NUMBER THEORY SEMINAR: | Yunqing TangHARVARD UNIVERSITY |
Algebraic solutions of differential equations over the projective line minus three points |

on Wednesday, April 01, 2015, at 3:00 PM in Science Center 507 | ||

The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will focus on this case and introduce a p-adic convergence condition, which would hold if the p-curvature is defined and vanishes. Using the algebraicity criteria established by Andr\'e, Bost, and Chambert-Loir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all p, then its monodromy is trivial. |

INFORMAL DYNAMICS AND GEOMETRY SEMINAR: | Kevin PilgrimINDIANA UNIVERSITY |
On the classification of critically fixed rational maps |

on Wednesday, April 01, 2015, at 4:00 - 5:00 PM in Science Center 507 | ||

I will discuss the classification of rational maps each of whose critical points is also a fixed-point. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Hong-Jian HeTSINGHUA UNIVERSITY |
Higgs Boson as the Origin of Mass and Inflation |

on Wednesday, April 01, 2015, at 4:00 PM in Science Center Hall A | ||

The discovery of a spin-0 Higgs boson (125GeV) at CERN Large Hadron Collider (LHC) is a milestone in fundamental science, and marks a historical turning point of particle physics. Why is it expected not to be the same Higgs boson as predicted by the Standard Model of particle physics? Does it provide the origin of masses for all elementary particles? Can it serve as the inflaton to drive cosmological inflation for hot big bang? How would the new physics associated with this Higgs boson be probed at high energy colliders and cosmic observations? In this colloquium, I will discuss some clues and implications of the above questions. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Sunil ChhitaUNIVERSITY OF BONN, ICERM |
The two-periodic Aztec diamond |

on Wednesday, April 01, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

Random domino tilings of the Aztec diamond shape exhibit interesting features and statistical properties related to random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. This model is considerably harder to analyze than the uniform model which only has two phases, liquid and solid. I will give an overview of some of the asymptotic results for the two-periodic Aztec diamond that have been obtained in joint work with Kurt Johansson |

THURSDAY SEMINAR: | Mike HopkinsHARVARD UNIVERSITY |
Motivic vector bundles |

on Thursday, April 02, 2015, at 3:30 - 5:30 pm in Science Center Hall E |

GAUGE THEORY, TOPOLOGY, AND SYMPLECTIC GEOMETRY SEMINAR: | Tara HolmCORNELL UNIVERSITY |
The Topology of Toric Origami Manifolds |

on Friday, April 03, 2015, at 3:30 - 4:30 PM in Science Center 507 | ||

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and much about their topological invariants can be read directly from the polytope. In this talk we examine the toric origami case: we will describe how toric origami manifolds can also be classified by their combinatorial moment data, and present some results about the topology of toric origami manifolds. This is joint work with Ana Rita Pires. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS PHYSICAL MATHEMATICS SEMINAR - TALK 2: | Anda DegeratuUNIVERSITY OF FREIBERG |
Analysis on QAC Spaces |

on Monday, April 06, 2015, at 1:00 - 2:00 PM in Science Center 232 | ||

In this talk I will introduce the class of quasi-asymptotically conical (QAC) manifolds, a less rigid Riemannian formulation of the QALE geometries introduced by Joyce in his study of crepant resolutions of Calabi-Yau orbifolds. Our set-up is in the category of real stratified spaces and Riemannian geometry. Given a QAC manifold, we identify the appropriate weighted Sobolev spaces, for which we prove the finite dimensionality of the null space for generalized Laplacian as well as their Fredholmness. The methods we use are based on techniques developed in geometric analysis by Grigor'yan and Saloff-Coste, as well as Colding and Minicozzi, and Peter Li. We show that our geometries satisfy the volume doubling property and the Poincaré inequality, and we use these properties to analyze the heat kernel behaviour of a generalized Laplacian and to establish Li-Yau type estimates for it. With these estimates we construct a parametrix for our operator and establish our Fredholm results. This work is joint with Rafe Mazzeo. |

BASIC NOTIONS SEMINAR: | Shing-Tung YauHARVARD UNIVERSITY |
Period integrals of algebraic manifolds and their differential equations (Joint works with Bong Lian and An Huang) |

on Monday, April 06, 2015, at 3:00 PM in Science Center 507 |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Yuchen PeiUNIVERSITY OF WARWICK |
A q-weighted version of the Robinson-Schensted algorithm |

on Wednesday, April 08, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

The Robinson-Schensted algorithm is a combinatorial algorithm which plays a fundamental role in the theory of Young tableaux and symmetric functions. It also provides a combinatorial framework for the study of certain particle systems including the totally asymmetric simple exclusion process (TASEP). In this talk I will describe a q-weighted version, which can be viewed as a randomised version of this algorithm, and which provides a similar framework for the study of the q-Whittaker functions and the q-TASEP particle system. I will also describe how a well known symmetry property of the classical algorithm carries over to the q-setting, via a branching version of the growth diagram technique introduced by Sergey Fomin. Part of this talk is based on joint work with Neil O'Connell. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Lionel LevineCORNELL UNIVERSITY |
Circles in the Sand |

on Wednesday, April 15, 2015, at 11:30 am - 1:00 pm in Science Center 530 | ||

I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Wei-Kuo ChengUNIVERSITY OF CHICAGO |
Self-averaging and universality in spin glasses |

on Wednesday, April 22, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. In this talk, we will consider several spin glass models and study quenched self-avergaging properties of some physical observables including the magnetization, the Hamiltonian and the cross overlap in the disorder chaos problem. Second, we will discuss some universal properties of the Gibbs measures and conclude in particular that the Parisi ultrametric structure of the Gibbs measure in the mixed p-spin model is universal under relevant moment matching assumptions. Based on the joint works with A. Auffinger and D. Panchenko. |

NUMBER THEORY SEMINAR: | Alexander MerkurjevUCLA |
Suslin's Conjecture on the reduced Whitehead group of a simple algebra |

on Wednesday, April 29, 2015, at 3:00 PM in Science Center 507 | ||

I will report on the proof of Suslin's Conjecture on the generic non-triviality of the reduced Whitehead group of a simple algebra. |