem for strip-shrinking. This talk will be based partly on the preprint arXiv:1410.3834 and partly on an upcoming preprint with Wehrheim. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Ji Oon Lee - *1st speaker*KAIST |
Tracy-Widom Distribution for Real Sample Covariance Matrices with General Population |

on Friday, October 31, 2014, at 12:00 - 2:30 PM in Science Center 232 | ||

Consider a sample covariance matrix of the form XX^*. The sample X is an MxN real random matrix whose columns are independent multivariate Gaussian vectors with covariance Σ. We show that the fluctuation of the largest rescaled eigenvalue is given by Tracy-Widom distribution for a large class of general Σ. This is a joint work with Kevin Schnelli. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Mykhaylo Shkolnikov - *2nd speaker*PRINCETON UNIVERSITY |
Intertwinings, wave equations and beta ensembles |

on Friday, October 31, 2014, at 12:00 - 2:30 PM in Science Center 232 | ||

We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to wave equations and more general hyperbolic partial differential equations. The talk will be devoted to this recent development, as well as an algebraic perspective on intertwinings which, in particular, gives rise to a novel intertwining in beta random matrix theory. The talk is based on joint works with Soumik Pal and Vadim Gorin. |

BASIC NOTIONS SEMINAR: | Joe HarrisHARVARD UNIVERSITY |
Plane Curves |

on Monday, November 03, 2014, at 3:00 PM in Science Center 507 |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS PHYSICAL MATHEMATICS SEMINAR: | Kaoru IkedaHARVARD UNIVERSITY |
Symplectic structures of projective flag manifold and the unitary representations |

on Monday, November 03, 2014, at 12:00 - 2:00 PM in Science Center 232 |

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.) |

NUMBER THEORY SEMINAR: | Cristian PopescuUC SAN DIEGO |
Equivariant Iwasawa theory, Hecke characters and the K-theory of number fields |

on Wednesday, November 05, 2014, at 3:00 PM in Science Center 507 | ||

In recent work with Greither, we proved a main conjecture in equivariant Iwasawa theory, refining Wiles' results on the classical main conjecture over totally real number fields. Via Iwasawa co-descent this permitted us to prove a refinement of the classical Brumer-Stark conjecture (under certain hypotheses.) In joint work with Banaszak, we used these results to construct a family of algebraic Hecke characters for an arbitrary CM number field, generalizing Weil's Jacobi sum Hecke characters. Further, we used certain special values of these Hecke characters to construct "Stickelberger splitting" maps for the localization sequences in the Quillen K-theory of CM and totally real number fields. We will review these results and constructions and comment on further potential applications to the classical conjectures of Iwasawa and Kummer-Vandiver on class-groups of cyclotomic fields. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Washington TaylorMIT |
String theory vacua and the enumeration of elliptic Calabi-Yau threefolds |

on Wednesday, November 05, 2014, at 4:15 PM in Science Center 507 | ||

The quest for a systematic understanding of solutions to string theory leads to many interesting classification problems in geometry. A central example is the problem of classifying Calabi-Yau threefold geometries. While it is not known if the number of topological types of CY threefolds is finite, the restricted class of elliptically fibered CY threefolds has been shown by Gross to admit a finite number of distinct topological types (up to birational equivalence). This talk describes recent work in which insights from physics have been combined with basic algebraic geometry and combinatorics to give a systematic approach to bounding and enumerating all elliptically fibered Calabi-Yau threefolds. Empirical evidence suggests that in fact elliptically fibered Calabi-Yau threefolds may constitute the majority of possible CY threefold geometries, particularly at large Hodge numbers. Classification of fourfolds and implications for physics will also be discussed. |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Daniel LittSTANFORD UNIVERSITY |
Non-abelian Lefschetz Hyperplane Theorems |

on Tuesday, November 11, 2014, at 3:00 PM in Science Center 507 | ||

Work of Lefschetz (in 1924) and Grothendieck (in SGA II) provides many relationships between properties of a smooth projective variety X and an ample divisor D in X. For example, the singular or l-adic cohomology of X agrees with that of D in low degree; X and D have the same Picard group if X has dimension at least 4; and X and D have the same fundamental group if X has dimension at least 3. I’ll describe a general result which encompasses some of these Lefschetz hyperplane theorems and many new ones, comparing maps out of X to maps out of D. The case when the target of these maps is a moduli scheme or stack is of particular interest; for example, one may take the target to be Mg, and thus compare families of curves over X to families over D. |