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.
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.

THURSDAY SEMINAR: Jacob Lurie
Harvard University
Etale motivic cohomolgy II
on Thursday, March 30, 2017, at 3:00 - 5:00 PM in Science Center 507

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.

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS MATHEMATICAL PHYSICS SEMINAR: Nathan Haouzi
UC Berkeley
Little Strings and Classification of surface defects
on Monday, April 03, 2017, at 12:00 pm in CMSA Building, 20 Garden St, G02
The so-called 6d (2,0) conformal field theory in six dimensions, labeled by an ADE Lie algebra, has become of great interest in recent years. Most notably, it gave new insights into lower dimensional supersymmetric field theories, for instance in four dimensions, after compactification. In this talk, I will talk about a deformation of this CFT, the six-dimensional (2,0) little string theory: its origin lies in type IIB string theory, compactified on an ADE singularity. We further compactify the 6d little string on a Riemann surface with punctures. The resulting defects are D-branes that wrap the 2-cycles of the singularity. This construction has many applications, and I will focus on one: I will provide the little string origin of the classification of surface defects of the 6d (2,0) CFT, for ADE Lie algebras. Furthermore, I will give the physical realization of the so-called Bala-Carter labels that classify nilpotent orbits of these Lie algebras.

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 SEMINAR: John Loftin
Rutgers University
Families of projective domains and neck-separating degenerations of convex $RP^2$ surfaces
on Thursday, April 06, 2017, at 3:00 - 4:30 pm in Science Center 232
Given a bounded convex domain $\Omega$ in $RP^n$, Cheng-Yau provide a unique solution to the Monge-Ampere equation $\det v_{ij} = (-1/v)^{n+2}$, for convex $v$ and Dirichlet boundary value $v=0$. This solution $v$ then determines a unique hypersurface, a hyperbolic affine sphere, asymptotic to the boundary of the cone over $\Omega$ in $R^{n+1}$. The hyperbolic affine sphere carries tensors, the Blaschke metric and cubic form, which are invariant under special linear automorphisms of the cone. These tensors descend to $\Omega$ to be projectively invariant. Benoist-Hulin show that if $\Omega_i\to\Omega$ in the Hausdorff topology, then these tensors converge in the local $C^\infty$ topology. A 2-dimensional hyperbolic affine sphere carries a conformal structure induced by the Blaschke metric, and the cubic tensor is equivalent to a holomorphic cubic differential. A quotient of such a domain $\Omega$ by a subgroup of $PGL(3,R)$ acting discretely and properly discontinuously then is called a real projective surface, and the conformal structure and cubic differential descend to the quotient. We will discuss a recent result relating degenerations of convex real projective surfaces along necks in terms of the geometry of the bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces. In addition to the Benoist-Hulin convergence results mentioned above, the proof also uses analytic techniques of Dumas-Wolf and Wolpert.

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.

DIFFERENTIAL GEOMETRY SEMINAR: Fangyang Zheng
Ohio State University
On compact Kahler manifolds with positive or negative holomorphic sectional curvature
on Tuesday, April 18, 2017, at 3:15 pm in CMSA Building, 20 Garden St, G10
In this talk, I will discuss some recent progress in the study of holomorphic sectional curvature H on compact Kahler manifolds. In the positively curved case, a goal is to understand what kind of manifolds can admit such a metric, and in the negatively curved case, there is the recent breakthrough by Wu and Yau which establishes the ampleness of the canonical line bundle for such manifolds. We will discuss an existence result for compact Kahler manifolds with positive H and a splitting result for projective Kahler manifolds with semi-negative H, in terms of the nef dimension.

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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