9/12/24
Kean Fallon
Title: MsGT: Multi-scale Geometry and Topology Feature Extraction for Point Clouds
Abstract: In recent years, LiDAR (Light Detection And Ranging) devices are being used more and more frequently in industrial applications, including automated driving, quality control, and urban planning. These systems collect information as point clouds, or collections of points in Euclidean space. While the human eye may be able to discern what image is behind a point cloud, without help a computer cannot. So we provide help by extracting various mathematical measurements, or features, from a point cloud. The choice and arrangement of these features is key to the computers ability to process this information accurately. This talk introduces a new feature extraction method, abstinent of deep learning, that provides good accuracy in point cloud classification tasks. This research is ongoing and was thanks to a summer spent in the GRIPS Sendai program.
9/19/24
Sydney Miyasaki
Title: The Miracle of Quasirandom Graphs
Abstract: We will define the notion of a "random-like" graph. We will also discuss the astonishing robustness of this concept, and a remarkable application of this notion to a seemingly intractable problem.
9/26/24
Chad Berner
Title: Frames and applications to Fourier series
Abstract: In the Hilbert space setting, an orthonormal basis is a rigid structure that allows for stable reconstruction of all vectors. However, it is often the case in practice that this structure is too rigid to deal with noise and error. Additionally, constructing an orthonormal basis in general can be computationally challenging. Frames on the other hand, are more forgiving in signal recovery because of their redundancy. We will study frames as a relaxion of an orthonormal basis and try to understand their most important approximation properties. Furthermore, we apply this theory to Fourier approximation and even consider Fourier approximations that are more relaxed than frame approximations.
10/3/24
Tony Vuolo
Title: Everyone’s Special: All-Kings Digraphs
Abstract: A “king” in a tournament T is a vertex from which every other vertex in T is reachable by a path of length at most 2. This definition can be easily extended to simple digraphs. We call an “n-court” any n-vertex digraph where all vertices are kings. Define Υ(n) = min{|E(C)| : C is an n-court}. In this talk, we show that there is an n-court for all n ∈ Z +\{1, 2, 4} and provide upper bounds on Υ(n).
10/10/24
Alice Kessler
Title: Propositions as Types
Abstract: There are several foundations of math, easily the most accessible of which being set theory. Other foundations such as Topos Theory have their advantages and disadvantages. We will discuss a third foundation; Type Theory. There are many variants of type theory, we will use a version of Intensional Martin-Löf Intuitionistic type theory. This version ends up providing a rich logical theory in which we consider types as propositions, and inhabitants of a type as a proof of that proposition. This outlook has many uses. For example, it is often used in proof checkers, like the languages Lean and Coq. This is due to the fact that verifying a proof is formalized as type checking, something easily done in a computational manor. We will introduce the basic idea of type theory, some basic types and constructions, and then do some basic proofs to get a feel for it.
10/17/24
Split Talk (25 mins each)
Alex Parker
Title: New bounds on a generalization of Tuza's conjecture
Abstract: We will discuss an old conjecture of Tuza about packing triangles in graphs, generalize the conjecture to hypergraphs, and prove some small cases of the conjecture.
Coy Schwieder
Title: Proper rainbow saturation for cycles, paths, and cliques
Abstract: Given a graph $H$, we say that a graph $G$ is \emph{properly rainbow $H$-saturated} if there exists some proper edge-coloring of $G$ which has no rainbow copy of $H$, yet the addition of any edge $e$ to $G$ forces any proper edge-coloring of $G + e$ to have a rainbow copy of $H$. We denote the minimum number of edges in a properly rainbow $H$-saturated graph on $n$ vertices by $\rsat(n,H)$. In this talk, we discuss new bounds on the proper rainbow saturation number for odd cycles, paths, and cliques.
10/24/24
No Talk (Graduate Student Forum)
10/31/24
Split Talk (25 minutes each)
Alex White & Micky Santiago-Zayas
Title: An Introduction to Orbifold Notation
Abstract: Roughly speaking, an orbifold (short for “orbit manifold”) is the quotient of a manifold by a discrete group acting on it. The orbifold notation (or orbifold signature) is a notational system which encodes topological information about a symmetry pattern’s corresponding orbifold. The groups which can be represented by the orbifold notation include the point groups on a sphere, the frieze groups, wallpaper groups of the Euclidean plane and their analogues in the hyperbolic plane. We give a brief history of the invention of the notation and discuss the orbifold notation in general with a particular emphasis on the orbifold notation for wallpaper groups of the Euclidean plane. We also prove Conway’s Magic Theorem for plane repeating patterns and how it can be used to identify the orbifold signature for wallpaper groups of the Euclidean plane.
11/7/24
Kimberly Hadaway
Title: Research Opportunities and Resources at Iowa State
Abstract: In this presentation, we share some ways to get involved with research in small groups in the larger mathematical community (including free trips and collaboration funding), and we also highlight ways to get support at Iowa State through different campus offices.
11/14/24
Micah Coats
Title: Subharmonic in the distributional sense implies mean value subharmonic
Abstract: Subharmonic functions can be equivalently defined in a mean value sense and using the usual definition that -Δu≤0 for twice differentiable u . However, these definitions can be extended by integrating a function u against the Laplacians of non-negative test functions and defining the result as the distributional Laplacian of u. In this proof by Caffarelli, it is shown that subharmonic functions in the distributional sense are also equivalent to subharmonic functions in the mean value sense.
11/21/24
Billy Duckworth
Title: An Ultrafilter Proof of Hindman's Theorem
Abstract: A subset of the natural numbers is called an IP set if it contains an infinite set that is closed under taking sums of distinct elements. Hindman's Theorem states that any finite coloring of the naturals will have a monochromatic IP set. We will go over some basic properties about filters before using them to prove the theorem in a relatively straightforward way.
11/28/24
No Talk (Holiday)
12/5/24
Enrique Gomez-Leos
Title: Tiling randomly perturbed dense multipartite graphs
Abstract: Many results in extremal graph theory concern minimum degree conditions that guarantee the existence of a spanning subgraph, for instance, a perfect tiling in a host graph G by a fixed subgraph H. Recall that the Erdős-Rényi random graph G_{n,p} consists of the vertex set [n] where each edge is present, independently, with probability p=p(n). In this regime, a key question is to determine the threshold for which G_{n,p} contains a perfect H-tiling. In 2003, Bohman, Frieze, and Martin introduced the randomly perturbed graph model which connects the two questions together. In this talk we discuss a multipartite variant of this graph model and determine the threshold for a perfect clique tiling. Those interested in applications of the Regularity Lemma are encouraged to attend! This is joint work with Ryan R. Martin.
12/12/24
Split Talk (25 mins each)
Md Zulfiqur Haider
Title: A Robust Finite Differences-Based Scheme for Reliable PDE Model Reduction
Abstract: The search for robust finite-dimensional model reductions for PDE models highlights significant limitations in widely used computational techniques. Standard methods like Finite Differences and Finite Elements, when applied without careful consideration, often introduce spurious unobservable (indistinguishable) high-frequency vibrational modes into the dynamics. As a result, these reduced models fail to retain exact observability uniformly as the discretization parameter approaches zero. Traditionally, these spurious modes are addressed using “direct Fourier filtering” or “indirect filtering” methods, which add complexity to the process.
In this talk, we present a new Finite Differences-based scheme that addresses these challenges directly. By utilizing equidistant grid points and averaging operators, this approach eliminates the need for filtering, providing a simpler, more reliable, and mathematically rigorous alternative. We will demonstrate how this new method outperforms the standard Finite Differences approach and enhances control design and implementation.
Sycamore Herlihy
Title: Independent sets in trees
Abstract: Given a graph G, a subset S ⊆ V(G) is an independent set if no two vertices in S are adjacent in G. We let i(G) denote the number of independent sets in G. We will calculate i(G) exactly for some specific classes of trees, and discuss for which positive integers n we can construct a tree T with i(T)=n.
1/30/25
Zachary Brennan
Title: Tiling randomly perturbed dense multipartite graphs
Abstract: Probabilistic zero forcing (PZF) is a graph coloring process in which blue vertices infect (color blue) white vertices with a probability proportional to the number of neighboring blue vertices. This talk introduces reversion probabilistic zero forcing (RPZF), which shares the same infection dynamics as PZF but also allows for blue vertices to revert to being white in each round. We establish a tool which, given a graph's RPZF Markov transition matrix, calculates the probability that the graph turns all white or all blue as well as the time at which this is expected to occur. For specific graph families we product a threshold number of blue vertices for the graph to become entirely blue in one step with high probability.
2/6/25
Sydney Miyasaki
Title: Rainbow Turan Numbers of Brooms
Abstract: How many edges can a graph have if certain sub-structures are forbidden? This is a central question in extremal combinatorics, often called a Turan-type problem. In this talk, we will look at Turan-types problems when the additional structure of edge-coloring is added. We will focus on a family of fairly simple sub-structures called brooms. Along the way, we will find various pleasing connections to and applications of group theory.
2/13/25
Enrique Gomez-Leos
Title: Tilings in randomly perturbed dense multipartite graphs
Abstract: Many results in extremal graph theory concern minimum degree conditions that guarantee the existence of a perfect tiling in a host graph $G$ by a fixed subgraph $H$. Recall that the Erdős-Rényi random graph $G_{n,p} consists of the vertex set $[n]$ where each edge is present, independently, with probability $p=p(n)$. In this regime, a key question is to determine the threshold for which $G_{n,p}$ contains a perfect $H$-tiling. In 2003, Bohman, Frieze, and Martin introduced the randomly perturbed graph model which connects the two questions together. In this talk we discuss a multipartite variant of this graph model and determine the threshold for a perfect clique tiling. This is joint work with Ryan R. Martin.
2/20/25
Matt Burnham
Title: Some extremal properties of strong tournaments
Abstract: A tournament is an orientation of the complete graph; in other words, a tournament is a directed graph with exactly one edge between every pair of vertices and no other edges. A directed graph is called strong (or strongly connected) if there is a directed path from every vertex to every other vertex. A score sequence of a tournament is a list (s_1,s_2,...,s_n) of outdegrees of the vertices. We explore two extremal problems. (1) Imbuing the set of strong score sequences with the lexicographical ordering, which strong tournaments appear at the extremes? (2) Which strong tournaments have the maximum/minimum number of 3-cycles? This talk will be accessible to graduate students of all backgrounds (not just the combinatorists).
2/27/25
Thomas Griffin
Title: Icosystems – Modeling a Swarm
Abstract: Swarms are collections of loosely coupled distinct agents each following "simple" rules. Swarms do not use central coordination, and individual agents need not be aware of the swarm’s overall function/goal. As each agent performs its task, the swarm collective can display unusual and unexpected emergent behaviors. For those swarms defying analytic evaluation, simulation remains as the only method to reveal these emergent behavior. A number of swarm behaviors will be demonstrated with different simple rules to follow, and then combined to discuss some more interesting dynamics and real world applications.
3/13/25
Coy Schwieder
Title: Conflict-free hypergraph matchings and odd Ramsey numbers
Abstract: Given a graph $G$ and a subgraph $H$, an $H$-odd coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ (not necessarily induced) has an odd color class. Let $r_{odd}(G,H)$ be the minimum number of colors needed for an $H$-odd edge-coloring of $G$. This notion of odd Ramsey theory was introduced in 2023 by Noga Alon in relation to what he termed ``Graph Codes", and has since been gaining interest. In this talk, we discuss the relations between these odd Ramsey numbers, graph codes, and the Erd\H{o}s-Gy\'arf\'as function, recent improvements to the Hypergraph Matching Method and how it can be used to obtain bounds on odd Ramsey numbers, and extend recent results on $r_{odd}(K_{n,n}, K_{2,2})$ by Bennett, Heath, and Zerbib. Indeed, we use the hypergraph matching method to show $r_{odd}(K_{n,n}, K_{2,t}) \leq \frac{n}{t} + o(n)$ and $r_{odd}(\mathcal{K}_{n, \dots, n}^{(k)}, \mathcal{K}_{1,\dots, 1, 2,2}) \leq \frac{n}{2} + o(n)$.
3/20/25
No Talk: Spring Break
3/27/25
Micah Coats
Title: The Nikodym Metric Space: treating sets as points in a metric space.
Abstract: If (X, M, µ) is a finite measure space, then a metric on sets can be defined by p(A,B)= µ(AΔB) (where AΔB is the set difference of A and B). This so called Nikodym Metric is complete and it turns finite absolutely continuous measures with respect to µ into uniformly continuous functions with respect to the Nikodym metric. An application of the Baire category theorem gives the Vitali-Hahn-Saks Theorem, a result remarkably similar to the continuity of pointwise limits of uniformly bounded linear operators.
4/3/25
Billy Duckworth
Title: An Ergodic Theory Proof of Szemeredi's Theorem
Abstract: Szemeredi's theorem states that any set of natural numbers that has positive density contains arithmetic progressions of arbitrary lengths. He proved it by an "exceedingly intricate but elementary" combinatorial argument; later, Furstenberg used tools from ergodic theory to formulate a shorter (but not necessarily easier) proof.
4/10/25
Nick Layman
Title: A Brief Introduction to Flag Algebra
Abstract: Flag algebras were developed in 2007 and provide a novel way to solve many extremal combinatorics problems. Primarily, flag algebras allow us to investigate the maximum subgraph density of a fixed subgraph in a given family. The simplest example of this is maximizing the number of edges among triangle free graphs. We provide context for related areas where flag algebras have already been useful, then describe the rigorous underpinnings, and ultimately attempt to impart some intuition of how to work with them.
4/24/25
Conor Thompson
Title: Canonical Ramsey theory and size canonical Ramsey on small bipartite graphs
Abstract: Erdős and Rado developed the concept of “canonical colorings” to extend Ramsey’s Theorem to situations involving an infinite number of colors. We provide a brief overview of the basics of canonical Ramsey theory, discuss its application to graph-coloring problems, and present some results (and conjectures) involving bipartite graphs in which one half is small.
5/1/25
Kean Fallon
Title: Equiangular Tight Frames in Real Symplectic Vector Spaces
Abstract: Equiangular line packings in real and complex inner product spaces present a mathematical structure that has gathered significant interest in recent years. Zauner's conjecture, which states that a maximal packing of $d^2$ complex lines exists in $\mathbb{C}^d$ for every dimension $d$, is considered one of the most important problems in quantum information theory and may be related to Hilbert's 12th problem. In this talk, we develop the theory of equiangular lines over real symplectic space. In particular, we show that when ported to the symplectic setting, Zauner's conjecture is equivalent to the Skew Hadamard conjecture.
5/8/25
Laura Gamboa Guzman
Title: Formalizing Time: Temporal Logics and the Challenge of Visualizing MLTL
Abstract: Temporal logics are a family of modal logics that reason about timelines. They are usually obtained by expanding classical propositional logic with modal operators that can qualify the value of a proposition over time, such as "p will always be true" or "q is true until p becomes true." However, different concepts of time are often captured by significant logical systems, as these tend to encode the various characteristics that define them, such as continuous vs. discrete time and linear vs. branching time. The use and development of these logics have been increasing significantly over the last 50 years, as researchers and engineers in fields related to computer science have been using them to verify safety-critical systems in a formal and precise manner.
In this talk, I will introduce some of the better-known temporal logics that aim to formalize different concepts of time and briefly explain the different properties that make them good candidates for use in different computer-based environments. After that, I will focus on a logic I have been working on during my PhD known as Mission-time (Linear) Temporal Logic (MLTL), which is a logic that reasons about finite and discrete timelines (called traces) where finite intervals bound the temporal operators. Although MLTL is only as expressive as classical propositional logic, it has been capturing the attention of multiple research groups in recent years, and its succinctness has shown to become a challenge for engineers easily when trying to validate the formulas they believe are capturing the desired behaviors. For that, at the Laboratory for Temporal Logic at ISU, we have been developing algorithms that allow us to take an MLTL formula and produce a visual representation for the traces that satisfy the formula.