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Colloquium

Mathematical Sciences colloquium

This colloquium is co-coordinated by Daniel Glasscock (email: Daniel_Glasscock@uml.edu), Amanda Redlich (email: Amanda_Redlich@uml.edu), and Joris Roos (email: Joris_Roos@uml.edu). Contact any of them if you would like to speak as part of this seminar.

Fall 2021

Distinct Dyadic Systems

  • Tess Anderson, Purdue University
  • September 8, 11 a.m. - Noon, Virtual Meeting

Abstract: In analysis and related fields, separating the real numbers into dyadic "chunks" is often used as a key tool in many proofs. Dyadic versions of objects define are often easier to analyze, and in many senses, can completely substitute for the continuous objects. To represent a continuous object as a dyadic one, we need a "distinct dyadic system". In this talk we completely characterize all distinct dyadic systems, motivating their study and underlining some interesting number theory lurking under the surface.


Byzantine Dispersion on Graphs

  • William K. Moses Jr., University of Houston
  • September 22, 11 a.m. - Noon, Virtual Meeting

Abstract: In this talk, we consider the problem of Byzantine dispersion and extend previous work along several parameters. The problem of Byzantine dispersion asks: given n robots, up to f of which are Byzantine, initially placed arbitrarily on an n node anonymous graph, design a terminating algorithm to be run by the robots such that they eventually reach a configuration where each node has at most one non-Byzantine robot on it. Previous work solved this problem for rings and tolerated up to n-1 Byzantine robots. In this paper, we investigate the problem on more general graphs. We first develop an algorithm that tolerates up to n-1 Byzantine robots and works for a more general class of graphs. We then develop an algorithm that works for any graph but tolerates a lesser number of Byzantine robots. We subsequently turn our focus to the strength of the Byzantine robots. Previous work considers only "weak" Byzantine robots that cannot fake their IDs. We develop an algorithm that solves the problem when Byzantine robots are not weak and can fake IDs. Finally, we study the situation where the number of the robots is not n but some k. We show that in such a scenario, the number of Byzantine robots that can be tolerated is severely restricted. Specifically, we show that it is impossible to deterministically solve Byzantine dispersion when \lceil k/n \rceil > \lceil (k-f)/n \rceil.


Fundamentally Semidistributive Lattices

  • Emily Barnard, DePaul University
  • October 6, 11 a.m. - Noon, Virtual Meeting

Abstract: A partially ordered set is a set together with a relation that allows us to compare some but possibly not all elements. In this talk we discuss a special class of posets called semidistributive lattices. Such lattices are ubiquitous in algebraic combinatorics, in the representation theory of quivers, and the study of polyhedra. Motivated by a classic theorem of Birkhoff which classifies all finite distributive lattice, we will discuss a recent classification of finite semidistributive lattice by Reading, Speyer and Thomas. The talk will be self-contained with many examples and connection to recent research projects. All are welcome.

de Finetti Lattices and Magog Triangles

  • Andrew Beveridge, Macalester College
  • October 20, 11 a.m. - Noon, Virtual Meeting

Abstract: The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. (These rules are a special case of de Fineti's axiom from probability.) We give a bijection from a family of poset refinements of $F_{n,2}$ to magog triangles. We then adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles. This talk is based on joint work with Ian Calaway (Stanford University) and Kristin Heysse (Macalester College).


Geometric Incidence Theory and Uniform Distribution

  • Ayla Gafni, University of Mississippi
  • November 3, 11 a.m. - Noon, Virtual Meeting

Abstract: The Szemeredi-Trotter Incidence Theorem, a central result in geometric combinatorics, bounds the number of incidences between n points and m lines in the Euclidean plane.  Replacing lines with circles leads to the unit distance problem, which asks how many pairs of points in a planar set of n points can be at a unit distance.  The unit distance problem breaks down in dimensions 4 and higher due to degenerate configurations that attain the trivial bound.  However, nontrivial results are possible under certain structural assumptions about the point set.  In this talk, we will give an overview of the history of these and other incidence results.  Then we will introduce a quantitative notion of uniform distribution and use that property to obtain nontrivial bounds on unit distances and point-hyperplane incidences in higher-dimensional Euclidean space.  This is based on joint work with Alex Iosevich and Emmett Wyman.


Hilbert Geometries and Entropy Rigidity

  • Dave Constantine, Wesleyan University
  • December 1, 11 a.m. - Noon, Virtual Meeting

Abstract: Any convex, bounded subset of Euclidean space can be equipped with a metric that reflects the shape of its boundary. This metric (or way of measuring distance) is simple to write down and there are many things you can do with it using only tools from linear algebra. The result is a Hilbert geometry, and these can be both familiar and novel. For instance, the familiar Euclidean and hyperbolic spaces can themselves be seen as specific examples of Hilbert geometries. On the other hand, there is a wide world of Hilbert geometries which are not Euclidean or hyperbolic and share some of the properties of these familiar spaces, but differ from them in others. In this talk I'll start with an introduction to Hilbert geometries accessible to anyone who has had some linear algebra. Then in the second half of the talk I will discuss some recent work (joint with Adeboye and Bray) on entropy rigidity for these geometries. In short, we show that examples which minimize volume entropy (a measurement of the geometric complexity of the geometry) must be very special - in fact they are the familiar hyperbolic space we'll encounter at the start of our discussion.