1:30pm Jake Bennett Fiedler (UW Madison): Assouad dimension and point-to-set principles.
Abstract: Hausdorff and packing dimension both have "effective" analogues, which quantify the information content of individual points. Through the point-to-set principle, these effective notions can be used to establish new classical results in geometry measure theory. Assouad-type dimensions give additional information about the structure of a fractal set by quantifying how rapidly it can appear to grow between suitably separated scales. We discuss effective versions of this family of notions as well as some connections to classical fractal geometry.
1:50pm Josh Hinman (University of Washington): Fat Polytopes and Spheres.
Abstract: A four-dimensional polytope has four types of faces: vertices, edges, polygons, and facets. Combinatorialists are on the hunt for fat polytopes: those with many edges and polygons but few facets and vertices. In this talk, I'll share my construction of shellable spheres---objects slightly more general than polytopes---with unbounded fatness. These spheres are a new piece of evidence that arbitrarily fat polytopes may exist.
2:10pm Olga Medrano Martin del Campo (UChicago): Interactions of finite VC and Littlestone dimensions.
Abstract: This talk gives a brief overview of part of my thesis work. Stable graphs, as well as Littlestone classes, were characterized by the existence of linear-sized 'good' sets, a kind of strongly homogeneous set, in the work of Malliaris-Shelah and Malliaris-Moran. We show that VC classes are characterized by the existence of linear-sized symmetric or asymmetric good pairs (which we define). We give three proofs using different methods and give different kinds of bounds.
3:00pm Hannah Sheats (UIC): On the linear complexity of sets of bounded VC_2 dimension in F_p^n.
Abstract: A theorem from 2018 of Alon, Fox and Zhao shows that a set of bounded VC-dimension in an abelian group of bounded exponent can be well approximated by a union of cosets of a subgroup of bounded index. Results from 2021 and 2025 due to Terry and Wolf show that a subset of $\mathbb{F}_p^n$ with bounded VC_2-dimension (a higher arity analogue of VC-dimension) can be well approximated by a union of fibres of a bilinear form. While they obtain efficient bounds on the quadratic complexity of this approximation, their method requires an application of the general arithmetic regularity lemma and therefore the bounds obtained for the linear complexity are inefficient. In this talk we present results, joint with Terry, that give a new proof with efficient bounds on the linear complexity of subsets of $\mathbb{F}_p^n$ with bounded VC_2 dimension.
3:20pm Clay Mizgerd (UIC): Sampling permutations of F_q via random translations.
Abstract: A classical result of Carlitz says that linear functions and the field inversion x |-> x^{q-2} generate the symmetric group on the finite field F_q. Restricting from linear functions to just translations still generates the alternating group. We analyze the question of how fast an alternating sequence of inversions and random translations converges to a uniformly random permutation. We prove matching upper and lower bounds of order q log q many steps via a connection with Mobius transformations.
3:40pm Chris Wilson (UChicago): The Hausdorff dimension of the intersection of two Kakeya sets.
Abstract: In this talk, I will prove that given two Besicovitch sets in the plane whose associated line-parameter sets are Borel, the Hausdorff dimension of the intersection is 2. The proof combines recent bounds for planar radial projections with bounds for planar Furstenburg sets. I will also discuss how this result is connected to the following open question: given three Borel functions on the real line, is there a non-vertical line intersecting all three graphs?
Rachel Greenfeld
Email