Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.

On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

N-terminal modifications improve the receptor affinity and pharmacokinetics of radiolabeled peptidic gastrin-releasing peptide receptor antagonists: examples of 68Ga- and 64Cu-labeled peptides for PET imaging

UNLABELLED Gastrin-releasing peptide receptors (GRPrs) are overexpressed on a variety of human cancers, providing the opportunity for peptide receptor targeting via radiolabeled bombesin-based peptides. As part of our ongoing investigations into the development of improved GRPr antagonists, this study aimed at verifying whether and how N-terminal modulations improve the affinity and pharmacokinetics of radiolabeled GRPr antagonists. METHODS The potent GRPr antagonist MJ9, Pip-d-Phe-Gln-Trp-Ala-Val-Gly-His-Sta-Leu-NH(2) (Pip, 4-amino-1-carboxymethyl-piperidine), was conjugated to 1,4,7-triazacyclononane, 1-glutaric acid-4,7 acetic acid (NODAGA), and 1,4,7-triazacyclononane-1,4,7-triacetic acid (NOTA) and radiolabeled with (68)Ga and (64)Cu. The GRPr affinity of the corresponding metalloconjugates was determined using (125)I-Tyr(4)-BN as a radioligand. The labeling efficiency of (68)Ga(3+) was compared between NODAGA-MJ9 and NOTA-MJ9 in acetate buffer, at room temperature and at 95°C. The (68)Ga and (64)Cu conjugates were further evaluated in vivo in PC3 tumor xenografts by biodistribution and PET imaging studies. RESULTS The half maximum inhibitory concentrations of all the metalloconjugates are in the high picomolar-low nanomolar range, and these are the most affine-radiolabeled GRPr antagonists we have studied so far in our laboratory. NODAGA-MJ9 incorporates (68)Ga(3+) nearly quantitatively (>98%) at room temperature within 10 min and at much lower peptide concentrations (1.4 × 10(-6) M) than NOTA-MJ9, for which the labeling yield was approximately 45% under the same conditions and increased to 75% at 95°C for 5 min. Biodistribution studies showed high and specific tumor uptake, with a maximum of 23.3 ± 2.0 percentage injected activity per gram of tissue (%IA/g) for (68)Ga-NOTA-MJ9 and 16.7 ± 2.0 %IA/g for (68)Ga-NODAGA-MJ9 at 1 h after injection. The acquisition of PET images with the (64)Cu-MJ9 conjugates at later time points clearly showed the efficient clearance of the accumulated activity from the background already at 4 h after injection, whereas tumor uptake still remained high. The high pancreas uptake for all radiotracers at 1 h after injection was rapidly washed out, resulting in an increased tumor-to-pancreas ratio at later time points. CONCLUSION We have developed 2 GRPr antagonistic radioligands, which are improved in terms of binding affinity and overall biodistribution profile. Their promising in vivo pharmacokinetic performance may contribute to the improvement of the diagnostic imaging of tumors overexpressing GRPr.

4D/3D reduction of dualities: mirrors on the circle

We engineer a brane picture for the reduction of Seiberg dualities from 4D to 3D, valid also in the presence of orientifold planes. We obtain effective 3D dualities on the circle by T-duality, geometrizing the non-perturbative superpotential which is an affine Toda potential. When reducing to pure 3D, we define a double-scaling limit which creates a sector of interacting singlets, giving a unified mechanism for the brane reduction of dualities.