CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS HOMOLOGICAL MIRROR SYMMETRY SEMINAR: | Philip EngelHarvard University |
Mirror symmetry in the complement of an anticanonical divisor |

on Thursday, October 27, 2016, at 2:00 - 4:00 pm in CMSA Building, 20 Garden St, G10 | ||

According to the SYZ conjecture, the mirror of a Calabi-Yau variety can be constructed by dualizing the fibers of a special Lagrangian fibration. Following Auroux, we consider this rubric for an open Calabi-Yau variety X-D given as the complement of a normal crossings anticanonical divisor D in X. In this talk, we first define the moduli space of special Lagrangian submanfiolds L with a flat U(1) connection in X-D, and note that it locally has the structure of a Calabi-Yau variety. The Fukaya category of such Lagrangians is obstructed, and the degree 0 part of the obstruction on L defines a holomorphic function on the mirror. This “superpotential" depends on counts of holomorphic discs of Maslov index 2 bounded by L. We then restrict to the surface case, where there are codimension 1 “walls” consisting of Lagrangians which bound a disc of Maslov index 0. We examine how the superpotential changes when crossing a wall and discuss how one ought to “quantum correct” the complex structure on the moduli space to undo the discontinuity introduced by these discs. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL SEMINAR: | Bong LianBrandeis University |
Tautological systems |

on Friday, October 28, 2016, at 12:45 pm in CMSA Building, 20 Garden St, G10 | ||

I will give an introduction to this class of D-modules whose construction is inspired by the B-model side of mirror symmetry. This theory has led to solutions to a number of old problems on period integrals and mirror symmetry. In this talk, I will review the construction and explain some of the basic properties of the D-modules. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Joseph MinahanUppsala University |
Supersymmetric gauge theories on $d$-dimensional spheres |

on Monday, October 31, 2016, at 12:00 pm in CMSA Building, 20 Garden St, G10 | ||

In this talk I discuss localizing super Yang-Mills theories on spheres in various dimensions. Our results can be continued to non-integer dimensions, at least perturbatively, and can thus be used to regulate UV divergences. I will also show how this can provide a way to localize theories with less supersymmetry. |

DIFFERENTIAL GEOMETRY SEMINAR: | Robert HaslhoferUniversity of Toronto |
The moduli space of 2-convex embedded spheres |

on Tuesday, November 01, 2016, at 3:00 - 4:00 PM *Special Time* in Science Center 507 | ||

We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture M_2 is contractible, but the topology of M_n for n\geq 3 is highly nontrivial. In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants. I will start by explaining the topological background and sketching a geometric analytic proof of Smale’s theorem that M_1 is contractible. In the second half of my talk, I’ll describe recent work on space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery. *Organized by Prof. S.T. Yau* |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Ramon van HandelPrinceton University |
Inhomogeneous random matrices |

on Wednesday, November 02, 2016, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

How do random matrices behave when the entries can have an arbitrary variance pattern? New problems arise in this setting that are almost completely orthogonal to classical random matrix theory. I will illustrate such problems by describing one of my favorite conjectures on this topic due to R. Latala, and the various mathematical techniques and questions that are emerging from its investigation. |

LOGIC COLLOQUIUM: | Theodore A. SlamanUC Berkeley |
Recursion Theory and Diophantine Approximation |

on Thursday, November 03, 2016, at 4:00 pm in 2 Arrow St, Rm 420 | ||

Recursion Theory deals with the definability of sets, especially sets of natural numbers or equivalently real numbers. Diophantine Approximation deals with the approximation of real numbers by rational numbers, which can be viewed as a number theoretic form of definability. We will discuss connections between these areas. |

DIFFERENTIAL GEOMETRY SEMINAR: | Nicolaos KapouleasBrown University |
Gluing constructions for minimal surfaes and other geometric objects |

on Tuesday, November 08, 2016, at 4:15 pm in Science Center 507 |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Norden E. HuangNational Central University |
On Holo-Hilbert Spectral Analysis |

on Wednesday, November 09, 2016, at 4:30 pm in CMSA Building, 20 Garden St, G10 | ||

Traditionally, spectral analysis is defined as transform the time domain data to frequency domain. It is achieved through integral transforms based on additive expansions of a priori determined basis, under linear and stationary assumptions. For nonlinear processes, the data can have both amplitude and frequency modulations generated by intra-wave and inter-wave interactions involving both additive and nonlinear multiplicative processes. Under such conditions, the additive expansion could not fully represent the physical processes resulting from multiplicative interactions. Unfortunately, all existing spectral analysis methods are based on additive expansions, based either on a priori or adaptive bases. While the adaptive Hilbert spectral analysis could accommodate the intra-wave nonlinearity, the inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase lock modulations are left untreated. To resolve the multiplicative processes, we propose a full informational spectral representation: The Holo-Hilbert Spectral Analysis (HHSA), which would accommodate all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions, through additional dimensions in the spectrum to account for both the variations in frequency and amplitude modulations (FM and AM) simultaneously. Applications to wave-turbulence interactions and other data will be presented to demonstrate the usefulness of this new spectral representation. |

DIFFERENTIAL GEOMETRY SEMINAR: | Sebastien PicardColumbia University |
Geometric Flows and Strominger systems |

on Tuesday, November 15, 2016, at 4:15 pm in Science Center 507 | ||

The anomaly flow is a geometric flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. I will discuss criteria for long time existence and convergence of the flow on toric fibrations with the Fu-Yau ansatz. This is joint work with D.H. Phong and X.W. Zhang. |

LOGIC COLLOQUIUM: | Haim HorowitzHewbrew University of Jerusalem |
On the non-existence and definability of mad families |

on Thursday, November 17, 2016, at 4:00 pm in 2 Arrow St, Rm 408 | ||

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+"there are no mad families" is actually equiconsistent with ZFC. I'll present the ideas behind the proof in the first part of the talk. In the second part of the talk, I'll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I'll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I'll show how large cardinals must be involved in such a solution. This is joint work with Saharon Shelah. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MEMBERS' SEMINAR: | Henri GuenanciaStony Brook University |
Singular varieties with trivial canonical bundle |

on Friday, November 18, 2016, at 5:00 pm in CMSA Building, 20 Garden St, G10 | ||

If X is a smooth projective variety (or compact Kähler manifold) with trivial first Chern class, then a famous result of Beauville and Bogomolov asserts that up to a finite étale cover, X is a product of varieties of three possible type: abelians varieties (or tori), Calabi-Yau varieties or Hyperkähler varieties. These last two classes are defined using properties of the algebra of global holomorphic forms. If X is singular though (say with torsion canonical bundle and klt singularities) this result is not known and presumably very difficult. In this talk, we will explain that if in addition X is assumed to be strongly stable (which is an infinitesimal version of irreducibility) then X falls into one of the singular analogues of the two categories above (Calabi-Yau and Hyperkähler varieties). This is ongoing joint work with Stefan Kebekus and Daniel Greb. |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | James LeeUniversity of Washington |
Conformal growth rates, spectral geometry, and distributional limits of graphs |

on Wednesday, November 30, 2016, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

Given a graph, one can deform its geometry according to a function that assigns non-negative weights to the vertices. We refer to this as a "conformal" deformation of the graph metric. For a finite graph, it makes sense to define the area of such a weight as the average of the squared weights of the vertices. One can similarly define the area of a conformal weight for a unimodular random graph. The conformal growth exponent is the smallest rate of volume growth of balls achievable by a conformal weight of unit area. We show that if a unimodular (rooted) random graph (G,x) has quadratic conformal growth (QCG) and the law of deg(x) is sufficiently well-behaved, then the random walk on G is almost surely recurrent. We also argue that our joint with Kelner, Price, and Teng (2011) can be used to show that every distributional limit of finite planar graphs has QCG. More generally, this holds for H-minor-free graphs, and other interesting families like string graphs (the intersection graph of continuous arcs in the plane). These methods do not rely on circle packings, and can instead be thought of as directly uniformizing the underlying graph metric. They yield a short proof of Benjamini and Schramm's result that a distributional limit of finite, bounded-degree planar graphs is almost surely recurrent, and provide a positive answer to their conjecture that the same should hold for H-minor-free graphs. Gurel-Gurevich and Nachmias recently solved a central open problem by showing that the uniform infinite planar triangulation (UIPT) and quadrangulation (UIPQ) are almost surely recurrent. By combining QCG with methods from spectral geometry, we present a new proof of this fact that follows from strong quantitative bounds on the heat kernel. A similar phenomenon holds beyond dimension two: Assuming the law of deg(x) has tails that decay faster than any inverse polynomial, the almost sure spectral dimension of a unimodular random graph (G,x) is equal to its conformal growth exponent. This has consequences for limits of graphs that can be sphere-packed in R^d for d > 2. |