Office: BA6103 (St. George campus); DH3021 (Mississauga campus)
Email: eagamse [at] math.toronto.edu
I am a Postdoctoral Fellow using combinatorial techniques to study geometry at the Mathematics department of the University of Toronto. I received my PhD from Northeastern University under the supervision of Jonathan Weitsman.
A moduli space is a geometric space that parametrises a class of objects. By studying the geometric and topological properties of the moduli space, we hope to gain a better understanding of the classification of these objects. In our case, we start with a curve Σ, and ask what are the possible vector bundles of rank n that lie over Σ? In order for our moduli space to be "nice", we need to restrict our attention to vector bundles with certain properties. Recall that the slope μ of a vector bundle is its degree divided by its rank. A vector bundle E over M is said to be stable if μ(D) < μ(E) for all holomorphic subbundles D of E. If p is a point on Σ, then a parabolic bundle on (Σ,p) is a stable bundle over Σ together with a flag 0=E0 ⊂ E1 ⊂ ... ⊂ Em = Ep in the fibre at p. I use combinatorial techniques to study the cohomology ring of the moduli space of parabolic bundles over (Σ,p). In this paper we prove some results about relations between Chern classes of certain line bundles over this moduli space. I am currently preparing a paper exploring the case where Σ has more than one marked point.
Let Σ be a Riemann surface of genus g ≥ 2, and let p ∈ Σ be a point. Choose a loop c ∈ π1(Σ \ p) that represents the boundary of Σ \ p. Let G be a compact Lie group, and pick a maximal torus T ⊂ G and a "generic" element t ∈ T. Consider the space Sg(t):={ρ ∈ Hom(π1(Σ \ p),G) | ρ(c) ~ t}/G (where ~ denotes conjugacy in G). This space is the moduli space of flat G-connections over Σ (with prescribed holonomy around the marked point p). When G = SU(n), this space is also the moduli space of parabolic vector bundles over Σ mentioned above. In this paper, we let G=SO(2n+1) and prove that certain products of Chern classes of line bundles over Sg(t) must vanish.
Given a torus acting on a manifold, geometers often study the behaviour of the action near its fixed points. In particular, applying the Atiyah-Bott-Berline-Vergne localisation formula gives us identities that the weights of the torus action at the fixed points must satisfy. We ask whether we can reverse this process; that is, given a collection of sets of torus weights, can we find a manifold with a torus action whose fixed point data is precisely that collection? Of course, it is necessary for the torus weights to satisfy the ABBV identities; we show that in certain contexts satisfying these identities is also sufficient. A more technical description is available here, and a preliminary version of our paper is here.
A common theme in mathematics is that linear things are easier to study than non-linear ones. Thus, when faced with a problem, one hopes to be able to convert it into a linear one, without losing too much information in the process. Geometric quantisation is one way of converting a group action on a suitable manifold into a representation of that group on a vector space; I study the characters of these representations. Here is a more detailed summary of my work on this topic; the full paper is available on the arXiv.
I am the current organiser of the sympelctic geometry seminar at the University of Toronto.
Website under construction... watch this space!