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The lack of oestrogen receptor, progesterone receptor and human epidermal growth factor receptor-2 expression in breast cancer (BC) is the basis for the categorization of the tumour as triple negative breast carcinoma (TNBC). The majority of TNBCs are aggressive tumours with common metastases and decreased expression of markers that could help in identifying the metastatic lesion as of mammary origin. Breast markers, such as gross cystic disease fluid protein-15 (GCDPF-15), GATA binding protein 3 (GATA3), mammaglobin (MGB) and SOX10, are not uniquely specific to BC. Our aim was to evaluate trichorhinophalangeal syndrome type 1 (TRPS1) protein as a breast marker in a series of cytokeratin-5-expressing TNBC, mostly corresponding to basal-like TNBCs, previously characterized for the expression of other breast markers. One hundred seventeen TNBCs in tissue microarrays were immunostained for TRPS1. The cut-off for positivity was ≥ 10%. The reproducibility of this classification was also assessed. TRPS1 positivity was detected in 92/117 (79%) cases, and this exceeded the expression of previously tested markers like SOX10 82 (70%), GATA3 11 (9%), MGB 10 (9%) and GCDFP-15 7 (6%). Of the 25 TRPS1-negative cases, 11 were positive with SOX10, whereas 5 to 6 dual negatives displayed positivity for the other makers. The evaluation showed substantial agreement. Of the five markers compared, TRPS1 seems the most sensitive marker for the mammary origin of CK5-expressing TNBCs. Cases that are negative are most often labelled with SOX10, and the remainder may still demonstrate positivity for any of the 3 other markers. TRPS1 has a place in breast marker panels.

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$., Comment: 12 pages, no figures. A mistake in the main theorem is fixed; supersedes published version

Ginzburg-Landau fields are the solutions of the Ginzburg-Landau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\mathbb{R}^2$, spheres, tori, etc.) with de Gennes-Neumann boundary conditions. We also prove that, for each such manifold and all positive $\alpha$ and $\beta$, the Ginzburg-Landau free energy is a Palais-Smale function on the space of gauge equivalence classes, Ginzburg-Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg-Landau fields is compact., Comment: 16 pages, 1 figure

We analyze 2-dimensional Ginzburg-Landau vortices at critical coupling, and establish asymptotic formulas for the tangent vectors of the vortex moduli space using theorems of Taubes and Bradlow. We then compute the corresponding Berry curvature and holonomy in the large volume limit., Comment: 28 pages, 2 figures, published version