Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, March 16, 2021.

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Tuesday, March 16, 2021

11:00 am in Zoom,Tuesday, March 16, 2021

An asymptotic local-global principle for integral Kleinian sphere packings

Edna Jones (Rutgers University)

Abstract: We will discuss an asymptotic local-global principle for certain integral Kleinian sphere packings. Examples of Kleinian sphere packings include Apollonian circle packings and Soddy sphere packings. Sometimes each sphere in a Kleinian sphere packing has a bend (1/radius) that is an integer. When all the bends are integral, which integers appear as bends? For certain Kleinian sphere packings, we expect that every sufficiently large integer locally represented as a bend of the packing is a bend of the packing. We will discuss ongoing work towards proving this for certain Kleinian sphere packings. This work uses the circle method, quadratic forms, spectral theory, and expander graphs.

For Zoom info, please email obeck@illinois.edu

2:00 pm in Zoom,Tuesday, March 16, 2021

Expressing graphs as symmetric differences of cliques of the complete graph

Puck Romback (University of Vermont)

Abstract: Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathcal{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathcal{C}$. We are interested in the minimum cardinality of such a collection. In this talk, we will discuss properties of this invariant and its close connection to the minimum rank problem. This talk will be accessible to students. Joint work with Calum Buchanan and Christopher Purcell.

For Zoom information, please contact Sean at SEnglish (at) illinois (dot) edu.

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 16, 2021

Solution to Enflo's problem

Paata Ivanishvili (North Carolina State University)

Abstract: Pick any finite number of points in a Hilbert space. If they coincide with vertices of a parallelepiped then the sum of the squares of the lengths of its sides equals the sum of the squares of the lengths of the diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube {0,1}^n. In this case the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have parallelogram law. Back in 1978 Enflo asked: in an arbitrary Banach space if the sum of the squares of diagonals is bounded by the sum of the squares of its sides for all parallelepipeds (up to a universal constant), does the same estimate hold for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we positively resolve Enflo's problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points are called of Enflo type 2. In particular, we show that Rademacher type and Enflo type coincide.