DIFFERENTIAL GEOMETRY SEMINAR: | Hiro Lee TanakaHARVARD UNIVERSITY |
Lagrangian Cobordisms and Fukaya Categories |

on Tuesday, September 23, 2014, at 4:15 PM in Science Center 507 | ||

Given an exact symplectic manifold with some extra decorations, one can construct two categories whose objects are (exact, decorated) Lagrangians: The Fukaya category, and a category whose morphisms are cobordisms. Both can be triangulated, and there is even a functor between them respecting their triangulated structures. In this talk we discuss some work-in-progress drawing parallels between the two categories, and if time allows, we will discuss the relation of our work with that of Biran-Cornea, or discuss possible applications to the Nearby Lagrangian Conjecture and Heegard-Floer invariants for 2-, 3-, and 4-manifolds. |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Gabriel BujokasHARVARD UNIVERSITY |
Covers of an elliptic curve E and curves in ExP^1 |

on Tuesday, September 23, 2014, at 4:00 PM in MIT E-17-122 | ||

**This seminar meets at MIT this week** |

NUMBER THEORY SEMINAR: | Alison MillerHARVARD UNIVERSITY |
Counting Simple Knots via Arithmetic Invariant Theory |

on Wednesday, September 24, 2014, at 3:00 - 4:00 PM in Science Center 507 | ||

Certain knot invariants coming from the Alexander module have natural number-theoretic structure: they can be interpreted as ideal classes in certain rings. I will explain the connection between these invariants and the arithmetic invariant theory of Sp_2g acting on its adjoint representation. The associated counting questions have a knot-theoretic interpretation. In fact, they count certain families of high-dimensional knots and related objects. I will give asymptotic answers to these questions in the case of genus 1 assuming standard number-theoretic heuristics. |

HARVARD LOGIC SEMINAR: | Gil SagiLudwig-Maximilians University, Munich |
What is a Fixed Term? |

on Wednesday, September 24, 2014, at 4:30 PM in Logic Center, 2 Arrow St, Rm 408 | ||

In standard model-theoretic semantics, logical terms are said to be fixed in the system while nonlogical terms remain variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing them precisely amounts to. My proposal is that when a term is considered logical in a system, what gets fixed is its intension rather than its extension. I provide a rigorous way of spelling out this idea. Further, I show that under certain natural assumptions, some paradigmatic examples of nonlogical terms cannot be fixed in a standard system: they require more structure than such a system affords. We thus obtain a precondition for logical terms. I then propose a graded account of logicality: the less structure a term requires, the more logical it is. Finally, I relate this idea to invariance criteria for logical terms. Invariance criteria can be used as a tool in determining how much structure a term needs in order to be fixed. Thus, rather than settling on one criterion for logicality, I use invariance conditions as a measure for logicality. |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS COLLOQUIUM: | Asaf MadiHARVARD MEDICAL SCHOOL |
Using big genomic data to analyze inner workings of immune cells |

on Wednesday, September 24, 2014, at 4:00 PM in Science Center 507 |

CENTER OF MATHEMATICAL SCIENCES & APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Amir DemboSTANFORD UNIVERSITY |
Statistical Mechanics on Sparse Random Graphs: Mathematical Perspective |

on Wednesday, September 24, 2014, at 12:00 - 2:00 PM in Science Center 232 | ||

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on random finite graphs that converge locally to trees we review recent progress in validating the `cavity' prediction for the limiting free energy per vertex and the approximation of local marginals by the belief propagation algorithm. This talk is based on joint works with Anirban Basak, Andrea Montanari, Nike Sun and Alan Sly. |

GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Joanna NelsonCOLUMBIA/IAS |
Cylindrical contact homology in dimension 3 via intersection theory and more |

on Friday, September 26, 2014, at 3:30 - 4:30 PM in Science Center 507 | ||

After reviewing the difficulties in proving d²=0 for dynamically convex contact manifolds, we explain how intersection theory and automatic transversality come to the rescue, as implemented by Hutchings-Nelson. So that these methods do not remain shrouded in mystery, we provide an overview of the relevant ideas from intersection theory for pseudoholomorphic curves in 4-dimensional symplectizations a la Siefring and Hutchings. Time permitting we end with a brief sketch of how to obtain invariance of cylindrical contact homology over Q via nonequivariant formulations, obstruction bundle gluing, domain dependent almost complex structures, and S¹-equivariant constructions. The discussion of invariance is based on work in progress by Hutchings-Nelson. |

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 |

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.) |