2024 U-M REU Seminars

Zizhao Ding and Jason Zheng: Sidorenko's Conjecture and Graph Homomorphism Density

Mentor: Daniel Altman

Sidorenko’s Conjecture states, roughly, that for all bipartite graphs H, the homomorphism density of H in a large graph G is asymptotically minimized when the edges of G are chosen randomly and independently. The conjecture is of significant interest in additive combinatorics, with many classes of bipartite graphs already known to be Sidorenko. In this talk, we survey Sidorenko’s Conjecture and related properties. We also present graph entropy and Razborov’s flag algebras as tools to study graph homomorphism density in general.

Chenglu Wang:  Representation Stability of Vertical Configuration Spaces, Part 1

Mentor: Jenny Wilson

In this first talk of the three-part series, we aim to make sense of every word that appears in the title of our project. Given a family of topological spaces or algebraic structures \{ X_n \}_{n \in \mathbb{N}}, one often wants to ask the following question: Does the (co)homology group (which I will informally define in the talk) stabilizes as n \rightarrow \infty? If so, we can just compute a small number of (co)homology groups and infer the rest through this stability! Such a phenomenon is called homological stability, and appears in many areas of mathematics. However, such stability is sometimes too good to hope for, leading Farb and Church to propose a more nuanced concept called representation stability in 2010. In this framework, although the (co)homology groups may not stabilize as groups, they exhibit stability “up to actions by symmetric groups S_n.” In this talk, we will introduce key concepts of representation stability, with a focus on illustrative examples from configuration spaces and a variant known as vertical configuration spaces. 

Chunye Yang: Representation Stability of Vertical Configuration Spaces, Part 2

Mentor: Jenny Wilson

In the second talk of the three-part series, I will introduce the underlying algebraic structure that accounts for the representation stability phenomenon defined in Chenglu’s talk, called FI-modules. I will explain how FI-modules are essentially a nice-behaving sequence of   representations, and they are algebraically analogous to R-modules. Then, I will give several concrete examples of this FI-module structure, and then explain the notion of finite generation of FI-modules, which turns out to be essential in proving representation stability. 

David Baron: Representation Stability of Vertical Configuration Spaces, Part 3

Mentor: Jenny Wilson

In the third and last talk on representation stability of vertical configuration spaces, we introduce the concept of the wreath product S_d \wr S_n and show how to construct a complete list of irreducible representations. Bianchi and Kranhold in their introduction of vertical configuration spaces showed that there is a natural action of products of wreath products on the space by permuting “clusters” and “particles”. This action descends to the (co)homology groups. Our aim is to show that the sequence of (co)homology groups under this action has an FI_G-module structure (an extension of the classical FI-module) and prove that the sequence stabilizes in the representation stability sense. Thus, it is essential to understand the structure of the wreath product groups and their irreducible representations.

David Cates and Kevin Le: Gamma Factors and Bessel Functions of GL2

Mentor: Elad Zelingher

The representations of GLn and various matrix subgroups are an important class of objects of study in the work on the Local Langlands correspondence. In this talk, we will give a simple example of this, working in finite fields instead of local fields, and show some tools which are helpful in the classification of representations.  

Bohang Yu: Classifying Uncommon Systems of Two Linear Equations

Mentor: Daniel Altman

A linear system of equation is called common if as n → ∞, any 2-coloring of F_p^n gives asymptotically at least as many monochromatic solutions to L as a random 2-coloring.
In the paper from Dingding Dong, Anqi Li and Yufei Zhao, they classified all 2-by-5 equations except for two special cases.
My presentation is going to summarize their idea and technique on this classification and brought up some new ideas regarding this topic.

Andy Chen: Phase representations of population distributions in dissipative dynamical systems

Mentor: Daniel Forger

The asymmetric particle population density method allows for simulation of the evolution of a probability density with coupling and noise in a dissipative dynamical system. Such systems attract the population over time towards a stable limit cycle, provided that the coupling is weak. We can then consider the mapping from a limit cycle point to its corresponding phase. This allows for a simpler asymptotic representation of the population's dynamics in terms of phase instead of space.

Sean Ku and Yuyang Wang: Geodesic and sympelectyic reduction of SE(2)

Mentor: Alejandro Bravo Doddoli

We are going to introduce the sympelectic reduction of cotangent bundle at first. Then, for the specific cotangent bundle T*SE(2), we will explain that there is a sypmelectic map between T*SE(2)//R^2 and T*SE(2)//SE(2)

Mitchell Godek: Enhancing Heat Transfer in a Channel by Unsteady Flow Perturbations

Mentor: Silas Alben

Understanding how to maximize heat transfer in a heated channel is vital for designing energy efficient processes, such as those in data centers or chemical reactors. In this talk, we will explore the case of a 2D rectangular channel with time periodic, incompressible unsteady flows constructed from a previously derived optimal steady flow. We will discuss how to construct these flows, and the current progress towards finding unsteady flows that maximize heat transfer for a fixed time-averaged power.

Hao Pan: Gravitational Stability of a Rigid Ring and a Sphere

Mentor: Tony Bloch

In celestial mechanics,  it is interesting to make the stability analysis of the orbital equilibria of a solid ring-planet system at certain spin rate. This is a classical problem which can be traced back to Lagrange and Maxwell.  We will construct the ideal model of rigid body system of Ring+ Sphere to study this problem mathematically. In this talk, I will follow the "Amended Potential method" raised from D.J. Scheeres' paper to find  the relative equilibria of several formulations. I will first give a new proof of the classical results already proven by Maxwell . If time permits, the new orbital equilibria would be discussed. 

Julia Hastings: Modeling the Phase Separation of Polymers

Mentor: Robert Krasny

The phase separation of polymers is an important process in chemical engineering. We begin by discussing Fick's law and the diffusion equation using Fourier Series and the finite-difference method. This is followed by an analysis of the Allen-Cahn equation, which is a phase field model combining the effects of diffusion and the attraction/repulsion of two kinds of polymers. 

Isaac Viviano: Comparison of Continuum and Particulate Models for Chemical Diffusion and Phase Separation

Mentor: Robert Krasny

We model the physical phenomena of diffusion and phase separation. Differential equations provide a continuum approximation of these processes. The heat equation models diffusion of a spatial concentration gradient. The Allen-Cahn and Cahn-Hilliard equations model phase separation of a binary mixture into its components. Each differential equation is approximated using the finite difference method. For particle-based models, the software LAMMPS is used to simulate molecular dynamics. The diffusive system study was an ideal gas diffusing into a vacuum. For phase separation, a two-component Leonard-Jones fluid was modeled. The qualitative behavior of the time evolution of the models is compared. Additionally, the differential equation parameters are estimated from molecular dynamics data to quantitatively connect the timescales.

Max Natonson and Rohan Wadhwa: Eigenvalue Spectra and their Sparsest Matrix Representation

Mentor: Ben Gould

The Jordan Normal Form is a useful way to write any finite-dimensional linear operator as a matrix, containing all the information about your operator's eigenvalues and their multiplicities. In particular it is a very sparse matrix representation. In this presentation, we will explore the question "What is the sparsest non-diagonal matrix with a given spectrum of eigenvalues, algebraic, and geometric multiplicities?". For a spectrum with 3 eigenvalues the Jordan Normal Form is the sparsest matrix representation, but things get more complicated when we have 4 or more eigenvalues. We will discuss a bound on how sparse the matrices we're looking for can be, some classes of matrices which are sparser than their Jordan Normal Form, and how large the gap between the number of zeros in a matrix and its Jordan Normal Form can be. Finally we will say a few words about the case in which zero is an eigenvalue, and where our research is headed. 

Marcin Sobotka: Singular Values and Dynamic Mode Decomposition

Mentor: Tony Bloch

Karina Dovgodko and Jack Westbrook: Hilbert—Kunz Multiplicity in a Two-Parameter Family

Mentor: Austyn Simpson

The Hilbert—Kunz multiplicity is a numerical invariant used to measure singular points on varieties over a field of prime characteristic. Introduced by Monsky in 1983, this invariant is defined in terms of the asymptotic behavior of the Frobenius endomorphism and is notoriously difficult to compute. In this talk we present an example exhibiting that the Hilbert—Kunz multiplicity may vary in a two-parameter family.

Emma Cardwell: Toric Structure in Colored Gaussian Graphical Models

Mentor: Aida Maraj

Given a colored graph G on $n$ nodes, we consider the linear space $\mathcal{L}_G$ of symmetric $n$ by $n$ matrices with linear conditions on the entries given by symmetries in the graph. This parameterizes the concentration matrices of the colored Gaussian graphical model given by G. We investigate when the space of inverses of these matrices is toric, possibly after a linear transformation. Our techniques are motivated by Laplacian transformations in Brownian motion tree models.

Amer Goel: Shortest Path Polytopes Arising from Graphical Models

Mentor: Aida Maraj

Here we investigate polytopes of toric statistical models. Our polytopes are paramtereized by pathways on a graph, so the vertices of the polytope are indicator vectors of the graph's shortest paths between nodes. We would like characterize some properties of these polytopes, like their halfspace descriptions, as well as find isomorphisms between these and better-known polytopes. This will allow us to understand the combinatorial structure of these polytopes, and connect them to other polytopes in the literature. We hope to provide characterizations for star graphs, path graphs, and their toric fiber products. 

Andrew Meyer: Patient-specific computational modeling and missing data imputation for phenotyping cardiovascular disease

Mentor: Indika Rajapakse and Dan Beard

Extensive data available in electronic health records have the potential to guide improved disease prognosis and treatment. However, available data records are messy—key measurements can be contradictory, redundant, incorrect, or totally missing. Some of these problems may be addressed by imputing the missing values in a rigorous fashion. Yet, even a complete set of measurements does not resolve data inconsistencies, nor would the availability of complete data alone be sufficient to inform care in many cardiovascular disease cases. Patient-specific cardiovascular computational modeling, which also depends on having a sufficiently complete input dataset, can resolve these inconsistencies and provide phenotypic insight beyond what is described by the raw clinical data, potentially informing diagnostic and prognostic decisions for physicians. 

Sabine Chu, Christine Gao, and Ananya Prasanna: Nonhyberbolic Cayley graphs of the free group

Mentor: Alex Wright

The free group on n generators is a basic, but highly fundamental object in group theory. In particular, we can associate it to a Cayley graph structure with an infinite generating set. We will prove a geometric fact about this Cayley graph: namely, there exist lattice-like structures in it, meaning it has infinite asymptotic dimension.

Daniel Cross: Rough Isometries of Nilpotent Groups

Mentor: Yanlong Hao

Rough isometries generalize the notion of isometries between metric spaces, and imposing an equivalence relation on the set of rough isometries from a metric space X to itself gives rise to a group RI(X), generalizing the isometry group Isom(X). In geometric group theory, we can give finitely generated groups a metric associated with a fixed generating set, called the word metric, and in many cases the structure of RI(G) for such a metric is not known. In this talk, I examine the problem of classifying RI(G) for finitely generated nilpotent G in greater detail, and introduce some constructions which may be useful in solving this problem.

David Cheng, Beinuo Guo, and Haran Mouli: The Realizability of Saturated Transfer Systems by Linear Isometries Operads

Mentor: Bar Roytman

A complex G-universe is a complex G-representation with countably infinite dimensional isotypical components. The realizability problem for linear isometries operads studies the stabilizer groups on spaces of linear isometries between complex G-universes. We explain how to reduce this to a problem in combinatorics, provide new proofs for the case of odd cyclic groups, and our progress for even cyclic groups.

Nicole Vuong: Optimizing unsteady wall-to-wall heat transfer

Mentor: Silas Alben

The efficiency of many real-life appliances relies on its ability to cool effectively. Thus, optimizing heat transfer in fluids via advection (the movement of a fluid) is a point of interest. In this talk, we explore the effect of adding unsteady perturbations to a previously derived optimal steady flow, with a fixed time-averaged power, on the heat transfer in a 2D channel. We also discuss current progress towards finding optimal unsteady flows that maximize heat transfer in the 2D channel via the genetic algorithm.

Isabel Trindade: Non-Axiomatizability of Categories in Finitary Logic

Mentor: Ronnie Chen

Lieberman et al. (2022) show that the category of Hilbert spaces is not axiomatizable in any finitary logic. This result is attained by proving there is no faithful, directed-colimit preserving functor from the category of HIlbert spaces into the category of sets. The result follows from a previous result that every category axiomatizable in finitary logic admits a faithful, directed colimit-preserving functor into Set. Here, we attempt to apply the same method to determine the axiomatizability or non-axiomatizability of other related categories.