JOINT DPT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Chiranjib MukherjeeCourant Institute |
Compactness and Large Deviations |

on Friday, March 24, 2017, at 2:00 in Science Center Room 232 | ||

In a reasonable topological space, large deviation estimates essentially deal with probabilities of events that are asymptotically (exponentially) small, and in a certain sense, quantify the rate of these decaying probabilities. In such estimates, upper bounds for such small probabilities often require compactness of the ambient space, which is often absent in problems arising in statistical mechanics (for example, distributions of local times in Brownian motion in the full space Rd). Motivated by such a problem, we present a robust theory of “translation-invariant compactification” of probability measures in Rd. Thanks to an inherent shift-invariance of the underlying problem, we are able to apply this abstract theory painlessly and solve a long standing problem in statistical mechanics, the mean-field polaron problem. This talk is based on joint works with S. R. S. Varadhan (New York), as well as with Erwin Bolthausen (Zurich) and Wolfgang Koenig (Berlin). |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: | Agnese BissiHarvard University |
Loops in AdS from conformal symmetry |

on Monday, March 27, 2017, at 12:00 pm in CMSA Building, 20 Garden St, G02 | ||

In this talk I will discuss a new use for conformal field theory crossing equation in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. I will revisit this problem and the dual 1/N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1/N^2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. The second approach involves Mellin space. As an example, I’ll show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of \phi^4 theory in AdS, the crossing solution reproduces a previous computation of the one-loop bubble diagram. I will end with a discussion on open problems and new developments. |

MATHEMATICAL PHYSICS SEMINAR: | Hannes PichlerHarvard University |
Photonic tensor networks produced by a single quantum emitter |

on Tuesday, March 28, 2017, at 2:45 pm in Jefferson 453 | ||

We discuss a protocol to generate two dimensional tensor network states using a single quantum system that sequentially interacts with a 1D string of qubits. This is accomplished by using parts of the string itself as a quantum queue memory. As a simple physical implementation, we consider a single atom or atom like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits while the mirror allows the implementation of the queue memory. We show that this allows for a realization of a universal quantum computer using a single atom coupled to light. To this end we first review the basic concepts of measurement based quantum computation, and then discuss an explicit protocol to deterministically create a 2D cluster state in a quantum nanophotonic experiment. We then classify the many-body quantum states that can be produced in this approach in terms of tensor network states. |

DIFFERENTIAL GEOMETRY SEMINAR: | Jordan KellerColumbia University |
Linear stability of Schwarzschild spacetime |

on Tuesday, March 28, 2017, at 3:00 - 4:00 PM in CMSA Building, 20 Garden St, G10 | ||

I will discuss recent work, joint with Pei-Ken Hung and Mu-Tao Wang, on the linear stability of the Schwarzschild spacetime. Our method employs Hodge decomposition to split linearized solutions into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For each portion, we derive Regge-Wheeler type equations for decoupled, gauge-invariant quantities at the linearized connection level. With the choice of an appropriate gauge, decay estimates on these decoupled quantities are used to establish decay of the linearized metric coefficients of the solution. |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | David StapletonStony Brook University |
Hilbert schemes of points on surfaces and their tautological bundles |

on Tuesday, March 28, 2017, at 3:00 pm in Science Center 507 | ||

Fogarty showed in the 1970s that the Hilbert scheme of n points on a smooth surface is itself smooth. Interest in these Hilbert schemes has grown since it has been shown they arise in hyperkahler geometry, geometric representation theory, and algebraic combinatorics. In this talk we will explore the geometry of certain tautological bundles on the Hilbert scheme of points. In particular we will show that these tautological bundles are (almost always) stable vector bundles. We will also show that each sufficiently positive vector bundles on a curve C is the pull back of a tautological bundle from an embedding of C into the Hilbert scheme of the projective plane. |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Nina HoldenMIT |
Percolation-decorated triangulations and their relation with SLE and LQG |

on Wednesday, March 29, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G02 | ||

The Schramm-Loewner evolution (SLE) is a family of random fractal curves, which is the proven or conjectured scaling limit of a variety of two-dimensional lattice models in statistical mechanics, e.g. percolation. Liouville quantum gravity (LQG) is a model for a random surface which is the proven or conjectured scaling limit of discrete surfaces known as random planar maps (RPM). We prove that a percolation-decorated RPM converges in law to SLE-decorated LQG in a certain topology. This is joint work with Bernardi and Sun. We then discuss a work in progress where we try to strengthen the topology of convergence of a RPM to LQG by considering conformal embeddings of the RPM into the complex plane. This is joint work with Sun and with Gwynne, Miller, Sheffield, and Sun. |

NUMBER THEORY SEMINAR: | Martin OlssonUC Berkeley |
Local fundamental groups and reduction mod $p$ |

on Wednesday, March 29, 2017, at 3:00 pm in Science Center 507 | ||

I will discuss various finiteness results for local fundamental groups, and how positive characteristic techniques can be used to deduce results in characteristic $0$. In particular, I will explain how to deduce a result of Xu on finiteness of local fundamental groups for klt singularities using positive characteristic methods. This is joint work with Bhargav Bhatt and Ofer Gabber. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Leslie GreengardCourant Institute |
Inverse problems in acoustic scattering and cryo-electron microscopy |

on Wednesday, March 29, 2017, at 4:00 pm in CMSA Building, 20 Garden St, G10 | ||

A variety of problems in image reconstruction give rise to large-scale, nonlinear and non-convex optimization problems. We will show how recursive linearization combined with suitable fast solvers are bringing such problems within practical reach, with an emphasis on acoustic scattering and protein structure determination via cryo-electron microscopy. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL SEMINAR: | Rak-Kyeong SeongUppsala University |
The Mirror and the Elliptic Genus of Brane Brick Models |

on Thursday, March 30, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G02 | ||

I will present recent progress in improving with the help of mirror symmetry our understanding of Type IIA brane configurations that encode 2d (0,2) gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We call these brane configurations brane brick models. The mirror of brane brick models consists of D5-branes wrapping 4-spheres whose intersections determine the corresponding 2d gauge theories. I will show how in the mirror picture 2d (0,2) phenomena such as Gadde-Gukov-Putrov triality have a natural description in terms of geometric transitions. In this context, I will also illustrate the computation of the elliptic genus for brane brick models and match it with the elliptic genus of the corresponding non-linear sigma model whose target space is the probed Calabi-Yau 4-fold. |

BRANDEIS, HARVARD, MIT, NORTHEASTERN JOINT MATHEMATICS COLLOQUIUM AT HARVARD: | Alexander GoncharovYale University |
Quantum Hodge Field Theory |

on Thursday, March 30, 2017, at 4:30 pm, Tea at 4 pm in the Math Common Room in Science Center Hall A | ||

We introduce quantum Hodge correlators. They have the following format. Take a family X → B of compact Kahler manifolds. Let S be an oriented topological surface with special points on the boundary, considered modulo isotopy. We assign to each interval between special points an irreducible local system Li on X , and to each special point an Ext between the neighboring local systems. A quantum Hodge correlator is assigned to this data and lives on the base B. It is a sum of finite dimensional convergent Feynman type integrals. The simplest Hodge correlators are given by the Rankin-Selberg integrals for L-functions. Quantum Hodge correlators can be perceived as Hodge-theoretic analogs of the invariants of knots and threefolds provided by the perturbative Chern-Simons theory. Here is an example. Hodge theory suggests to view a Riemann surface Σ as a threefold, and a point x on Σ as a knot in the threefold. Then the Green function G(x, y) on Σ - the basic Hodge correlator, is an analog of the linking number - the simplest Chern-Simons type invariant. What do the quantum Hodge correlators do? Let B be a point, and Li are constant sheaves. 1. Hodge correlators (S is a disc) describe an action of the Hodge Galois group by A∞- automorphisms of the cohomology algebra H∗ (X , C) preserving the Poincare pairing. 2. Quantum Hodge correlators (S is any surface) describe an action of the Hodge Galois group by quantum A∞-automorphisms of the algebra H∗ (X , C) with the Poincare pairing. |

DIFFERENTIAL GEOMETRY SEMINAR: | Chiu-Chu Melissa LiuColumbia University |
GW theory, FJRW theory, and MSP fields |

on Tuesday, April 04, 2017, at 2:45 pm in CMSA Building, 20 Garden St, G10 | ||

Gromov-Witten (GW) invariants of the quintic Calabi-Yau 3-fold are virtual counts of parametrized holomorphic curves to the quintic 3-fold. Fan-Jarvis-Ruan-Witten (FJRW) invariants of the Fermat quintic polynomial are virtual counts of solutions to the Witten equation associated to the Fermat quintic polynomial. In this talk, I will describe the theory of Mixed-Spin-P (MSP) fields interpolating GW theory of the quintic 3-fold and FJRW theory of the Fermat quintic polynomial, based on joint work with Huai-Liang Chang, Jun Li, and Wei-Ping Li. |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Steven HellmanUCLA |
Noncommutative Majorization Principles and Grothendieck's Inequality |

on Wednesday, April 05, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

The seminal invariance principle of Mossel-O'Donnell-Oleszkiewicz implies the following. Suppose we have a multilinear polynomial Q, all of whose partial derivatives are small. Then the distribution of Q on i.i.d. uniform {-1,1} inputs is close to the distribution of Q on i.i.d. standard Gaussian inputs. The case that Q is a linear function recovers the Berry-Esseen Central Limit Theorem. In this way, the invariance principle is a nonlinear version of the Central Limit Theorem. We prove the following version of one of the two inequalities of the invariance principle, which we call a majorization principle. Suppose we have a multilinear polynomial Q with matrix coefficients, all of whose partial derivatives are small. Then, for any even K>1, the Kth moment of Q on i.i.d. uniform {-1,1} inputs is larger than the Kth moment of Q on (carefully chosen) random matrix inputs, minus a small number. The exact statement must be phrased carefully in order to avoid being false. Time permitting, we discuss applications of this result to anti-concentration, and to computational hardness for the noncommutative Grothendieck inequality. (joint with Thomas Vidick) https://arxiv.org/abs/1603.05620 |

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: | Subhajit GoswamiUniversity of Chicago |
Liouville first-passage percolation and Watabiki's prediction |

on Wednesday, April 12, 2017, at 3:00 pm in CMSA Building, 20 Garden St, G10 | ||

In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model of random metric on a finite planar grid graph. It was studied primarily as a way to make sense of the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will also discuss about the apparent disagreement of these estimates with a prediction made in the physics literature about LQG metric. The talk is based on a joint work with Jian Ding. |

SPECIAL BASIC NOTIONS SEMINAR: | Jean-Pierre SerreCollège de France |
Some simple facts on lattices and orthogonal group representations |

on Wednesday, May 03, 2017, at 3:00 pm in Science Center Hall D | ||

Afternoon tea will follow at 4:15 pm in the Math Department Common Room, 4th floor. |

SPECIAL LECTURE SERIES: | Jean-Pierre SerreCollège de France |
Cohomological invariants mod 2 of Weyl groups, Pt. 1 |

on Monday, May 08, 2017, at 3:00 - 4:00 PM in Science Center 507 | ||

The first lecture will mostly be a résumé of the first part of AMS ULECT 28; the second lecture will give an explicit description of the cohomological invariants of the Weyl groups. *Afternoon tea will follow the talks at 4:15 pm in the Math Department Common Room, 4th Floor. |

SPECIAL LECTURE SERIES: | Jean-Pierre SerreCollège de France |
Cohomological invariants mod 2 of Weyl groups, Pt. 2 |

on Tuesday, May 09, 2017, at 3:00 - 4:00 PM in Science Center 507 | ||

The first lecture will mostly be a résumé of the first part of AMS ULECT 28; the second lecture will give an explicit description of the cohomological invariants of the Weyl groups. *Afternoon tea will follow the talks at 4:15 pm in the Math Department Common Room, 4th Floor. |