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Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.
We will explain the properties of quantum K-theory in comparison with quantum cohomology, and we will discuss the examples of projective space and the quintic hypersurface in P^4.

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

In the first part of the talk, I will establish the rationality of generating series formed from Euler characteristics of tautological bundles over Hilbert schemes of points on surfaces. In the second part, I will present results on virtual invariants of Quot schemes parameterizing rank zero quotients of trivial bundles on surfaces. The second part of the talk is based on work with Y. Kononov and work with D. Johnson, W. Lim, D. Oprea and R. Pandharipande.

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

TITLE: On the History of quantum cohomology and homological mirror symmetry

ABSTRACT: About 30 years ago, string theorists made remarkable discoveries of hidden structures in algebraic geometry.  First, the usual cup-product on the cohomology of a complex projective variety admits a canonical multi-parameter deformation to so-called quantum product, satisfying a nice system of differential equations (WDVV equations).  The second discovery, even more striking,  is Mirror Symmetry, a duality between families of Calabi-Yau varieties acting as a mirror reflection on the Hodge diamond.

Later it was realized that the quantum product belongs to the realm of symplectic geometry, and a half of mirror symmetry (called Homological Mirror Symmetry) is a duality between complex algebraic and symplectic varieties. The search of correct definitions and possible generalizations lead to great advances in many domains, giving mathematicians new glasses, through which they can see familiar objects in a completely new way.

I will review the history of major mathematical advances in the subject of HMS, and the swirl of ideas around it.

Talk chair: Paul Seidel

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

The hyperfinite II1 factor contains a wealth of subfactors, many of which give rise to new and fascinating mathematical structures. For instance, the standard representation of a subfactor generates a certain unitary tensor category that Jones described as (what he called) a “planar algebra.” It is a complete invariant for amenable, hyperfinite subfactors due to a deep result of Popa. However, generic subfactors are not amenable, and one typically does not know how to distinguish them. I will discuss a notion of “noncommutativity” for a subfactor that provides an invariant that is complementary to the planar algebra. Bare hand constructions of hyperfinite subfactors generally lead to “commutative” examples, and I will explain a theorem that allows us to produce “very noncommutative” ones as well. It involves actions of suitable groups on the hyperfinite II1 factor.

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

This describes joint work in progress with M. Mustata, in which we study the filtration on local cohomology sheaves induced by their natural mixed Hodge module structure. Special properties of this filtration, for instance strictness, lead to a number of different applications, including an injectivity theorem for dualizing complexes, local vanishing for forms with log poles, and especially a characterization of the local cohomological dimension of a closed subscheme in terms of  data arising from a log resolution of singularities.

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

The magnetoroton is the neutral excitation of a gapped fractional quantum Hall state. We argue that at zero momentum the magnetoroton has spin ±2, and show how the spin of the magnetoroton can be determined by polarized Raman scattering. We suggest that polarized Raman scattering may help to determine the nature of the ν=5/2 state. Ref: D.X. Nguyen and D.T. Son, arXiv:2101.02213.

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

The O’Connell-Yor polymer is a fundamental model of a polymer in a random environment. It corresponds to the positive temperature version of Brownian Last Passage percolation. Although much is known about this model thanks to remarkable algebraic structure uncovered by O’Connell, Yor and others, basic estimates for the behavior of the tails of the centered partition function for finite N that are available for zero temperature models are missing. I will present an iterative estimate to obtain strong concentration and localization bounds for the O’Connell-Yor polymer on an almost optimal scale N^{1/3+\epsilon}.

In the second part of the talk, I will introduce a system of interacting diffusions describing the successive increments of partition functions of different sizes. For this system, the N^{2/3} variance upper bound known for the OY polymer can be proved for a general class of interactions which are not expected to correspond to integrable models.

Joint work with Christian Noack and Benjamin Landon.

Zoom: https://harvard.zoom.us/j/99333938108?pwd=eklLTS9qaGVrWWx5elJWb2IrS284Zz09

Are proof assistants relevant to mathematics? One approach to this question is to explore the breadth of mathematical topics that can be formalised. The partition calculus was introduced by Erdös and R. Rado in 1956 as the study of “analogues and extensions of Ramsey’s theorem”. Highly technical results were obtained by Erdös-Milner, Specker and Larson (among many others) for the particular case of ordinal partition relations, which is concerned with countable ordinals and order types. Much of this material was formalised last year (with the assistance of Džamonja and Koutsoukou-Argyraki). Some highlights of this work will be presented along with general observations about the formalisation of mathematics, including ZFC, in simple type theory.

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

Since the discovery of high-temperature superconductors in cuprates in 1986, many theoretical ideas have been proposed which have enriched condensed matter theory. Especially, the resonating valence bond (RVB) state for (doped) spin liquids is one of the most fruitful idea. In this talk, I would like to describe the development of RVB idea to broader class of materials, especially more conventional magnets. It is related to the noncollinear spin structures with spin chirality and associated quantal Berry phase applied to many phenomena and spintronics applications. It includes the (quantum) anomalous Hall effect, spin Hall effect, topological insulator, multiferroics, various topological spin textures, e.g., skyrmions, and nonlinear optics. I will show that even though the phenomena are extensive, the basic idea is rather simple and common in all of these topics.

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

Many models in equilibrium statistical physics produce random fractal curves “at criticality.”  I will discuss one particular model, the loop-erased random walk, which is closely related to uniform spanning trees and Laplacian motion, and survey what is known today including some more recent results.  I will also discuss some of the important open problems and explain why the problem is hardest in exactly three dimensions. This talk is intended for a general mathematics audience and does not assume the audience knows the terms in the previous sentence.

Zoom: https://northeastern.zoom.us/j/95962897745?pwd=UFFPV2sxUitpWGFZbVErM1kwY284Zz09

For password email Andrew McGuinness