Quantum anomalies can arise when regularized determinants and traces of infinite dimensional operators in quantum field theories lose their multiplicative and tracial property. I will explain the basic ideas and methods and if time allows, will go over the celebrated Polyakov conformal anomaly formula and how the anomaly cancellation dictates the critical dimension D=26 for bosonic string theory.

The classical Adams spectral sequence can be computed via higher order operations in mod p cohomology. Baues and Jibladze carried out computations of the differential $d_2$ using the algebra of secondary operations. Baues and Blanc described an algebro-combinatorial structure which encodes enough information about $n^{th}$ order operations to compute the differential $d_n$. In joint work with Baues, we specialize that work to the case $n=3$ and describe a more concrete algebraic structure which suffices to compute the differential $d_3$.

Study of equivalences and symmetries of real submanifolds in complex space goes back to the classical work of Poincare and Cartan and was deeply developed in later work of Tanaka and Chern and Moser. This work initiated far going research in the area (since 1970's till present), which is dedicated to questions of regularity of mappings between real submanifolds in complex space, unique jet determination of mappings, solution of the equivalence problem, and study of automorphism groups of real submanifolds.

Current state of the art and methods involved provide satisfactory (and sometimes complete) solution for the above mentioned problems in nondegenerate settings. However, very little is known for more degenerate situations, i.e., when real submanifolds under consideration admit certain singularities of the CR-structure (such as non-constancy of the CR-dimension or that of the CR-orbit dimension).

The recent CR (Cauchey-Riemann Manifolds) -- DS (Dynamical Systems) technique, developed in our joint work with Shafikov and Lamel, suggests to replace a real submanifold with a CR-singularity by appropriate complex dynamical systems. This technique has recently hepled to solve a number of long-standing problems in CR-geometry, related to regularity of CR-mappings.

In this talk, we give an overview of the technique and the results obtained recently by using it. We also discuss a possible development in this direction, in particular, new sectorial extension phenomena for CR-mappings.

Higher inductive types are a generalization of inductive types. While an inductive type is generated by certain terms, a higher inductive type may be generated by terms, paths between terms, paths between paths between terms, etc. In the homotopy theoretic model of type theory, this corresponds to constructing cell complexes. We will see many other uses of higher inductive types, and will sketch the argument that $\pi_1(S^1)$ is isomorphic to the integers.

Over many fields, including finite fields and imaginary number fields, any central simple algebra of exponent 2 can be decomposed as a product of quaternion algebras. However, over arbitrary fields there exist examples of indecomposable algebras of exponent 2. For division algebras, indecomposability can be detected by non-trivial torsion in the Chow group of the associated Severi-Brauer variety. In this talk, we consider the analogous problem of decomposability for algebras with involution. We replace the Severi-Brauer variety with G/B, the variety of Borel subgroups of an algebraic group G of inner type Dn, and estimate the Chow group of G/B via the gamma-filtration on its Grothendieck group. In particular, we look at groups of type D4 and their associated trialitarian triples.