The 2019-2020 CMSA Members’ Seminar will occur every Friday at 5pm in CMSA G10. The Schedule will be updated below. Previous seminars can be found here.
|9/6/2019||Spiro Karigiannis||Title: Constructions of compact torsion-free $G_2$-manifolds
Abstract: Compact torsion-free $G_2$-manifolds are 7-dimensional analogues of Calabi-Yau threefolds, being compact Ricci-flat Riemannian manifolds with reduced holonomy that are important ingredients in theories of physics. All known constructions use an abstract existence theorem of Dominic Joyce to perturb “almost” solutions of a quasilinear elliptic PDE to honest solutions, and construct the “almost” solutions via glueing methods. I will first summarize some basic facts about $G_2$-manifolds and Joyce’s existence theorem, and then briefly mention the previous constructions by Joyce (1994), Kovalev (2003), and Corti-Haskins-Nordstrom-Pacini (2014). Then I will focus on a new construction (joint work of myself and Joyce, to appear in JDG) that is significantly more involved for several reasons, which I will elucidate. In particular one key step in our construction involves solving a linear first order elliptic PDE on a noncompact 4-manifold with prescribed asymptotics at infinity. (arXiv: 1707.09325)
|9/13/2019||Wei Gu||Title: Sigma models and mirror symmetry
Abstract: In this talk, I will roughly review why physicists be interested in Calabi-Yau manfiolds, and will introduce some tools we used to probe Calabi-Yaus and other spaces like Fanos which we called sigma models. I will also briefly mention how physicists using sigma models to study mirror symmetry. This is not a technical talks, rather, I will just focus on the pictures of the connections between math and physics from sigma models.
|9/20/2019||Ryohei Kobayashi||Title: Fermionic phases of matter on unoriented spacetime
Abstract: We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin− invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the Z8 classification of (1+1)d topological superconductors. We also compute the indicator formula of Z16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.
|9/27/2019||Yun Shi||Title: On motivic Donaldson-Thomas theory on local P2
Abstract: Donaldson-Thomas (DT) theory is an enumerative theory which counts ideal sheaves of curves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory is a categorification of the DT theory. This categofication contains more refined information of the moduli space, just like the topological space or cohomology contains more information than an Euler characteristic. In this talk, I will give a brief introduction to motivic DT theory. I will also discuss some results on this theory for moduli spaces of sheaves on the local projective plane.
|10/4/2019||Yoosik Kim||Title: Towards SYZ mirror symmetry of flag varieties.
Abstract: SYZ mirror symmetry has provided a geometric way of understanding mirror symmetry via T-duality. In this talk, I will discuss how to obtain SYZ mirrors of partial flag varieties using Floer theory.
|10/11/2019||Rongxiao Mi||Title: On the change of Gromov-Witten theory under extremal transitions.
Abstract: Extremal transitions are a topological surgery that conjecturally connects the moduli space of Calabi-Yau 3-folds (often known as “Reid’s Fantasy”). Through extremal transitions, we may be able to build new mirror pairs from old ones, provided we understand how mirror symmetry is preserved. In this talk, I will outline a conjectural framework that relates the genus zero Gromov-Witten theory under an extremal transition. I will explain how it works for a large family of extremal transitions among toric hypersurfaces.
|10/25/2019||Ruth J Williams||Title: Reflected Diffusions and (Bio)Chemical Reaction Networks
Abstract: Continuous-time Markov chain models are often used to describe them stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. Discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations (e.g., linear noise and Langevin), do not respect the constraint that chemical concentrations are never negative.
In this talk, we propose an approximation for such Markov chains, via reflected diffusion processes, that respects the fact that concentrations of chemical species are non-negative. This fixes a difficulty with Langevin approximations that they are frequently only valid until the boundary of the positive orthant is reached. Our approximation has the added advantage that it can be written down immediately from the chemical reactions. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.
|11/8/2019||Zhengping Gui||Title: Deformation quantization and Algebraic index theorem
Abstract: Deformation quantization is one approach to encapsulating the algebraic aspects of observables in a quantum mechanical system. By constructing a trace map on the algebra of quantum observables, correlation functions are defined. Using this paradigm, an algebraic analogue of the Atiyah-Singer index theorem was established by Fedosov and jointly by Nest and Tsygan.
In this talk, I will discuss how to use topological quantum mechanics to prove the algebraic index theorem when quantum algebra is twisted by vector bundles.
|11/15/2019||Ryan Thorngren||Title: Introduction to Bulk-Boundary Correspondences in Condensed Matter Physics
Abstract: A hallmark of topological phases are featureless insulators which are metallic at their edges. The bulk-boundary correspondence relates the ground state entanglement of the bulk with the anomalous properties of the boundary. I will give a gentle introduction to these ideas in a couple of simple models relevant to graphene and superconducting nanowires, respectively. If time permits, I will describe some recent work extending these ideas to bulk “phases” (actually critical points) described by conformal field theory.