Calendar

For a moduli space $M$ of stable sheaves over a K3 surface $X$, we propose a series of conjectural identities in the Chow rings $CH_\star (M \times X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring $R_\star (M) \subset CH_\star (M).$ We prove the proposed identities when $M$ is the Hilbert scheme of points on a K3 surface. This is based on joint work with L. Flapan, A. Marian and R. Silversmith.

I will construct LG mirrors for the Johnson-Kollár series of anticanonical del Pezzo surfaces in weighted projective 3-spaces. The main feature of these surfaces is that their anticanonical linear system is empty. Thus they fall outside of the range of the known mirror constructions. For each of these surfaces, the LG mirror is a pencil of hyperelliptic curves. I will exhibit the regularised I-function of the surface as a period of the pencil and I will sketch how to construct the pencil starting from a work of Beukers, Cohen, and Mellit on finite hypergeometric functions. This is joint work with Alessio Corti.

**CANCELED**

Most 4-manifolds do not admit symplectic forms, but most admit 2-forms that are “nearly” symplectic. Just like the Seiberg-Witten (SW) invariants, there are Gromov invariants that are compatible with the near-symplectic form. Although (potentially exotic) 4-spheres don’t admit them, there is still a way to bring in near-symplectic techniques and I will describe my ongoing pseudo-holomorphic attempt(s) at analyzing them.

We provide a new parton theory for hole doped cuprates. We will describe both a pseudogap metal with small Fermi surfaces and the conventional Fermi liquid with large Fermi surfaces within mean field level of the same framework.  For the pseudogap metal,  “Fermi arc” observed in ARPES can be naturally reproduced.  We also provide a theory for a critical point across which the carrier density jumps from x to 1+x.   We will also discuss the generalization of the theory to Kondo breaking down transition in heavy fermion systems and  generic SU(N) Hubbard model.

There is a large literature about points of bounded height on varieties, and about number fields of bounded discriminant. We explain how to unify these two questions by means of a new definition of height for rational points on (certain) stacks over global fields. I talked about some aspects of this work at Barry’s birthday conference, and will try in this talk to emphasize different points, including a conjecture about the asymptotic counting function for points of bounded height on a stack X which simultaneously generalizes the Manin conjectures (the case where X is a variety) and the Malle conjectures (the case where X is a classifying stack BG.)

I’ll discuss a recent connection between two seemingly unrelated
problems: how to measure a collection of quantum states without
damaging them too much (“gentle measurement”), and how to provide
statistical data without leaking too much about individuals
(“differential privacy,” an area of classical CS). This connection
leads, among other things, to a new protocol for “shadow tomography”
of quantum states (that is, answering a large number of questions
about a quantum state given few copies of it).

A ray in a graph G = (V, E) is a sequence X (possibly infinite) of distinct vertices x0, x1, . . . such that, for every i, E(xi , xi+1). A classical theorem of graph theory (Halin [1965]) states that if a graph has, for each k 2 N, a set of k many disjoint (say no vertices in common) infinite rays then there is an infinite set of disjoint infinite rays.

The proof seems like an elementary argument by induction that uses the finite version of Menger’s theorem at each step. One would thus expect the theorem to follow by very elementary (even computable) methods plus a compactness argument (or equivalently arithmetic comprhension, ACA0). We show that this is not the case.

Indeed, the construction of the infinite set of disjoint rays is much more complicated. It occupies a level of complexity previously inhabited by a number of logical principals and only one fact from the mathematical literature. Such theorems are called theorems of hyperarithmetic analysis. Formally this means that they imply (in !-models) that for every set A all transfinite iterations (through well-orderings computable from A) of the Turing jump beginning with A exist. On the other hand, they are true in the (!-model) consisting of the subsets of N generated from any single set A by these jump iterations.

There are many variations of this theorem in the graph theory literature that inhabit the subject of ubiquity in graph theory. We discuss a number of them that also supply examples of theorems of hyperarithmetic analysis as well as classical variations that are proof theoretically even stronger.

This work is joint with James Barnes and Jun Le Goh.

If time permits we will also discuss a new class of theorems suggested by a “lemma” in one of the papers in the area. We call them almost theorems of hyperarithmetic analyisis. These are theorems that are proof theoretically very weak over Recursive Comprehension (RCA0) but become theorems of hyperarithmetic analysis once one assumes ACA0.

Imagine the universe is a periodic crystal. If gravity makes space negatively curved, the thin walls of the crystalline structure might trace out a pattern of circles in the sky, visible at night. In this talk we will describe how to generate pictures of these patterns and how to think like a hyperbolic astronomer. We also touch on the connection to knots and links and arithmetic groups. The lecture is accompanied by an exhibit of prints in the Science Center lobby. (This talk will be accessible to members of the department at all levels.)

No additional detail for this event.

I will give an introduction to the topic of this semester’s seminar: the automorphisms of E_n-operads. Our main goal is to understand the computation of Fresse, Turchin and Willwacher of the rational homotopy of Map^h(E_m,E_n^Q) in terms of graph homology. We then discuss some potential applications, and lines of inquiry opened up by this result. This connects the topic to differential topology and number theory.

In this talk, we will introduce the concept of improvabilty of the dominant energy scalar and discuss strong consequences of non-improvability. We employ new, large families of deformations of the modified Einstein constraint operator and show that, generically, their adjoint linearizations are either injective, or else one can prove that kernel elements satisfy a “null-vector equation”. Combined with a conformal argument, we make significant progress toward Bartnik’s stationary conjecture. More specifically, we prove that a Bartnik minimizing initial data set can be developed into a spacetime that both satisfies the dominant energy condition and carries a global Killing field. We also show that this spacetime is vacuum near spatial infinity. This talk is based on the joint work with Dan Lee.

I’ll discuss the Viterbo transfer functor from the (partially) wrapped Fukaya category of a Liouville domain to that of a subdomain. It is a localization when everything in sight is Weinstein, and I’ll explain how much of that survives if we drop the assumption that the cobordism is Weinstein. The result allows us to turn natural questions about exact Lagrangians into interesting questions in homotopical algebra.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar