• Maria Alfonseca (North Dakota)

Methods of harmonic analysis in convex geometry and tomography (mini-course)

Harmonic analysis provides a powerful framework for addressing central problems in geometric tomography. Tools such as the Fourier transform, spherical harmonics, and Radon-type transforms allow one to translate geometric questions about sections, projections, and volumes of convex bodies into analytic statements about functions and measures. The talk will outline some of these tools and present several applications.

  • Károly Bezdek (Calgary)

Towards a theory of ball-bodies and ball-polyhedra (mini-course)

The theory of ball-polyhedra and ball-bodies essentially reconstructs classical convex geometry by replacing linear elements (lines, half-spaces, and hyperplanes) with spherical ones (arcs, balls, and spheres of a fixed radius r). While a ball-polyhedron is the intersection of a finite number of closed r-balls (with a non-empty interior), a ball-body (or r-convex body) is the intersection of an arbitrary family of closed r-balls. One can think of ball-bodies as the spherical analogue to general convex bodies, just as ball-polyhedra correspond to convex polytopes. In this talk we plan to survey the foundational pillars of this theory and explore developments that extend beyond it including the following topics: Spindle Convexity (r-Convexity); Separation and Supporting Spheres; Face Lattices and Combinatorics; Rigidity and Uniqueness; Extremal Problems (Volume, Covering, and Illumination).

  • Károly J. Böröczky (Alfréd Rényi Institute of Mathematics, Hungary)

Affine Invariant Inequalities (mini-course)

Recent advances and conjectures on inequalities invariant under linear transformations are reviewed including the proof of Keith Ball’s four decades old Blaschke-Santaló-type conjecture, update on the logarithmic Brunn-Minkowski conjecture, results related to fractional isoperimetric inequality and chord integrals, or Assaf Naor’s isomorphic isoperimetric conjecture.

  • Julián Haddad (Seville, Spain)

Applications of topological methods to convex geometry (mini-course)

We will review some notions of LS-category, Morse theory and Degree theory, and present some of their applications to problems on convexity such as properties of centroids of non-central sections of a convex body.