UMASS Boston Mathematics Colloquium Series
Spring 2018

ABSTRACT:I will discuss some current joint work with Helmut Hofer, in which we define and establish properties of a new class of pseudoholomorhic curves (feral curves) to study certain divergence free flows in dimension three. In particular, we show that if \(H\) is a smooth, proper, Hamiltonian in \(R^4\), then no energy level of \(H\) is minimal. That is, the flow of the associated Hamiltonian vector field has a trajectory which is not dense.

ABSTRACT: We report on the recent rigorous and general construction of the deformation-obstruction theories and virtual fundamental classes of nested (flag) Hilbert scheme of one dimensional subschemes of a smooth projective algebraic surface. This construction will provide one with a general framework to compute a large class of already known invariants, such as Poincare invariants of Okonek et al, or the reduced local invariants of Kool and Thomas in the context of their local surface theory. We show how to compute the generating series of deformation invariants associated to the nested Hilbert schemes, and via exploiting the properties of vertex operators, prove that in some cases they are given by modular forms. We finally establish a connection between the Vafa-Witten invariants of local-surface threefolds (recently analyzed Tanaka and Thomas) and such nested Hilbert schemes. This construction (via applying Mochizuki's wall- crossing techniques) enables one to obtain a relations between the generating series of Seiberg-Witten invariants of the surface, the Vafa-Witten invariants and some modular forms. This is joint work with Amin Gholampour and Shing-Tung Yau following arXiv:1701.08902 and arXiv:1701.08899.

ABSTRACT: Interacting excitatory and inhibitory neuronal populations often generate oscillations in electrical fields in the brain. I will briefly review this mechanism and the reasons to believe that it is important in brain function. Most of the talk will be focused on the effects of recurrent excitation, i.e., of the neurons of a local network in the brain exciting each other. Recurrent excitation can sustain activity in a network that would otherwise be quiescent; this is believed to be the basis of working memory. It can also lead to a runaway process, with excitation generating more excitation etc., much as the presence of a quadratic term on the right-hand side of a differential equation can lead to blow-up in finite time; this may be related to epileptic seizures. For model problems, we prove that abrupt transitions to runaway activity require recurrent excitation with fast kinetics, while working memory activity is more robust with recurrent excitation with slow kinetics.

Abstract: Early theorems in algebra, such as Jordan-Holder decomposition, provide ways of breaking down complicated structures into elementary, or at least more fundamental, instances. Analogs of such a decomposition occur in more advanced contexts, and in this talk I will explain how they arise in one version of mirror symmetry. Mirror symmetry relates invariants of the algebraic geometry of a variety X to invariants of the symplectic geometry of a mirror manifold Y along with a potential function W. In algebraic geometry, one may adopt a birational approach and consider a minimal model sequence starting at X and ending in a minimal model. In a concrete sense, such a sequence can be considered as a decomposition of your original space X. Likewise, for the symplectic mirror, one may degenerate Y and its potential, resulting in an actual decomposition of Y into a sequence of contact cobordisms. I will describe how such decompositions come in mirror pairs and discuss their categorical incarnations in homological mirror symmetry.

Abstract: Epilepsy, the propensity toward recurrent, unprovoked seizures is a devastating disease affecting 65 million people worldwide. Understanding and treating this disease remains a challenge, as seizures manifest through mechanisms and features that span spatial and temporal scales. In this talk, we will examine some aspects of this challenge through the statistical analysis and mathematical modeling of human brain voltage activity recorded simultaneously across microscopic and macroscopic spatial scales. We will focus on how seizure activity propagates over the brain’s surface, and how seizures stop. When possible, we will describe a corresponding computational model to propose specific mechanisms that support the observed spatiotemporal dynamics.

Abstract: By a polyhedron we understand a connected structure consisting of vertices (points in space), edges (line segments) and polygons, where every edge belongs to two polygons. We do not assume that the polygons are convex, planar or even finite. A 4-polytope is a connected collection of polyhedra where every polygon belongs to two polyhedra. A 4-polytope is regular whenever every local combinatorial reflection can be extended to an isometry of space preserving the polytope. If this is possible for every local combinatorial rotation, but not for the reflections we say that the 4-polytope is chiral. In 2004 Peter McMullen claimed that there are no chiral 4-polytopes in Euclidean space. This result turned out to be false. In this talk we describe the three chiral 4-polytopes in Euclidean space

Abstract: Tensor categories are mathematical structures that appear in various fields of mathematics (representation theory, low-dimensional topology, operator algebras, homotopy theory, etc.). They also provide good models in mathematical physics (conformal field theory, quantum statistics) and theoretical computer science (quantum computation). In this talk I will discuss tensor categories from the viewpoint of Hopf algebras and show how our knowledge of the latter can aid us in our understanding of the former. More specifically, I will present a recent result on the classification of pointed braided finite tensor categories admitting a fiber functor. This is based on joint work with Dmitry Nikshych.

Abstract: Understanding the symmetries of a topological space is a classical problem in mathematics. In this talk we will discuss how methods from topology can be used to approach this. We will review classical results such as those describing the finite groups that can act without fixed points on a sphere. This will be used as a motivation for the more modern notion of "group actions up to homotopy", which leads to interesting interactions between topology, group theory and representations. A number of examples will be provided to illustrate this.