This talk will introduce a method for learning to visualize 4‑dimensional space, give participants a chance to work on some 4D visualization exercises in small groups, and then present a few solutions using interactive 4D graphics software. The exercises range from elementary to advanced, so everyone from first-year undergraduates to experienced geometers should find something they like.

# Seminars

Given a complex reductive group G, the moduli space M(G) of G-Higgs bundles on a curve has a natural hyperkähler structure and it comes equipped with an algebraically completely integrable system through the Hitchin fibration. These moduli spaces have played an important role in mirror symmetry and in the geometric Langlands program and thus it has become of particular interest the study of certain decorated special subvarieties (branes) of M(G).

The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form.

In this seminar we shall introduce the Conley index of an isolated invarant set of a flow on a locally compact metric space. The Conley index is a homotopical tool which encapsulates dynamical information near the isolated invariant set. The definition of this invariant involves the use of some external objects, namely isolating blocks (or, more generally, index pairs). We will give a way to compute this index in "intrinsic terms" for flows defined on surfaces. To do this we will deepen into the structure of the unstable manifold of an isolated invariant set.

In this talk, we explore a notion that sits between the concept of locally finite variety and that of periodic variety, using the inescapable Green's relations. Namely, a variety is said to be K-finite, where K stands for any of the Green's relations, if every finitely generated semigroup in this variety has but finitely many K-classes. Our characterization uses the language of "forbidden objects".

In 1975, Sheehan conjectured that every d-regular Hamiltonian graph contains a second Hamiltonian cycle. This conjecture has been verified for all d greater than 22. In the light of Sheehan’s conjecture, it is natural to ask if regularity is genuinely necessary to force the existence of a second Hamiltonian cycle, or if a minimum degree condition is enough.

Let *X* be a set, *X'* be a disjoint copy of *X *and $\bar{X}\wedge\bar{X}=\{(x\wedge y): x,y\in X\cup X'\}$. We look at $\hat{X}=X\cup X'\cup(\bar{X}\wedge \bar{X})$ as a set of letters and consider the free semigroup $\hat{X}^+$ on the set $\hat{X}$. Auinger [1] constructed a model for the bifree locally inverse semigroup on *X* as a quotient semigroup of $\hat{X}^+$. This result enables us to talk about presentations $\langle X;R\rangle$ of locally inverse semigroups (LI-presentations) where $R\subseteq \hat{X}^+\times\hat{X}^+$.

Os célebres resultados de S.Newhouse ([N]) mostram que a bifurcação de uma tangência homoclínica asociada a uma sela numa superfície gera tangências homoclínicas robustas (isto é, tangências homoclínicas que persistem por pequenas perturbações) associadas a um conjunto hiperbólico especial chamado ferradura espessa. Além disso, a continuação (hiperbólica) da sela inicial está contida nesse conjunto hiperbólico.

In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.

We shall discuss residual properties of groups and their interpretation in connection with the profinite completion of groups of geometric nature.