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Casa Matemática Oaxaca (CMO)

Eduard Helly proved in 1913 one of the most celebrated results in geometry that gives conditions for the members of a family of convex objects (with convex boundary and without holes) to have a common point. Helly's theorem gives rise to numerous generalizations and variants, and is very close related with the classical theorems of Radon, Caratheodory and Tverberg, which is the r-partite version of Radon's theorem. All this theorems have a deep connection with many other areas in mathematics such as algebraic topology, discrete geometry, combinatorial geometry, and analysis and could be seen as a multidisciplinary area since it uses tools from, topology, geometry, computer sciences, probability etc. and, in resent years have been useful tool for applications to model problems in real life. The proposed workshop will assemble the key people senior and students working in this area, in order to explore recent progress and to help focus on future directions of research.

Casa Matemática Oaxaca (CMO)

A mathematical knot can be thought of as a piece of knotted string with ends attached. This knotted circle is allowed to move freely in space as long as it does not pass through itself. In recent years the study of knots has gained momentum through a series of applications, for instance in physics and in molecular biology. Such applications bring new urgency to certain classical questions pertaining to knots and how they sit inside 3-dimensional space.

Casa Matemática Oaxaca (CMO)

In addition to further developing equivariant bordism theory directly, the workshop participants will apply this theory to understand symmetries of high-dimensional objects in terms of lower-dimensional objects with similar symmetries, and, conversely, to understand when it is possible for an object with symmetry to be used to build a higher dimensional object with the same symmetry. The participants will also apply equivariant bordism theory to certain problems arising in quantum physics. Algebraic aspects of symmetry have been exploited to great effect in recent years in order to understand certain quantum mechanical systems called ``topological quantum field theories," and the workshop will push these applications further by coupling them to the geometric techniques enabled by equivariant bordism theory. The workshop activities have been carefully organized with the goal of nurturing new cosmopolitan research collaborations that unite experts and students from diverse backgrounds---both mathematical and personal.

Casa Matemática Oaxaca (CMO)

The proposed Workshop aims to study bridges between representations of algebras on one side and algebraic geometry and singularity theory on the other. One such bridge is provided by a process called `silting', allowing for example the transfer of combinatorial or homological structure in either direction. It is expected that the joint efforts of a group of internationally leading experts from many different areas of mathematics is yielding new insight in the difficult nature of these objects of study.