The Geometry Day is a meeting intended to explore cutting-edge aspects
of contemporary geometry and offer at the same time a panoramic view on a number of topics,
by bringing together scholars with different backgrounds.

It takes place roughly twice a year at the Instituto de Matemática e Estatística,
campus do Gragoatá, bloco H (rua professor Marcos Waldemar de Freitas Reis s/n,
24210-201 Niterói/RJ, Brazil).

Hyperpolygons spaces are a family of (finite dimensional, non-compact)
hyperkähler spaces, that can be obtained from coadjoint orbits by hyperkähler
reduction. In joint work with L.Godinho, we show that these space are diffeomorphic
(in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this
talk I will describe this relation and use it to analyse the fixed points locus of
a natural involution on the moduli space of parabolic Higgs bundles. The fixed point
locus of this involution is identified with the moduli spaces of polygons in
Minkowski 3-space and the identification yields information on the connected
components of the fixed point locus.
This is based on joint works with Leonor Godinho and with Indranil Biswas,
Carlos Florentino and Leonor Godinho.

Instantons are "finite energy" solutions to a geometric PDE for a connection on a vector
bundle. These have their origin in Physics but have also been extensively studied by
Mathematicians.
The first instanton on the Euclidean Schwarzschild manifold was found by Charap and Duff
in 1977, only 2 years later than the famous BPST instantons on ℝ^{4}
were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on ℝ^{4},
the case of the Euclidean Schwarzschild
manifold resisted many efforts for the past 40 years.
vI shall explain, how using a duality between vortices and "spherically symmetric" instantons, Akos Nagy and I, recently gave a complete description of a connected component of the moduli space of unit energy instantons on the Euclidean Schwarzschild manifold.
If time permits I will also explain how to use our techniques to:
1. Find new examples of instantons with non-integer energy;
2. Completely classify "spherically symmetric" instantons.
3. Give a counterexample to a conjectured possible non-Abelian extension of Birkhoff's Theorem.
(This is joint work with Akos Nagy)

The total inflection of a linear series (L,V) on a complex curve X is explicitly
computed by a classical formula of Plücker that generalizes the well-known
Riemann-Hurwitz formula for branched covers. When X is a curve equipped with an
anti-holomorphic involution, the situation is more interesting: the inflection now
depends on the topology of the real locus of X. This dependence is well-understood
when X is an elliptic curve and V=|L| is complete, essentially because complex
inflection points are precisely torsion points of the Jacobian, and X is topologically
a torus. In this talk we address a natural generalization, in which X is a
hyperelliptic curve. Our primary tool is an analytic degeneration of X to a
metrized complex of elliptic curves; the results are joint with Cristhian Garay
López (Cinvestav) and Indranil Biswas (TIFR).

We will review a program to construct special vertex operator algebras attached to
special holonomy manifolds and explain how their symmetries reflect the underlying geometry
of the manifold. Special attention would be given to the Calabi-Yau case in general and K3 in
particular.

Let G(r,n) be the complex Grassmann manifold parametrizing r-dimensional subspaces
of ℂ^{n}, and M a free Abelian group of rank n. The purpose of the talk is to
discuss a connection of the boson-fermion correspondence with classical Schubert calculus
for G(r,n). The latter can be phrased by noticing that Λ^{r}M is naturally an
H^{*}(G(r,n);ℤ)-module, which has an elegant description in terms of a
suitable Hasse-Schmidt derivation on Λ^{r}M, called Schubert
derivation. The Schubert derivation and its transpose provide suitable approximations of the
classical vertex operators generating the fermionic vertex superalgebras, whose bosonic expressions
we can now compute in an alternative way.

Among the special holonomy manifolds G_{2} and Spin(7) manifolds remain the most
untrattables, cause there is no known analogue of the powerful tools of complex geometry. I
will discuss how the structure and representation theory of certain vertex operator superalgebras
naturally associated to these manifolds can be used to get information about the topology
and geometry of the space.

Advisory board:
Alex Massarenti, Asun Jiménez, Cristhabel Vásquez, Daniele Sepe, Zhou Detang,
Maria Amélia Salazar, Matias del Hoyo, Sérgio Almaraz, Simon Chiossi;
Andrew Clarke, Mike Deutsch, Misha Verbitsky.