CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Tom HouCalifornia Institute of Technology |
Computer-assisted analysis of singularity formation of a regularized 3D Euler equation |

on Monday, February 26, 2018, at 4:30 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Whether the 3D incompressible Euler equation can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. In a recent joint work with Dr. Guo Luo, we provided convincing numerical evidence that the 3D Euler equation develops finite time singularities. Inspired by this finding, we have recently developed an integrated analysis and computation strategy to analyze the finite time singularity of a regularized 3D Euler equation. We first transform the regularized 3D Euler equation into an equivalent dynamic rescaling formulation. We then study the stability of an approximate self-similar solution. By designing an appropriate functional space and decomposing the solution into a low frequency part and a high frequency part, we prove nonlinear stability of the dynamic rescaling equation around the approximate self-similar solution, which implies the existence of the finite time blow-up of the regularized 3D Euler equation. This is a joint work with Jiajie Chen, De Huang, and Dr. Pengfei Liu. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Jordan KellerBHI |
Linear Stability of Schwarzschild Black Holes |

on Monday, February 26, 2018, at 12:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

The Schwarzschild black holes comprise a static, spherically symmetric family of black hole solutions to the vacuum Einstein equations. The physical relevance of such solutions is intimately related to their stability under gravitational perturbations. We present results on the linear stability of the Schwarzschild black holes, joint work with Pei-Ken Hung and Mu-Tao Wang. |

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

on Tuesday, February 27, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

Small Quantum Cohomology (Definition and Properties) I |

CMSA HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Colin DiemerIHES |
Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS |

on Tuesday, February 27, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Tuesday, Feb. 28 and Thursday, March 1: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau's the role of the mirror map is well-appreciated. In these talks I'll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we'll focus on the case of del Pezzo surfaces (due to unpublished work of Pantev) and the toric case (due to the speaker with Katzarkov and G. Kerr). Time permitting, we may make some speculations on the role of LG moduli in the work of Gross-Hacking-Keel (in progress work of the speaker with T. Foster). |

MATHEMATICAL PHYSICS SEMINAR: | Hans WenzlUniversity of California, San Diego |
Coideal Algebras and Subfactors |

on Tuesday, February 27, 2018, at 4:00 PM in Jefferson 356 | ||

It is well-known that any subgroup of a group G defines a module category over Rep G. Analogs of special embeddings of Lie groups, symmetric spaces, also exist for quantum groups as coideal subalgebras. This should also provide large classes of examples of module categories of fusion categories coming from quantum groups. This has been worked out for a number of important cases where one can explicitly calculate the corresponding algebra objects and indices of the corresponding subfactors. |

LOGIC SEMINAR: | Alexander Van AbelCity University of New York |
The Feferman-Vaught Theorem and the Product of All Prime Finite Fields |

on Tuesday, February 27, 2018, at 5:15 pm in Science Center 507 | ||

The Feferman-Vaught Theorem in model theory gives a sort of upper bound on the complexity of definable subsets in a product structure. We show how this theorem implies that subsets whose boundary is dense (in the product topology where every factor structure has the discrete topology) are undefinable. We also show how the upper bound in the F.V. Theorem is the best possible, in the case of the language of rings where each factor structure is an integral domain. Finally, we apply these results to the ring $\prod_{p prime} F_p$, the product of all finite prime fields, and obtain a quantifier elimination result for the structure. |

NUMBER THEORY SEMINAR: | Wei ZhangMIT |
Special cycles on simple Shimura varieties |

on Wednesday, February 28, 2018, at 3:00 PM in Sci Center 507 | ||

This is a work in progress inspired by the the arithmetic Gan--Gross--Prasad conjecture where one is interested in the arithmetic diagonal cycle on the product of two Shimura varieties. We study special cycles on simple Shimura varieties attached to central simple algebras with an involution of second kind. We study some local questions arising from the relative trace formula approach. |

INFORMAL GEOMETRY & DYNAMICS SEMINAR: | Richard SchwartzBrown University |
5 points on the sphere |

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

The question of which configurations of N points on the sphere minimize the total potential energy (say with respect to a power law) is a very difficult one except when N=1,2,3,4,6,12. I'll explain the progress I made on the problem when N=5. My main result is a computer-assisted proof that the triangular bi-pyramid is the optimal configuration with respect to a power law with exponents if and only s |

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

on Thursday, March 01, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

Small Quantum Cohomology (Definition and Properties) II |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Colin DiemerIHES |
Moduli spaces of Landau-Ginzburg models and (mostly Fano) HMS |

on Thursday, March 01, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Tuesday, Feb. 28 and Thursday, March 1: Mirror symmetry as a general phenomenon is understood to take place near the large complex structure limit resp. large radius limit, and so implicitly involves degenerations of the spaces under consideration. Underlying most mirror theorems is thus a mirror map which gives a local identification of respective A-model and B-model moduli spaces. When dealing with mirror symmetry for Calabi-Yau's the role of the mirror map is well-appreciated. In these talks I'll discuss the role of moduli in mirror symmetry of Fano varieties (where the mirror is a Landau-Ginzburg (LG) model). Some topics I expect to cover are a general structure theory of moduli of LG models (follows Katzarkov, Kontsevich, Pantev), the interplay of the topology of LG models with autoequivalence relations in the Calabi-Yau setting, and the relationship between Mori theory in the B-model and degenerations of the LG A-model. For the latter topic we'll focus on the case of del Pezzo surfaces (due to unpublis |

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

on Tuesday, March 06, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

Big Quantum Cohomology I |

CMSA HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Dmytro ShklyarovTechnische Universität Chemnitz |
On categories of matrix factorizations and their homological invariants |

on Tuesday, March 06, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

The talks will cover the following topics: 1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements. 2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion would be to try to attach similar Hodge-like data to abstract derived categories. I will talk about some recent results in this direction and illustrate the approach in the context of the LG B-models. 3. Hochschild cohomology of LG orbifolds. The scope of applications of the LG mod- els in mirror symmetry is significantly expanded once we include one extra piece of data, namely, finite symmetry groups of singularities. The resulting models are called orbifold LG models or LG orbifolds. LG orbifolds with abelian symmetry groups appear in mir- ror symmetry as mirror partners of varieties of general type, open varieties, or other LG orbifolds. Associated with singularities with symmetries there are equivariant versions of the matrix factorization categories which, just as their non-equivariant cousins, describe D-branes in the corresponding orbifold LG B-models. The Hochschild cohomology of these categories should then be isomorphic to the closed string algebra of the models. I will talk about an explicit description of the Hochschild cohomology of abelian LG orbifolds. |

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

on Thursday, March 08, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

Big Quantum Cohomology II |

CMSA HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Dmytro ShklyarovTechnische Universität Chemnitz |
On categories of matrix factorizations and their homological invariants |

on Thursday, March 08, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

The talks will cover the following topics: 1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements. 2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion wou |

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

on Tuesday, March 13, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

GW potential |

CMSA HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Dmytro ShklyarovTechnische Universität Chemnitz |
On categories of matrix factorizations and their homological invariants |

on Tuesday, March 13, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

The talks will cover the following topics: 1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements. 2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion wou |

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

on Thursday, March 15, 2018, at 1:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

WDVV equation |

CMSA HOMOLOGICAL MIRROR SYMMETRY FOCUSED LECTURE SERIES: | Dmytro ShklyarovTechnische Universität Chemnitz |
On categories of matrix factorizations and their homological invariants |

on Thursday, March 15, 2018, at 3:00 - 4:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

The talks will cover the following topics: 1. Matrix factorizations as D-branes. According to physicists, the matrix factorizations of an isolated hypersurface singularity describe D-branes in the Landau-Ginzburg (LG) B-model associated with the singularity. The talk is devoted to some mathematical implications of this observation. I will start with a review of open-closed topological field theories underlying the LG B-models and then talk about their refinements. 2. Semi-infinite Hodge theory of dg categories. Homological mirror symmetry asserts that the “classical” mirror correspondence relating the number of rational curves in a CY threefold to period integrals of its mirror should follow from the equivalence of the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. The classical mirror correspondence can be upgraded to an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the assertion wou |

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

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

The P^2 case |

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 |