Department of Mathematics FAS Harvard University One Oxford Street Cambridge MA 02138 USA Tel: (617) 495-2171 Fax: (617) 495-5132
To post a seminar which takes place at the Mathematics department, please email seminars@math.harvard.edu with date, time, room, title and possibly with an abstract.
MATHEMATICAL PHYSICS SEMINAR: Hannes Pichler
Harvard 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 Keller
Columbia 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 Stapleton
Stony 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 Holden
MIT
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 Olsson
UC 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 Greengard
Courant 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 Seong
Uppsala 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 Goncharov
Yale 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 Liu
Columbia 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 Hellman
UCLA
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

NUMBER THEORY SEMINAR: Frank Calegari
University of Chicago
Modularity lifting theorems beyond Shimura varieties
on Wednesday, April 05, 2017, at 3:00 pm in Science Center 507
Recent work of Caraiani-Scholze has opened the possibility of proving modularity lifting theorems for GL(n) with n>2. We discuss joint work in progress in this direction with Allen, Caraiani, Gee, Helm, LeHung, Newton, Scholze, Taylor, and Thorne, and give some applications of these results.

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL LECTURE SERIES ON DONALDSON-THOMAS AND GROMOV-WITTEN THEORIES: Artan Sheshmani
Aarhus University/CMSA
Stable pair PT invariants on smooth fibrations
on Wednesday, April 05, 2017, at 9:00 - 10:30 am in CMSA Building, 20 Garden St, G10
We study Pandharipande-Thomas’s stable pair theory on smooth K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka’s formula for the Euler characteristics of moduli spaces of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration.

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL LECTURE SERIES ON DONALDSON-THOMAS AND GROMOV-WITTEN THEORIES: Artan Sheshmani
Aarhus University/CMSA
Stable pair PT invariants on nodal fibrations: perverse sheaves, Wallcrossings, and an analog of fiberwise T-duality
on Friday, April 07, 2017, at 9:00 - 10:30 am in CMSA Building, 20 Garden St, G10
Following lecture 4, we continue the study of stable pair invariants of K3-fibered threefolds., We investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the K3-fibration. In the case that the fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.

JOINT DEPARTMENT OF MATHEMATICS AND CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX AND PROBABILITY THEORY SEMINAR: Subhajit Goswami
University 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.

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL LECTURE SERIES ON DONALDSON-THOMAS AND GROMOV-WITTEN THEORIES: Artan Sheshmani
Aarhus University/CMSA
DT versus MNOP invariants and S_duality conjecture on general complete intersections
on Wednesday, April 12, 2017, at 9:00 - 10:30 am in CMSA Building, 20 Garden St, G10
Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs Z⊂H in a Calabi–Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of Z⊂H, pushed forward to X and considered as a certain Joyce–Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS SPECIAL LECTURE SERIES ON DONALDSON-THOMAS AND GROMOV-WITTEN THEORIES: Artan Sheshmani
Aarhus University/CMSA
Proof of S-duality conjecture on quintic threefold I
on Friday, April 14, 2017, at 9:00 - 10:30 am in CMSA Building, 20 Garden St, G10
I will talk about an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. We use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture.

SPECIAL BASIC NOTIONS SEMINAR: Jean-Pierre Serre
Collège 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 Serre
Collège 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 Serre
Collège 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.

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