Calendar
- 14November 14, 2021No events
- 15November 15, 2021No events
- 16November 16, 2021
Holographic Algorithms
When are two mathematical functions the same? One might think that this can be generally answered immediately from their definitions. However, functions may have numerous dissimilar alternative definitions. Fortunately, sameness can be often demonstrated systematically by certain linear mappings internal to the function definitions. These mappings, called holographic transformations, offer a powerful tool for showing that a function class is equivalent to one known to be efficiently computable, or, alternatively, that it is equivalent to one known to be in a completeness class suspected to be computationally intractable. We shall survey these ideas and their applications in computational complexity.
https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09
CMSA Algebraic Geometry in String Theory Seminar: Gromov-Witten theory of complete intersections
I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).
https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09
**note change in time**
CMSA Combinatorics, Probability and Physics Seminar: A tale of two balloons
From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees.
Joint work with Omer Angel and Gourab Ray.https://harvard.zoom.us/j/99715031954?pwd=eVRvbERvUWtOWU9Vc3M2bjN3VndBQT09
Password: 1251442
*note unusual time*
Limits of Hodge structures via D-modules
1 Oxford Street, Cambridge, MA 02138 USAIt is well-known that each cohomology group of a compact K\”ahler manifold carries a Hodge structure. If we consider a degeneration of compact K\”ahler manifolds over a disk then it is natural to ask how the Hodge structures of smooth fibers degenerate. When the degeneration only allows a reduced singular fiber with simple normal crossings (i.e. semistable), Steenbrink constructed the limit of Hodge structure algebraically. A consequence of the existence of the limit of Hodge structure is the local invariant cycle theorem: the cohomology classes invariant under monodromy action come from the cohomology classes of the total space. In this talk, I will try to explain a method using D-modules to construct the limit of Hodge structure even when the degeneration is not semistable.
Webpage: https://sites.google.com/view/harvardmitag - 17November 17, 2021
CMSA Colloquium: Curve counting on surfaces and topological strings
Enumerative geometry is a venerable subfield of Mathematics, with roots dating back to Greek Antiquity and a present inextricably linked with developments in other domains. Since the early 90s, in particular, the interaction with String Theory has sent shockwaves through the subject, giving both unexpected new perspectives and a remarkably powerful, physics-motivated toolkit to tackle several traditionally hard questions in the field.
I will survey some recent developments in this vein for the case of enumerative invariants associated to a pair (X,D), with X a complex algebraic surface and D a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to (X,D), including the log Gromov–Witten invariants of the pair, the Gromov–Witten invariants of an associated higher dimensional Calabi–Yau variety, the open Gromov–Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar–Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.https://harvard.zoom.us/j/95767170359
(Password: cmsa)
Arithmetic volumes of unitary Shimura varieties
1 Oxford Street, Cambridge, MA 02138 USAThe integral model of a GU(n-1,1) Shimura variety carries a natural metrized line bundle of modular forms. Viewing this metrized line bundle as a class in the codimension one arithmetic Chow group, one can define its arithmetic volume as an iterated self-intersection. We will show that this volume can be expressed in terms of logarithmic derivatives of Dirichlet L-functions at integer points, and explain the connection with the arithmetic Siegel-Weil conjecture of Kudla-Rapoport. This is joint work with Jan Bruinier.
Arithmetic volumes of unitary Shimura varieties
1 Oxford Street, Cambridge, MA 02138 USAThe integral model of a GU(n-1,1) Shimura variety carries a natural metrized line bundle of modular forms. Viewing this metrized line bundle as a class in the codimension one arithmetic Chow group, one can define its arithmetic volume as an iterated self-intersection. We will show that this volume can be expressed in terms of logarithmic derivatives of Dirichlet L-functions at integer points, and explain the connection with the arithmetic Siegel-Weil conjecture of Kudla-Rapoport. This is joint work with Jan Bruinier.
Joint Harvard-CUHK-YMSC Differential Geometry Seminar
will speak on:
Quantum cohomology as a deformation of symplectic cohomology
Let X be a compact symplectic manifold, and D a normal crossings symplectic divisor in X. We give a criterion under which the quantum cohomology of X is the cohomology of a natural deformation of the symplectic cochain complex of X \ D. The criterion can be thought of in terms of the Kodaira dimension of X (which should be non-positive), and the log Kodaira dimension of X \ D (which should be non-negative). We will discuss applications to mirror symmetry. This is joint work with Strom Borman and Umut Varolgunes.
Zoom Link: https://cuhk.zoom.us/j/94377988344
Meeting ID: 943 7798 8344
Passcode: 20211117Tales of random projections: where probability meets geometry
1 Oxford Street, Cambridge, MA 02138 USAIn several areas of mathematics, including probability theory, asymptotic functional analysis, statistics and data science, one is interested in high-dimensional objects, such as measures, data or convex bodies. One common theme is to try to understand what lower-dimensional projections can say about the corresponding high-dimensional objects. I will describe several results that address this question, starting with classical results and moving on to more recent breakthroughs, my own research and some open questions. The talk will be self-contained and accessible to undergraduate students.
Website: https://people.math.harvard.edu/~ana/ons/
- 18November 18, 2021
CMSA Quantum Matter in Mathematics and Physics Seminar: Exact Eigenstates in Non-Integrable Systems: A violation of the ETH
*Note special time*
We find that several non-integrable systems exhibit some exact eigenstates that span the energy spectrum from lowest to the highest state. In the AKLT Hamiltonian and in several others “special” non-integrable models, we are able to obtain the analytic expression of states exactly and to compute their entanglement spectrum and entropy to show that they violate the eigenstate thermalization hypothesis. This represented the first example of ETH violation in a non-integrable system; these types of states have gained notoriety since then as quantum Scars in the context of Rydberg atoms experiments. We furthermore show that the structure of these states, in most models where they are found is that of an almost spectrum generating algebra which we call Restricted Spectrum Generating Algebra. This includes the (extended) Hubbard model, as well as some thin-torus limits of Fractional Quantum Hall states. Yet in other examples, such as the recently found chiral non-linear luttinger liquid, their structure is more complicated and not understood.
—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Wwhttps://harvard.zoom.us/j/977347126
Password: cmsaCMSA Quantum Matter in Mathematics and Physics Seminar: Exact Eigenstates in Non-Integrable Systems: A violation of the ETH
*Note special time*
We find that several non-integrable systems exhibit some exact eigenstates that span the energy spectrum from lowest to the highest state. In the AKLT Hamiltonian and in several others “special” non-integrable models, we are able to obtain the analytic expression of states exactly and to compute their entanglement spectrum and entropy to show that they violate the eigenstate thermalization hypothesis. This represented the first example of ETH violation in a non-integrable system; these types of states have gained notoriety since then as quantum Scars in the context of Rydberg atoms experiments. We furthermore show that the structure of these states, in most models where they are found is that of an almost spectrum generating algebra which we call Restricted Spectrum Generating Algebra. This includes the (extended) Hubbard model, as well as some thin-torus limits of Fractional Quantum Hall states. Yet in other examples, such as the recently found chiral non-linear luttinger liquid, their structure is more complicated and not understood.
—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists
https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Wwhttps://harvard.zoom.us/j/977347126
Password: cmsa - 19November 19, 2021
Harvard-MIT-MSR Combinatorics Seminar
1 Oxford Street, Cambridge, MA 02138 USAwill speak on:
Extended weak order in affine type
The extended weak order is a partial order associated to a Coxeter system (W,S). It is the containment order on “biclosed” sets of positive roots in the (real) root system associated to W. When W is finite, this order coincides with the (right) weak order on W, but when W is infinite, the weak order on W is a proper order ideal in the extended weak order. It is well-known that the weak order on W is a lattice if and only if W is finite. In contrast, it is a longstanding conjecture of Matthew Dyer that the extended weak order is a lattice for any W, which is open in the case that W is infinite. I will present joint work with David Speyer where we prove this conjecture for the affine Coxeter groups.
- 20November 20, 2021No events