CMSA HODGE AND NOETHER-LEFSCHETZ LOCI SEMINAR : | Hossein MovasatiIMPA |
A new model for modular curves |

on Wednesday, January 23, 2019, at 1:30 - 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

One of the non-trivial examples of a Hodge locus is the modular curve X_0(N), which is due to isogeny of elliptic curves (a Hodge/algebraic cycle in the product of two elliptic curves). After introducing the notion of enhanced moduli of elliptic curves, I will describe a new model for X_0(N) in the weighted projective space of dimension 4 and with weights (2,3,2,3,1). I will also introduce some elements in the defining ideal of such a model. The talk is based on the article arXiv:1808.01689. |

COLLOQUIUM: | Simion FilipHarvard and IAS |
Discrete groups, Lyapunov exponents, and Hodge theory |

on Thursday, January 31, 2019, at 4:00 pm in Science Center 507 | ||

Families of algebraic manifolds give interesting examples of discrete subgroups of Lie groups, via their monodromy. They also lead to differential equations, such as the hypergeometric ones, whose solutions have an arithmetic significance. After providing the necessary background I will explain a connection to dynamical invariants called Lyapunov exponents, which reveals special geometric features of the discrete groups and the corresponding differential equations. |

GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR: | Zoltán SzabóPrinceton University |
Algebraic methods in knot Floer homology |

on Friday, February 01, 2019, at 3:30 pm in Science Center 507 | ||

The aim of this talk is to present an algebraic description of knot Floer homology, discovered in a joint work with Peter Ozsváth. Future schedule is found here: https://scholar.harvard.edu/gerig/seminar |

CMSA SPECIAL LECTURE SERIES ON DERIVED ALGEBRAIC/DIFFERENTIAL GEOMETRY: TUESDAYS & THURSDAYS BEGINNING FEBRUARY 5, 2019: | Artan SheshmaniCMSA |
Lecture 1: Model and NA-categories |

on Tuesday, February 05, 2019, at 3:00 - 4:30 pm in CMSA Building, 20 Garden St, G10 | ||

Derived algebraic geometry is a branch of mathematics whose aim is to extend algebraic geometry to study of spaces which are locally defined over topological ring spectra (rather than commutative rings of polynomials) and hence they can come equipped with higher homotopical structure. An object in derived geometry comes with an infinity groupoid of internal symmetries as well as infinity algebra of functions (which we will define in detail during the course). One main application of study of derived algebraic spaces (derived schemes or stacks) has been to provide an algebraic description of configuration spaces in gauge theory. Take for instance the so-called BV-BRST complex which encodes the function algebra on the homological resolution of a locus of solutions to differential equation which encodes movement of particles in certain gauge theoretic mechanical system. The derived (or higher) differential geometry is also similar to differential geometry in higher homotopical setting. In derived differential geometry smooth manifolds are replaced with smooth infinity-stacks or (infinity, 1)-sheaves which live over their site. The course discusses both derived algebraic and derived differential geometries and aims to eventually study application of these two constructions in enumerative geometry of moduli stack of algebraic objects on Calabi-Yau or Fano manifolds (for instance Donaldson-Thomas theory of sheaves on Calabi-Yau 4 folds which have a priori ill-behaved non-perfect deformation-obstruction theories). Some of the main topics covered in the course are: Derived Artin Stacks and their cotangent complexes, De Rham complexes and S1 equivariant schemes (loop spaces), Cyclic homology, Shifted symplectic structures, Lagrangians and Lagrangian fibrations, Derived intersection theory and finally a derived analog of Uhlenbeck-Yau correspondences. The detail syllabus of the course and schedule is provided at: http://cmsa.fas.harvard.edu/derived-geometry/ |

CMSA SPECIAL LECTURE SERIES ON DERIVED ALGEBRAIC/DIFFERENTIAL GEOMETRY: TUESDAYS & THURSDAYS BEGINNING FEBRUARY 5, 2019: | Artan SheshmaniCMSA |
Lecture 2: Grothendieck topologies and homotopy descent |

on Thursday, February 07, 2019, at 3:00 - 4:30 pm in CMSA Building, 20 Garden St, G10 | ||

Derived algebraic geometry is a branch of mathematics whose aim is to extend algebraic geometry to study of spaces which are locally defined over topological ring spectra (rather than commutative rings of polynomials) and hence they can come equipped with higher homotopical structure. An object in derived geometry comes with an infinity groupoid of internal symmetries as well as infinity algebra of functions (which we will define in detail during the course). One main application of study of derived algebraic spaces (derived schemes or stacks) has been to provide an algebraic description of configuration spaces in gauge theory. Take for instance the so-called BV-BRST complex which encodes the function algebra on the homological resolution of a locus of solutions to differential equation which encodes movement of particles in certain gauge theoretic mechanical system. The derived (or higher) differential geometry is also similar to differential geometry in higher homotopical setting. In derived differential geometry smooth manifolds are replaced with smooth infinity-stacks or (infinity, 1)-sheaves which live over their site. The course discusses both derived algebraic and derived differential geometries and aims to eventually study application of these two constructions in enumerative geometry of moduli stack of algebraic objects on Calabi-Yau or Fano manifolds (for instance Donaldson-Thomas theory of sheaves on Calabi-Yau 4 folds which have a priori ill-behaved non-perfect deformation-obstruction theories). Some of the main topics covered in the course are: Derived Artin Stacks and their cotangent complexes, De Rham complexes and S1 equivariant schemes (loop spaces), Cyclic homology, Shifted symplectic structures, Lagrangians and Lagrangian fibrations, Derived intersection theory and finally a derived analog of Uhlenbeck-Yau correspondences. The detail syllabus of the course and schedule is provided at: http://cmsa.fas.harvard.edu/derived-geometry/ |

COLLOQUIUM: | Hector PastenPontificia Universidad Catolica de Chile |
Modular forms and the ABC conjecture |

on Monday, February 11, 2019, at 3:00 pm in Science Center 507 | ||

The ABC conjecture concerns a surprising relation between addition and factorization of integers, and it is well-known for having a number of unexpected consequences. The problem remains open and most of the literature around it is focused on conditional applications, but nonetheless there are some unconditional results. In this talk I will give an overview of the available unconditional progress on the ABC conjecture and the relevant techniques, with particular attention to its connection with modular forms. |

CMSA SPECIAL LECTURE SERIES ON DERIVED ALGEBRAIC/DIFFERENTIAL GEOMETRY: TUESDAYS & THURSDAYS BEGINNING FEBRUARY 5, 2019: | Artan SheshmaniCMSA |
Lecture 3: Derived Artin stacks |

on Tuesday, February 12, 2019, at 3:00 - 4:30 pm in CMSA Building, 20 Garden St, G10 | ||

Derived algebraic geometry is a branch of mathematics whose aim is to extend algebraic geometry to study of spaces which are locally defined over topological ring spectra (rather than commutative rings of polynomials) and hence they can come equipped with higher homotopical structure. An object in derived geometry comes with an infinity groupoid of internal symmetries as well as infinity algebra of functions (which we will define in detail during the course). One main application of study of derived algebraic spaces (derived schemes or stacks) has been to provide an algebraic description of configuration spaces in gauge theory. Take for instance the so-called BV-BRST complex which encodes the function algebra on the homological resolution of a locus of solutions to differential equation which encodes movement of particles in certain gauge theoretic mechanical system. The derived (or higher) differential geometry is also similar to differential geometry in higher homotopical setting. In derived differential geometry smooth manifolds are replaced with smooth infinity-stacks or (infinity, 1)-sheaves which live over their site. The course discusses both derived algebraic and derived differential geometries and aims to eventually study application of these two constructions in enumerative geometry of moduli stack of algebraic objects on Calabi-Yau or Fano manifolds (for instance Donaldson-Thomas theory of sheaves on Calabi-Yau 4 folds which have a priori ill-behaved non-perfect deformation-obstruction theories). Some of the main topics covered in the course are: Derived Artin Stacks and their cotangent complexes, De Rham complexes and S1 equivariant schemes (loop spaces), Cyclic homology, Shifted symplectic structures, Lagrangians and Lagrangian fibrations, Derived intersection theory and finally a derived analog of Uhlenbeck-Yau correspondences. The detail syllabus of the course and schedule is provided at: http://cmsa.fas.harvard.edu/derived-geometry/ |

SOCIAL SCIENCE APPLICATIONS FORUM: | Bobby Pakzad-HursonBrown University |
Crowdsourcing and Optimal Market Design |

on Friday, February 22, 2019, at 3:00 PM in CMSA Building, 20 Garden St, G02 |

Artan SheshmaniCMSA |
Lecture 4: Cotangent complexes | |

on Saturday, December 14, 2019, at 3:00 - 4:30 pm in CMSA Building, 20 Garden St, G10 | ||