Calendar

In two dimensions, one can study quantum field theories on unorientable manifolds by introducing crosscaps. This defines a class of states called crosscap states which share a few similarities with the notion of boundary states. In this talk, I will show that integrable theories remain integrable in the presence of crosscaps, and this allows to exactly determine the crosscap state.

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.

The notion of topological phases has dramatically changed our understanding of insulators. There is much to learn about a band insulator beyond the assertion that it has a gap separating the valence bands from the conduction bands. In the particular case of two dimensions, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. This topological twist is not just a mathematical observation, it has observable consequences—the transverse Hall conductivity is quantized and proportional to the 1st Chern number of the vector bundle of occupied states over the Brillouin zone. Finer properties of band insulators refer not just to the topology, but also to their geometry. Of particular interest is the momentum-space quantum metric and the Berry curvature. The latter is the curvature of a connection on the vector bundle of occupied states. The study of the geometry of band insulators can also be used to probe whether the material may host stable fractional topological phases. In particular, for a Chern band to have an algebra of projected density operators which is isomorphic to the W∞ algebra found by Girvin, MacDonald and Platzman—the GMP algebra—in the context of the fractional quantum Hall effect, certain geometric constrains, associated with the holomorphic character of the Bloch wave functions, are naturally found and they enforce a compatibility relation between the quantum metric and the Berry curvature of the band. The Brillouin zone is then endowed with a Kähler structure which, in this case, is also translation-invariant (flat). Motivated by the above, we will provide an overview of the geometry of Chern insulators from the perspective of Kähler geometry, introducing the notion of a  Kähler band which shares properties with the well-known ideal case of the lowest Landau level. Furthermore, we will provide a prescription, borrowing ideas from geometric quantization, to generate a flat Kähler band in some appropriate asymptotic limit. Such flat Kähler bands are potential candidates to host and realize fractional Chern insulating phases. Using geometric quantization arguments, we then provide a natural generalization of the theory to all even dimensions.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/new-technologies-in-mathematics-seminar-series/

The usual language of algebraic geometry is not appropriate for arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace s rings by the collection of “vectors” or by bi-operads, and another based on “matrices” or props. Once one understands the delicate commutativity condition one can proceed following Grothendieck’s footsteps exactly. The props, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.

This seminar will be held in Science Center 530 at 4:00pm on Wednesday, October 26th.

Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyond. They were present in Diophantus’ Arithmetica (3rd century AD) and, nowadays, they are more relevant than ever as a key ingredient in the algorithms that, for instance, secure Bitcoin transactions or encrypt WhatsApp messages. In this talk, we will introduce elliptic curves, explain their central role in mathematics, and discuss related open problems and applications.

Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction.  We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to.  This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.

For more information, please see: https://people.math.harvard.edu/~demarco/AlgebraicDynamics/

I will outline the construction of a 2-category associated to a hyperKahler moment map. The construction is based on partial differential equations in one, two and three dimensions, combined with a three-dimensional version of the Gaiotto–Moore–Witten web formalism.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/algebraic-geometry-in-string-theory/

We discuss recent joint work with Kyle Binder on defining the unipotent
fundamental group of tropical varieties. This fundamental group arises from the
Tannakian formalism using tropical vector bundles with integrable connection. By
employing the Orlik–Solomon theorem, we prove that this computes the unipotent
completion of the fundamental group of algebraic varieties with smooth
tropicalization.

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http://math.mit.edu/seminars/combin/

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