Calendar

The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

Title: Topological Recursion and Enumerative Geometry

Abstract: Given a holomorphic curve in the complex 2-plane together with a suitably normalized symmetric meromorphic bilinear differential, the Chekhov-Eynard-Orantin Topological Recursion defines an infinite sequence of symmetric meromorphic multilinear differentials W_{g,n} on the curve. In many examples, the invariants W_{g,n} provide answers to enumerative problems. I will describe Topological Recursion and present several examples in which the answers are Hodge integrals (which are intersection numbers on moduli of curves) or Gromov-Witten invariants (which are virtual counts of holomorphic maps from Riemann surfaces to a Kahler manifold).

Registration is required to receive the Zoom information

Register here to attend

One of the most fascinating aspects about non-commutative spaces (aka von Neumann algebras), is the way their building data, which is often geometric in nature, impacts on the properties of their quantized symmetries. This is particularly the case for II₁ factors, where symmetries are encoded by their subfactors of finite Jones index. I will discuss some results and open problems that illustrate the unique interplay between analysis and algebra/combinatorics entailed by this interdependence, that’s specific to subfactor theory.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Estimating the mean of a distribution from i.i.d samples is a fundamental statistical task. In this talk, we will focus on the high-dimensional setting where we will design estimators achieving optimal recovery guarantees in terms of all relevant parameters. While optimal one-dimensional estimators have been known since the 80s (Nemirovskii and Yudin ’83), optimal estimators in high dimensions have only been discovered recently beginning with the seminal work of Lugosi and Mendelson in 2017 and subsequent work has led to computationally efficient variants of these estimators (Hopkins 2018). We will discuss statistical and computational extensions of these results by developing optimal estimators for settings where the data distribution only obeys a finite fractional moment condition as opposed to the existence of a second moment as assumed previously.

Joint work with Peter Bartlett, Nicolas Flammarion, Michael I. Jordan and Nilesh Tripuraneni.

The talk will be based on the following papers: https://arxiv.org/abs/2011.12433, https://arxiv.org/abs/1902.01998.

Zoom: https://harvard.zoom.us/j/98231541450

We will report on a recent joint work with Junyan Cao, cf. arXiv:2012.05063. The main topics we will discuss are revolving around the extension of pluricanonical forms defined on the central fiber of a family of Kaehler manifolds. For our results to hold we need the divisor of zeros of the said forms to be sufficiently “nice”, in a sense that will become clear during the talk.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

For convex complete intersections, the Gromov-Witten (GW) invariants are often computed using the Quantum Lefshetz Hyperplane theorem, which relates the invariants to those of the ambient space. However, even in the genus 0 theory, the convexity condition often fails when the target is an orbifold, and so Quantum Lefshetz is no longer guaranteed. In this talk, I will showcase a method to compute these invariants, despite the failure of Quantum Lefshetz, for a class of orbifold complete intersections. This talk will be based on joint work with Felix Janda (Notre Dame) and Yang Zhou (Harvard) and upcoming work with Rachel Webb (Berkeley).

Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

Fracton phases show exotic properties, such as sub-extensive entropy, local particle-like excitation with restricted mobility, and so on. In order to find natural fermionic fracton phases, we explore supersymmetric quantum field theory with exotic symmetry. We use superfield formalism and write down the action of a supersymmetric version of one of the simplest models with exotic symmetry, the φ theory in 3+1 dimensions. It contains a large number of ground states due to the fermionic higher pole subsystem symmetry. Its residual entropy is proportional to the area instead of the volume. This theory has a self-duality similar to that of the φ theory. We also write down the action of a supersymmetric version of a tensor gauge theory, and discuss BPS fractons.

Zoom: https://harvard.zoom.us/j/977347126

I’ll explain a very simple proof of the fact that K3 surfaces of finite height admit (many) CM lifts, a result due independently to Ito-Ito-Koshikawa and Z. Yang, which was used by the former to prove the Tate conjecture for products of K3s. This will be done directly showing that the deformation ring of a polarized K3 surface of finite height admits as a quotient that of its Brauer group. The method applies more generally to many isogeny classes of points on Shimura varieties of abelian type.

Zoom: https://harvard.zoom.us/j/99334398740

Password: The order of the permutation group on 9 elements.

In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Some of the conjectures are in retrospect simple to prove, whereas others remain so far unproven. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

Quantum systems out of equilibrium can exhibit different dynamical phases that are fundamentally characterized by their entanglement dynamics and entanglement scaling. Random quantum circuits with non-unitarity induced by measurement or other sources provide a large class of systems for us to investigate the nature of these different entanglement phases and associated criticality. While numerical studies have provided a lot of insight into the behavior of such quantum circuit models, analytical understanding of the entanglement criticality in these models has remained challenging in many cases. In this talk, I will focus on the random non-unitary fermionic Gaussian circuits, namely non-unitary circuits for non-interacting fermions. I will first present a numerical study of an entanglement critical phase in this type of circuit. Then, I will discuss the analytical understanding of general entanglement phases in this type of circuit via a general correspondence among (1) non-unitary fermionic Gaussian circuits, (2) fermionic Gaussian tensor network, and (3) unitary non-interacting fermions subject to quenched disorder. In particular, we show that the critical entanglement phase numerically found in the non-unitary Gaussian circuit without any symmetry can be described by the theory of (unitary) disordered metal in the symmetry class DIII. I will comment on the entanglement critical phases that correspond to unitary disordered fermion critical points or unitary disordered metals in other symmetry classes.

Subscribe to Harvard CMSA seminar videos (more to be uploaded):

https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww

https://www.youtube.com/channel/UCBmPO-OK1sa8T1oX_9aVhAg/playlists (all in playlist)

Zoom: https://harvard.zoom.us/j/977347126