SPECIAL SEMINAR ON GEOMETRY: | Ryosuke TakahashiTohoku University |
A new parabolic flow approach to the Kahler-Einstein problem |

on Wednesday, November 22, 2017, at 4:00 - 5:15 pm in Science Center 507 | ||

We introduce the ``Ding-Mabuchi flow'', a new parabolic flow which is designed to deform a given Kahler metric to a Kahler-Einstein one, and fit Donaldson's new GIT picture. We provide three results about the Ding-Mabuchi flow: (1) Long-time existence and smooth convergence on canonically polarized manifolds (2) Long-time existence and strong convergence on anti-canonically polarized manifolds with Kahler-Einstein metrics, and discrete automorphism group (3) Relation between the Ding-Mabuchi flow and optimal destabilizer on Ding-unstable toric Fano manifolds. In the canonically polarized case, the large time behavior of the Ding-Mabuchi flow is similar to that of the Kahler-Ricci flow. On the other hand, the Ding-Mabuchi flow on Ding-unstable Fano manifolds has a different aspect. We observe that it encodes the optimal destabilizing information about Kahler/algebro-geometric structure in the case when the manifold is toric. This talk is based on the joint work with T.Collins (Harvard Univ) and T. Hisamoto (Nagoya Univ). |

NUMBER THEORY SEMINAR: | Renee BellMIT |
Local-to-Global Lifting for Curves in Characteristic p |

on Wednesday, November 29, 2017, at 3:15 pm *sharp in Science Center 507 | ||

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over a more general ground field. |

JOINT DEPARTMENT OF MATHEMATICS AND CMSA RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | David GamarinkMIT |
(Arguably) Hard on Average Constraint Satisfaction Problems |

on Wednesday, November 29, 2017, at 3:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

Many combinatorial optimization problems defined on random instances such as random graphs, exhibit an apparent gap between the optimal value, which can be estimated by non-constructive means, and the best values achievable by fast (polynomial time) algorithms. Through a combined effort of mathematicians, computer scientists and statistical physicists, it became apparent that a potential and in some cases a provable obstruction for designing algorithms bridging this gap is an intricate geometry of nearly optimal solutions, in particular the presence of chaos and a certain Overlap Gap Property (OGP), which we will introduce in this talk. We will demonstrate how for many such problems, the onset of the OGP phase transition indeed nearly coincides with algorithmically hard regimes. Our examples will include the problem of finding a largest independent set of a graph, finding a largest cut in a random hypergrah, random NAE-K-SAT problem, the problem of finding a largest submatrix of a random matrix, and a high-dimensional sparse linear regression problem in statistics. Joint work with Wei-Kuo Chen, Quan Li, Dmitry Panchenko, Mustazee Rahman, Madhu Sudan and Ilias Zadik. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Amitai ZernikInstitute for Advanced Study |
Computing the A∞ algebra of RP^2m ↪ CP^2m using open fixed-point localization |

on Wednesday, November 29, 2017, at 12:00 PM in CMSA Building, 20 Garden Street, Room G10 | ||

I'll explain how to compute the equivariant quantum A∞ algebra A associated with the Lagrangian embedding of RP^2m in CP^2m, using a new fixed-point localization technique that takes into account contributions from all the corner strata. It turns out that A is rigid, so its structure constants are independent of all choices. When m = 1 and in the non-equivariant limit, they specialize to give Welschinger's counts of real rational planar curves passing through some generic, conjugation invariant configurations of points in CP^2m. So we get a diagrammatic expression for computing Welschinger invariants, which I'll demonstrate with some examples. Time permitting, I'll discuss a formal extension to higher genus which satisfies string and dilation. |

JOINT DEPARTMENT OF MATHEMATICS AND CMSA RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Ankur MoitraMIT |
A New Approach to Approximate Counting and Sampling |

on Wednesday, December 06, 2017, at 2:00 - 3:00 pm in CMSA Building, 20 Garden Street, Room G10 | ||

Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints. Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm. |