Jarod Alper

Title: The Construction of Projective Compactifications of the Moduli Space of Riemann Surfaces
Abstract: We will introduce a systematic, intrinsic technique for constructing projective moduli spaces. The main result will be a generalization of the Keel-Mori theorem giving conditions on an algebraic stack to admit a good moduli space. This result will be applied to construct new modular projective compactifications of M_g.


Brendan Creutz

Title: 2-torsion Brauer classes on double covers of ruled surfaces.
Abstract: Let X be a smooth double cover of a geometrically ruled surface over a separably closed field of characteristic different from 2. I will discuss recent work in which we give a finite presentation of the two-torsion in the Brauer group of X with generators given by central simple algebras over the function field of X and relations coming from the Neron-Severi group of X. I will also discuss some of the motivation for this coming from arithmetic applications such as computing Brauer-Manin obstructions to the existence of rational points. This is joint work with Bianca Viray.


Norman Do

Title: Orbifold Hurwitz numbers and topological recursion
Abstract: How many ways are there to map a genus g Riemann surface to the Riemann sphere? We obtain Hurwitz numbers by counting such maps with prescribed branching conditions. In recent work with Oliver Leigh and Paul Norbury, we show that a certain class of Hurwitz numbers is generated by topological recursion. The talk will be an advertisement for topological recursion, which appears in matrix models, enumeration of surface tilings, intersection theory on moduli spaces, enumerative geometry of threefolds, statistical mechanical models, quantum invariants of knots, topological string theory, and probably a lot more!


Maksym Fedorchuk

Title: GIT stability of Hilbert and Syzygy points of canonical curves
Abstract: I will introduce the log minimal model program for the moduli space of Riemann surfaces M_g and explain how it gives rise to a sequence of modular compactifications of M_g. These include the celebrated Deligne-Mumford space of stable curves, but also much more recent compact moduli space of at worst tacnodal curves constructed by Hassett and Hyeon. I will then introduce a Geometric Invariant Theory problem concerning stability of the Hilbert points of (bi)canonical curves and sketch its solution in the generic case due to Alper, Smyth, and myself. Finally, I will speculate how Syzygy points and their GIT stability can be used to glimpse the final steps in the log minimal model program for M_g.


Michael Harrison

Title: Computational Algorithms for Gonal Maps and Minimal Degree Plane Models of Algebraic Curves of Genus <= 6.
Abstract: We will describe effective computational algorithms for explicitly computing minimal degree (gonal) covers of the projective line and minimal degree birational plane models for algebraic curves of genus up to 6. These make use of the Lie Algebra method for trigonal curves and a close study of the minimal polynomial resolution of the canonical coordinate ring in the 4-gonal case. The algorithms have been implemented by the speaker and his collaborators in the Magma computer algebra system.


Anthony Henderson

Title: The modular generalized Springer correspondence
Abstract: Given a connected reductive algebraic group $G$ with Weyl group $W$, the Springer correspondence realizes the category of representations of $W$ as a quotient of the category of $G$-equivariant perverse sheaves on the nilpotent cone. In the original definition, the representations and sheaves were over a field of characteristic zero, but we have recently shown that the same formalism works with modular coefficients, where the categories are no longer semisimple. In the characteristic-zero case, Lusztig defined a generalized Springer correspondence to interpret the whole category of $G$-equivariant perverse sheaves on the nilpotent cone in terms of representations of relative Weyl groups. We define and determine a modular generalized Springer correspondence in the case $G=\mathrm{GL}(n)$. This gives a geometric explanation for the fact that, in the modular case, the category of modules over the Schur algebra can be obtained by successive recollements of categories of representations of suitable products of symmetric groups.
(Joint work with P. Achar, D. Juteau and S. Riche)


Finnur Larusson

Title: An Oka principle for equivariant isomorphisms.
Abstract: I will discuss new joint work with Frank Kutzschebauch (Bern) and Gerald Schwarz (Brandeis), available on the arXiv. Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$ that are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$?

If the actions of $G$ on $X$ and $Y$ are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch.

We prove that $X$ and $Y$ are $G$-biholomorphic if $X$ is $K$-contractible, where $K$ is a maximal compact subgroup of $G$, or if there is a $G$-diffeomorphism $\psi:X\to Y$ over $Q$, which is holomorphic when restricted to each fibre of the quotient map $X\to Q$. We prove a similar theorem when $\psi$ is only a $G$-homeomorphism, but with an assumption about its action on $G$-finite functions. When $G$ is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of $G$-biholomorphisms from $X$ to $Y$ over $Q$. This sheaf can be badly singular, even for a low-dimensional representation of $\mathrm{SL}_2(\mathbb C)$.

Our work is in part motivated by the linearisation problem for actions on $\mathbb C^n$. It follows from one of our main results that a holomorphic $G$-action on $\mathbb C^n$, which is locally $G$-biholomorphic over a common quotient to a generic linear action, is linearisable.


Anthonty Licata

Title: Symplectic Duality
Abstract: Symplectic resolutions are central objects at the intersection of algebraic geometry and representation theory. The goal of this talk will be to describe a fairly rich relationship, known as "symplectic duality", that exists between pairs of symplectic resolutions. All of this is joint work with Tom Braden, Nick Proudfoot, and Ben Webster.


Paul Norbury

Title: Gromov-Witten invariants and cohomological field theories.
Abstract: I will give a short introduction to Gromov-Witten invariants and their relationship to the Deligne-Mumford compactification of the moduli space of curves. This gives rise to the notion of a cohomological field theory. Recently, Eynard and Orantin have introduced a spectral curve---an algebraic curve with extra data---that encodes a cohomological field theory. I will describe these ideas, particularly focusing on the well-studied problem of the Gromov-Witten invariants of the two-sphere.


Peter O'Sullivan

Title: The universal properties of pure motives
Abstract: Motives are usually regarded as giving a universal cohomology theory for algebraic varieties. We examine the sense in which this is so for pure motives, using a formal analogy with the passage from a commutative ring to its spectrum and from a module to its sheaf on the spectrum.


Brett Parker

Title: Integer counts of holomorphic curves
Abstract: Roughly speaking, Gromov-Witten invariants count holomorphic curves, but the `number' of holomorphic curves is often not a whole number. I shall outline a different method for defining an invariant which gives an integer count of holomorphic curves. Time permitting, I will then explain why in dimension 4, this count recovers Taubes' Gromov invariants- which are equivalent to the Seiberg-Witten invariants.


David Smyth

Title: New Compactifications of Moduli Spaces of Riemann Surfaces
Abstract: In 1969, Deligne and Mumford constructed a beautiful compactification of the moduli space of Riemann Surfaces, namely the moduli space of stable curves. This compactification now plays a fundamental role in many areas of geometry and topology. Recently, however, we have become aware that the Deligne-Mumford compactification is only one of many possible geometrically significant compactifications of the moduli space of Riemann surfaces. In this talk, we will discuss recent progress toward constructing and classifying alternative compactifications.


Mathai Varghese

Title: Geometry and Index theory of (pseudo)differential algebra bundles
Abstract: I will discuss a natural generalization of the families index theorem of Atiyah and Singer. Some results include:
* Identification of the Lie algebra of derivations and the automorphism group of the algebra of (pseudo)differential operators.
* Description of a natural class of connections, curvings and 3-curvatures of the bundle gerbe associated to a (pseudo)differential algebra bundle, in terms of regularized traces and residue traces.
* Brief description of the associated projective families index theorem. This is joint work with Richard Melrose.