Studies the Riemann mapping theorem, meromorphic and entire functions, topics in analytic function theory. Prerequisite: MATH 7340 or equivalent.

Seminars

Chris Manon (GMU) - Toric geometry of moduli spaces of principal bundles on a curve

For $C$ a smooth projective curve, and $G$ a simple, simply connected

complex group, let $M_C(G)$ be the moduli space of semistable

$G-$principal bundles on $C$. As the curve $C$ moves in the moduli

$\mathcal{M}_g$ of smooth curves, the spaces $M_C(G)$ are known to

define a flat family of schemes, and this family can be extended to

the Deligne-Mumford compactification $\bar{\mathcal{M}}_g$. We

describe the geometry of the fibers of this family which appear at the

stable boundary, in particular we discuss a recent result which shows

that the fibers over maximally singular curves contain an important

and ubiquitous moduli space, the free group character variety

$\mathcal{X}(F_g, G),$ as a dense, open subspace. The latter is a

moduli space of representations of the free group $F_g$ in $G$, and

naturally appears as an object of interest in Teichm\"uller theory,

the theory of geometric structures, and the theory of Higgs bundles.

For $G = SL_2(\C)$ and $SL_3(\C)$ we describe maximal rank valuations

on the coordinate rings of these spaces, and how the associated

Newton-Okounkov polyhedra can be used to study the geometry of both

$\mathcal{X}(F_g, G)$ and $M_C(G).$

For $G = SL_2(\C)$ and $SL_3(\C)$ we describe maximal rank valuations

on the coordinate rings of these spaces, and how the associated

Newton-Okounkov polyhedra can be used to study the geometry of both

$\mathcal{X}(F_g, G)$ and $M_C(G).$ " class="addtocalendar" target="_new">Add to Google Calendar

Tea Time

Mona Merling (JHU) - Equivariant algebraic K-theory

The first definitions of equivariant algebraic K-theory were given in the early 1980’s by Fiedorowicz, Hauschild and May, and by Dress and Kuku; however these early space-level definitions only allowed trivial action on the input ring or category. Equivariant infinite loop space theory allows us to define spectrum level generalizations of the early definitions: we can encode a G-action (not necessarily trivial) on the input as a genuine G-spectrum.

I will discuss some of the subtleties involved in turning a ring or space with G-action into the right input for equivariant algebraic K-theory or A-theory, and some of the properties of the resulting equivariant algebraic K-theory G-spectrum. For example, our construction recovers as particular cases equivariant topological real and complex K-theory, Atiyah’s Real K-theory and statements previously formulated in terms of naive G-spectra for Galois extensions.

I will also briefly discuss recent developments in equivariant infinite loop space theory from joint work with Guillou, May and Osorno (e.g., multiplicative structures) that should have long-range applications to equivariant algebraic K-theory.

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MATH 7350

Studies the Riemann mapping theorem, meromorphic and entire functions, topics in analytic function theory. Prerequisite: MATH 7340 or equivalent.