Department of Mathematics, Duke University
- Tuesday, April 25, 2017, 3:00pm, NCSU, SAS 1102, Triangle Topology Seminar
The Andrews-Curtis Conjecture and new handle decompositions of the 4-sphere
Alex Zupan (University of Nebraska-Lincoln)
- The Andrews-Curtis Conjecture, proposed in the 1960s, asserts that every balanced presentation of the trivial group can be simplified with a set of moves, called Andrew-Curtis moves. Every handle decomposition of the 4-sphere with no 3-handles induces such a presentation, with handle-slides corresponding to Andrews-Curtis moves. The most prominent examples in this setting are due to Gompf-Scharlemann-Thompson, building off work of Akbulut-Kirby. We describe a new construction that generalizes the work of Gompf-Scharlemann-Thompson, with intriguing connections to the Andrews-Curtis Conjecture. This is joint work in progress with Jeffrey Meier.
- Thursday, April 27, 2017, 3:00pm, 298 Physics (Physics Faculty Lounge), String Theory Seminar
Algebraic structures behind conformal blocks
- We will discuss a relation between conformal blocks, describing kinematics of a CFT, and integrable models of quantum-mechanical particles. I will show how the dependence of blocks on cross-ratios is encoded in equations of motion of a Calogero-Sutherland model and their dependence on conformal dimension and spin of the exchanged operator - in those of a relativistic Calogero-Sutherland model. Both are simultaneosly controlled by an integrable connection generalizing 2d Knizhnik-Zamolodchikov equations. The connection is associated to representations of degenerate double affine Hecke algebra, which suggests that a uniform treatment of conformal blocks with spin is possible via this approach.
- Thursday, April 27, 2017, 4:15pm, UNC, 125 Hanes Hall, Probability Seminar
A Dirichlet Form approach to MCMC Optimal Scaling
Wilfrid Kendall (U of Warwick)
- In this talk I will discuss the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form method has the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional diffusion.
Zanella, G., Kendall, W. S., & Bédard, M. (2016). A Dirichlet Form
approach to MCMC Optimal Scaling.
arXiv, 1606.01528, 22pp. URL: arxiv.org/abs/1606.01528
- Monday, May 1, 2017, 3:15pm, 119 Physics, Geometry Seminar
Normal Functions over Locally Symmetric Varieties
Matt Kerr (Washington U in St. Louis)
An algebraic cycle homologous to zero on a variety leads to an extension of Hodge-theoretic data. In a variational context, the resulting section of a bundle of complex tori is called a normal function, and is used to study cycles modulo rational or algebraic equivalence.
The archetype for interesting normal functions arises from the Ceresa cycle, consisting of the difference of two copies of a curve in its Jacobian. The profound geometric consequences of its existence are evidenced in work of Nori, Hain and (most recently) Totaro. In contrast, a theorem of Green and Voisin demonstrates the *absence* of normal functions arising from cycles on very general projective hypersurfaces of large enough degree.
Inspired by recent work of Friedman-Laza on Hermitian variation of Hodge structure and Oort's conjecture on special subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold, and where one expects interesting cycles (and generalized cycles) to occur. I will also give evidence that these predictions might be "sharp", and draw some geometric consequences.
- Tuesday, May 9, 2017, 3:15pm, 130 Physics, CTMS Adventures In Theory Lectures Seminar
- Friday, September 22, 2017, 3:15pm, 119 Physics, Algebraic Geometry Seminar
Rita Pardini (U Pisa)
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