Abstract. We introduce a new approach for computing the monodromy of the Hitchin map and use this to completely determine the monodromy for the moduli spaces of L-twisted G-Higgs bundles, for the groups G = GL(2, C), SL(2, C) and P SL(2, C). We also determine the twisted Chern class of the regular locus, which obstructs the existence of a section of the moduli space of L-twisted Higgs bundles of rank 2 and degree deg(L) + 1. By counting orbits of the monodromy action with Z 2 -coefficients, we obtain in a unified manner the number of components of the character varieties for the real groups G = GL(2, R), SL(2, R), P GL(2, R), P SL(2, R), as well as the number of components of the Sp(4, R)-character variety with maximal Toledo invariant. We also use our results for GL(2, R) to compute the monodromy of the SO(2, 2) Hitchin map and determine the components of the SO(2, 2) character variety.

Abstract. We construct triples of commuting real structures on the moduli space of Higgs bundles, whose fixed loci are branes of type (B, A, A), (A, B, A) and (A, A, B). We study the real points through the associated spectral data and describe the topological invariants involved using KO, KR and equivariant K-theory.

Abstract. Through the action of anti-holomorphic involutions on a compact Riemann surface Σ, we construct families of (A, B, A)-branes LG c in the moduli spaces MG c of Gc-Higgs bundles on Σ. We study the geometry of these (A, B, A)-branes in terms of spectral data and show they have the structure of real integrable systems.

Singular limits of 6D F-theory compactifications are often captured by Tbranes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

The first lecture shall introduce classical Higgs bundles and the Hitchin fibration, and describe the associated spectral data in the case of principal Higgs bundles for classical complex Lie groups. Whilst bibliography is provided in the text, the main references followed are Hitchin's papers [Hit87, Hit87a, Hit92, Hit07]. Lecture 2During the second lecture we shall construct Higgs bundles for real forms of classical complex Lie groups as fixed points of involutions, and describe the corresponding spectral data when known, as appearing in [Sch11, Sch13, Sch13b] and [HitSch]. Along the way, we shall mention different applications and open problems related to the methods introduced in both lectures. ExercisesEach lecture contains exercises of varying difficulty, whose solutions can be found in [Sch13], and commented in the .tex file. Open problems which might be tackled with methods similar to the ones introduced in the lectures shall also be mentioned, and appear indicated with ((*)). For each of these problems we suggest references which feature approaches that may be useful. BibliographyWe shall highlight the main references considered, as well as the precise places where the methods used were developed. Since it proves to be very difficult to give a comprehensive and exhaustive account of research in tangential areas, we shall restrain ourselves to mentioning related work only when it directly involves methods using spectral data. The reader should refer to references in the bibliography for further research in related topics (e.g., see references in [Ap09, P13, Sch13]).

We present a modified age-structured SIR model based on known patterns of social contact and distancing measures within Washington, USA. We find that population age-distribution has a significant effect on disease spread and mortality rate, and contribute to the efficacy of age-specific contact and treatment measures. We consider the effect of relaxing restrictions across less vulnerable age-brackets, comparing results across selected groups of varying population parameters. Moreover, we analyze the mitigating effects of vaccinations and examine the effectiveness of age-targeted distributions. Lastly, we explore how our model can applied to other states to reflect social-distancing policy based on different parameters and metrics.

Abstract. This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

Mid-dimensional (A, B, A) and (B, B, B)-branes in the moduli space of flat G C -connections appearing from finite group actions on compact Riemann surfaces are studied. The geometry and topology of these spaces is then described via the corresponding Higgs bundles and Hitchin fibrations.Date: March 30, 2018.