1）Historically, in toric topology two types of objects are studied: the actions of a compact torus Tn on manifolds and their real versions: the actions of discrete torus (Z/2)n on manifolds. A lot is known about the relation between moment angle manifolds and quasitoric manifolds, and there is a real version of this relation, namely, the relation between real moment angle manifolds and small covers. In 2012 Jeremy Hopkinson, a graduate student of Nigel Ray, introduced a quaternionic version of these stories. In quaternionic version a manifold is acted on by a noncommutative group (S3)n and a lot of interesting topology and combinatorics arise that deserves further study. In my talk, I will describe briefly the main points and difficulties which appear when you try to generalize from toric topology to "quoric" topology (this is the short name for "quaternionic toric" introduced by Hopkinson). I will concentrate on "quoric" surfaces, which are 8-dimensional manifolds acted on by (S3)2. It happens that there is an action of a compact 3-torus on each such manifold, which gives a series of examples of complexity one torus actions. The T3-orbit spaces of these actions are homeomorphic to 5-spheres.
2）It is known that the right sided multiplication action of a compact torus T3 on the Lie group U(3) of unitary matrices is free and the quotient is diffeomorphic to the complete flag variety Flag(C3). On the other hand, there exists the multiplication action of T3 on U(3) from the left side: it reduces to the non-free action of T3 on Flag(C3). Buchstaber and Terzic, using their theory of (2n,k)-manifolds, had proved that the orbit space Flag(C3)/T3 is homeomorphic to the 4-sphere. In other words, the double quotient T3\U(3)/T3 of a non-free two-sided multiplication action is a 4-sphere. There is a real version, which tells that the quotient of O(3) by the two-sided multiplication action of (Z/2)3 is a 3-sphere. It happens, that the two-sided quotients of SO(3) by a pair of discrete groups acting from different sides appear in crystallography and material science under the name "misorientation spaces". We study misorientation spaces for different pairs of proper point crystallography groups. In many cases the misorientation space is a 3-sphere according to Poincare conjecture. In the remaining cases the topology of the misorientation space can also be completely described by Thurston's elliptization conjecture. In some cases it is possible to avoid the hardcore 3-dimensional topology: for several misorientation spaces we provide precise coordinates. As we hope, these can be used in applications.
3）In the last years it became quite popular in the world to combine topology with the brain study. Most of research is concentrated on the application of homology and persistent homology (I guess, because these can be calculated more or less efficiently). There is, however, a subfield in the brain study, which appears to be very geometrical and topological in its nature. The question is: how the location of a mammal is encoded in its brain? Nobel prize 2014 in physiology was given for discovery of place cells and grid cells in a brain of a mammal. A neural cell of this type fires when a mammal comes inside certain location of space. We may wonder: is it possible to recover the geometry and topology of the environment, given some data about neural activity? This task is very complicated and requires the cooperation of biologists, cognitive scientists, mathematicians, and specialists in big data analysis. I will give a general survey, and, if time allows, try to explain some fascinating homotopy theory beyond these problems.
Each finite-dimensional complex semisimple Lie algebra has a so-called universal centralizer, a hyperkähler variety of interest to geometric representation theorists. This variety carries a completely integrable system whose construction resembles that of the Kostant--Toda system. I will promote this resemblance to a precise relationship between the two aforementioned integrable systems.
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