NUMBER THEORY SEMINAR: | Ananth ShankarMIT |
Exceptional splitting of reductions of abelian surfaces |

on Wednesday, March 21, 2018, at 3:00 pm in Science Center 507 | ||

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We will discuss this problem over number fields and function fields, mainly focusing on the case when the surface has real multiplication. This is joint work with Tang and Maulik-Tang. |

INFORMAL DYNAMICS & GEOMETRY SEMINAR: | Kenneth BrombergUniversity of Utah |
Bounds on renormalized volume and the volume of the convex core |

on Wednesday, March 21, 2018, at 4:00 pm in Science Center 530 | ||

A conformally compact hyperbolic 3-manifold will have infinite volume (at least if the conformal boundary is non-empty). Krasnov and Schlenker defined a renormalized volume (motivated by work of Graham and Witten on conformally compact Einstein manifolds) that assigns a finite volume to such manifolds. This defines a function on the space of all conformally compact hyperbolic 3-manifolds. We will discuss the gradient flow of this function which seems to reveal, in ways we will make precise, the geometry of the manifolds. In particular, we will show how this gradient flow can be used to give bounds on the volume of the convex core of the manifolds. This is joint work with M. Bridgeman and J. Brock. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Ramesh NarayanDepartment of Astronomy, Harvard University |
Black Holes and Naked Singularities |

on Wednesday, March 21, 2018, at 4:30 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Black Hole solutions in General Relativity contain Event Horizons and Singularities. Astrophysicists have discovered two populations of black hole candidates in the Universe: stellar-mass objects with masses in the range 5 to 30 solar masses, and supermassive objects with masses in the range million to several billion solar masses. There is considerable evidence that these objects have Event Horizons. It thus appears that astronomical black hole candidates are true Black Holes. Direct evidence for Singularities is much harder to obtain since, at least in the case of Black Holes, the Singularities are hidden inside the Event Horizon. However, General Relativity also permits Naked Singularities which are visible to external observers. Toy Naked Singularity models have been constructed, and some observational features of accretion flows in these spacetimes have been worked out. |

CMSA SPECIAL LECTURE SERIES ON QUANTUM COHOMOLOGY, NAKAJIMA VARETIES AND QUANTUM GROUPS: | Artan SheshmaniQGM & CMSA |
GW Invariants via Quantum Cohomology |

on Thursday, March 22, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

The Quintic threefold case |

THURSDAY SEMINAR: | Jacob LurieHarvard University |
Koszul Duality |

on Thursday, March 22, 2018, at 3:00 - 5:00 pm in Science Center 507 |

HARVARD LOGIC COLLOQUIUM: | Donald MartinUCLA |
Cantor's Grundlagen |

on Thursday, March 22, 2018, at 4:00 - 5:00 pm in Logic Center, 2 Arrow St, Rm 420 | ||

"Cantor’s early (1883) Grundlagen einer allgemeinen Mannigfaltigkeitslehre is badly organized and has important errors and omissions. Nevertheless it is rich in content, and its concepts are in some ways superior to Cantor’s later ones." "I will mainly concentrate on a few aspects of Grundlagen: (1) what might be called Cantor’s quasi- axiomatic, iterative account of ordinal numbers; (2) the role that something like a Replacement Axiom plays in this account; (3) the relation between the Grundlagen notion of absolute infinity and Cantor’s later notion of inconsistent multiplicities." |

MATHEMATICAL PHYSICS SEMINAR: | Wei ZhangMIT |
Superpositivity of L-functions and ‘completion of square’ |

on Friday, March 23, 2018, at 3:00 PM in Jefferson 250 | ||

We explain how the Riemann hypothesis and its generalization imply the positivity of the special values of all derivatives of normalized L-functions. This raises a question: can one interpret these positive values as squares? We present some examples of such interpretations, in the joint work with Xinyi Yuan and Shouwu Zhang over number fields on the Gross-Zagier formula, and with Zhiwei Yun over function fields on the Higher Gross-Zagier formula respectively. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Yi XieSCGP |
Surgery, Polygons and Instanton Floer homology |

on Monday, March 26, 2018, at 12:00 PM in CMSA Building, 20 Garden Street, G02 | ||

Many classical numerical invariants (including Casson invariant, Alexander polynomial and Jones polynomial) for 3-manifolds or links satisfy surgery formulas relating three different 3-manifolds or links. All those invariants are categorified by certain Floer homologies (or Khovanov homology) which also satisfy so-called surgery exact triangles. In this talk I will discuss the notion of "surgery exact polygons" which appears in the SU(N)-instanton Floer homology theory. Roughly speaking, an "n-gon" is a relation among the Floer homology groups of 3-manifolds obtained by n different Dehn surgeries on a fixed knot. It generalizes the surgery exact triangle in SU(2)-instanton Floer homology. If time permits, I will also talk about a homological-mirror-symmetry-type conjecture which motivates this work. This is joint work with Lucas Culler and Aliakbar Daemi. |

INFORMAL GEOMETRY & DYAMICS SEMINAR: | Curtis McMullenHarvard University |
Planes immersed in hyperbolic 3-manifolds |

on Wednesday, March 28, 2018, at 4:00 PM in Science Center 530 |

DIFFERENTIAL GEOMETRY SEMINAR: | D.H. PhongColumbia University |
The Anomaly Flow and Fu-Yau Hessian Equations |

on Friday, March 30, 2018, at 3:00 - 4:15 pm in Science Center 507 | ||

The Hull-Strominger system is a system of equations in 3 complex dimensions arising from string theory, but which is particularly interesting from the point of view of both non-K\”ahler geometry and partial differential equations: there, metrics are not required to be K\”ahler, but only conformally balanced, and they must satisfy a curvature condition which is quadratic in the curvature tensor. A key difficulty is to implement the conformally balanced condition, in the absence of a $d-\bar d$ Lemma as in K\”ahler geometry. The Anomaly Flow is a flow of metrics designed to achieve this. However, in doing so, it has to forego properties which are normally desirable in fully non-linear PDE theory, such as concavity. A crucial test case is the case of Calabi-Eckmann fibrations, solved by J.X. Fu and X.T. Yau in 2006 by elliptic methods. We show that the Anomaly Flow performs even better than elliptic methods in this case. In fact, the resulting techniques also lead to the solution of a whole family of Hessian equations in arbitrary dimensions, of which the generalization proposed by Fu-Yau of their 2006 equation is just the simplest example. This is joint work with S. Picard and X.W. Zhang. |