Schedule of Talks

Title: An Introduction to IPMs

Abstract: Integral projection models (IPMs) are a type of population model which characterizes a population’s distribution according to a continuous measurement, such as size. To do this vital rates, functions describing growth, survival, and reproduction, are modeled via regressions on individual level data. The regressions are then combined into a kernel function which is used to ‘project’ what the future population distribution will be with respect to the chosen measure. In this talk we will construct and analyze an IPM using data from a Soay sheep population. Additionally, we will discuss the benefits of IPMs in comparison to more historically used population models, matrix projection models (MPMs). Further, we will cover the current research directions of IPMs.

Title: Multishot Capacity of Adversarial Networks

Abstract: Adversarial network coding studies the transmission of data over networks affected by adversarial noise. In this context, the noise is modeled by an omniscient adversary who designs their attacks, but is restricted to corrupt a proper subset of the network edges. We assume that the adversary has full knowledge of the network topology. A combinatorial framework for adversarial networks was introduced by Ravagnani and Kschischang in 2018 and the study was recently furthered by Beemer, Kilic and Ravagnani, with particular focus on the one-shot capacity. The one-shot capacity measures the maximum number of symbols that can be sent in a single use of the network without errors. We present the problem of studying the capacity of a network in multiple transmission rounds for different assumptions on the adversarial model and some bounds on the multishot capacity in this direction.

Title: Optimal Tilings with the Minimal Tiling Property

Abstract: A tiling of the unit square is an MTP tiling if the smallest tile can tile all the other tiles. We look at the function f(n)= max (sum s_i) , where s_i is the side length of the i-th tile and the sum is taken over all MTP tilings with n tiles. There are good conjectures for the answer, but the problem appears to be difficult. If n = k^2+3, it was conjectured that f(n) = k+1/k. Using electrical network theory we show that in such an optimal tiling the smallest tile cannot be too small. Next, we show that any tiling that violates the conjecture must consist of at least three tile sizes and has exactly one minimal tile.

Title: A lower bound on the failed zero-forcing number of a graph.

Abstract:  Given a finite simple graph G=(V,E) and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set S is a zero-forcing set if filling S and applying all possible instances of the color-change rule causes all vertices in V to be filled. A failed zero-forcing set is a set of vertices that is not a zero-forcing set. Given a graph G, the failed zero-forcing number F(G) is the maximum order of a failed zero-forcing set. Fetcie, Jacob and Saavedra (Involve 8:1 (2015), 99–117) asked whether given any k there exists an ℓ such that all graphs with at least ℓ vertices must satisfy F(G)≥k. I will be presenting the answer to this question: for a graph G with n vertices, F(G)≥⌊(n−1)∕2⌋.

Title: The Discrete Spherical Maximal Function

Abstract:  Spherical averaging operators arise naturally in many settings, including PDEs and distance problems. One can ask what happens with spherical averaging operators in the discrete setting of the integers. This was first introduced and studied by Magyar-Stein-Wainger, who proved the l^p to l^p boundeness of the discrete spherical maximal function. Recently, Lyall, Magyar, Newman, and Woolfit published a new direct proof of the boundedness of the same function from l^2 to l^2 that does not rely on abstract transference theorems or delicate asymptotics for the Fourier transform of discrete spheres.  This presentation will include some background and motivations for the discrete spherical maximal function, and then will provide a brief overview of the proof by Lyall, Magyar, Newman, and Woolfitt of the boundedness of the discrete spherical maximal function.

Title: Approximate Deconvolution Leray Reduced Order Modeling

Abstract:  In the field of reduced order modeling, filtering methods have received significant recent attention for treating poorly performing models as a method of regularization. However, filtering by itself can be susceptible to over-smoothing of generated solutions. We propose new reduced order models that integrate the methods of approximate deconvolution with a goal of balancing accuracy and smoothing properties. The models are tested with particular application to fluid flows simulated by the Navier-Stokes equations.

Title: Structural and Practical Identifiability of Acute Viral Infection Models

Abstract: Structural and practical identifiability are important model features that ensure unique estimation of model parameters and replicability of modeling conclusions and predictions. Increasing biological realism in a model often increases model complexity and the size of the model parameter space, which, in turn, can affect model identifiability. We consider four within-host models of acute infections used to describe influenza A virus dynamics in mice. We perform structural identifiability analysis using a differential algebra approach for three different softwares, DAISY, StructuralIdentifiability.jl, and COMBOS. We then fit each model to either viral load or both viral load and CD8+ T cell data and calculate goodness of fit for each model. Finally, we implement profile likelihood analyses to determine practical identifiability. In the case of influenza A in mice, we find that data-av

Title: Graphs with Many Hamiltonian Paths

Abstract: A graph is hamiltonian-connected if every pair of vertices can be connected by a hamiltonian path, and it is hamiltonian if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio of pairs of vertices connected by hamiltonian paths to all pairs of vertices approaches 1. We then consider minimal graphs that are hamiltonian- connected. It is known that any order-n graph that is hamiltonian-connected must have ≥ 3n/2 edges. We construct an infinite family of graphs realizing this minimum.

Title: Quantum Popcorn Solves All Our Problems

Abstract: In 2023 QuEra unveiled Aquila, the first publicly available neutral atom quantum computer. Aquila performs computations using the geometry of atoms positioned on a plane and is in fact a natural solver for the maximal weighted independent set problem restricted to unit disk graphs (UDG-MWIS). Nguyen et al. (arXiv:2209.03965) developed a framework for using Aquila to solve MWIS for graphs with arbitrary connectivity. We extend their techniques to create an efficient encoding onto Aquila of the quadratic assignment problem (QAP), an NP-hard combinatorial optimization problem and generalization of the well-known traveling salesman problem.

Title: A Novel Insertion Algorithm: Mom Says We Have Sagan-Worley at Home

Abstract:  In this talk, we will define a new algorithm that constructs tableaux from permutations. We will discuss all of the necessary background knowledge, including similar preexisting insertion algorithms like RSK and Sagan-Worley. The new algorithm is much more like Sagan-Worley,as they both work with strict shifted partitions and create tableaux with numbers that may be marked. Much of the background information from this talk is from "Schensted Algorithms for Dual Graded Graphs” by Sergey Fomin.

Title: A Novel Insertion Algorithm: Mom Says We Have Sagan-Worley at Home

Abstract:  In this talk, we will define a new algorithm that constructs tableaux from permutations. We will discuss all of the necessary background knowledge, including similar preexisting insertion algorithms like RSK and Sagan-Worley. The new algorithm is much more like Sagan-Worley,as they both work with strict shifted partitions and create tableaux with numbers that may be marked. Much of the background information from this talk is from "Schensted Algorithms for Dual Graded Graphs” by Sergey Fomin.

Title: Lagrangian Grassmannian Computations using Checkerboards

Abstract: A major concern in Algebraic Geometry lies in understanding the multiplication of two Schubert Classes, which describe intersections of Richardson Varieties. In 2003, Ravi Vakil provided a combinatorial description of these multiplications. Vakil also provided a bijection between common flag degeneration methods for the Grassmannian and 'checkerboard games.' These proved useful for calculation and enhanced our geometric understanding of these intersections, allowing for many adjacent discoveries. It remains to determine if such a bijection exists in the Lagrangian Grassmannian (LG) case. In this talk, we describe some adapted rules for checkerboards arising in the LG case. We also present results from a variety of computations and an analysis of relevant invariants. This work draws heavily from Dr. Leonardo Mihalcea's conjectures about degenerations of flags in the Lagrangian Grassmannian.

Title: Gray bats

Abstract: Gray bats are a social species that use echolocation and fly in groups. Transfer entropy (TE) is a quantitative tool used in information theory to determine whether information from a source time series, X, is transferred to another time series, Y, given what we know about the past of Y. This research utilizes TE to determine whether the number of gray bats emerging from a wild colony at a given time, with and without the presence of introduced obstacles, influences different properties of the calls heard at the next time step. These properties include the number of calls per bat and the power (in decibels) used per bat at different frequencies.

Title: Sparse Approximate Inverse Smoother for Multigrid Preconditioning in GPU-Accelerated Poisson Solvers

Abstract: The p-multigrid preconditioner with a Chebyshev-accelerated Jacobi smoother has been proved as effective for high-order finite element discretizations. However, it still requires a significant number of iterations when applied to highly deformed meshes. The Sparse Approximate Inverse (SAI) preconditioner has been shown to effectively accelerate the convergence of iterative methods, with its construction being inherently parallel. In this work, we apply the SAI as a smoother within the p-multigrid preconditioning. We focus on three aspects: choosing an appropriate sparsity pattern for the SAI; coloring nodes to segregate them into several sets and solving a number of least squares problems simultaneously using the matrix-free operator; and implementing a thresholding strategy to reduce the number of nonzero entries of SAI. We evaluate the performance of the smoother on highly deformed meshes using GPUs. Our numerical results demonstrate the effectiveness of this approach.

Title: How to Succeed in Graduate School

Abstract: This talk focused on academic and mental health resources available to graduate students at VT and how to have a work-life balance.

Title: Professional Development Opportunities in Math

Abstract: This talk focused on professional development opportunities for VT graduate students and what is needed for future job applications. 

Title: Counting Pattern Avoiding Permutations by Big Descents

Abstract: One can define various statistics on permutations and study the distribution of these statistics over the symmetric group. Moreso, one may seek to describe the distribution of a permutation statistic over a set of restricted permutations which avoid predetermined patterns. In this talk, I will present a bijective proof of an equidistribution result regarding the bdes and pk statistics over 231-avoiding permutations. Further, I will highlight our results in finding the distribution of big descents over sets of permutations that avoid one pattern of length three and pairs of patterns of length three.

Title: An APOS Understanding of Complex Exponents

Abstract: Matt Park will discuss the results of a pilot study of his dissertation. His dissertation concerns how students taking complex analysis leverage their knowledge about the operations of real-valued exponents and logarithms to reason about the complex case. He will present findings related to responses to one question from his data. 

Title: Lattice-Based Cryptosystems for Resource-Constrained Devices

Abstract: Lattice-based cryptography is an area of post-quantum cryptography based on multi-dimensional lattices, and the difficulty of solving two well-studied hard problems: Shortest Integer Solution Problem (SIS) and Learning With Errors Problem (LWE). A key advantage of lattice-based cryptography over other types of post-quantum cryptography is the speed of lattice-based algorithms and reasonable key sizes. This makes lattice-based cryptography useful for key generation on resource-constrained devices, such as RFID tags, smart cards, and sensors. I will highlight current research in applying lattice-based cryptography to these types of devices, and where future work can be done to design lattice-based cryptosystems that are both secure and efficient.

Title: "Where there's symmetry, there are symmetric polynomials"

Abstract: My research is in symmetric polynomials, so that is what I will introduce the audience to in the talk. Even if you think they are cool, you might have a nagging feeling that they appear out of nowhere. To remedy that, I will be taking an unconventional route and talk about some topics in math where symmetric polynomials appear naturally. In such cases, the polynomial is often "a record" of the symmetry of the object. There will be Rubik's cubes, Viete formulae and the slaying of the evil cousin, anti-symmetrization. This will be a bored board talk and audience participation will be forever cherished :)

Title: Manifestations of Topology via the Euler Characteristic

Abstract: In this talk we explore different situations in which the global topology of a space manifests itself in unexpected ways. In particular, we will focus on a few cases in which the Euler characteristic of a surface appears.

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Title: A Taste of Arithmetic Combinatorics

Abstract: Arithmetic combinatorics is the study of combinatorial problems involving arithmetic operations. Specifically speaking, we study a series of problems of counting the number of certain combinatorial objects such as points and lines in a field with addition and multiplication. Arithmetic combinatorics relates to many branches of mathematics, especially harmonic analysis, number theory, and algebraic geometry. Moreover, there are numerous applications of arithmetic combinatorics in theoretical computer science, for instance, linearity testing and randomness extractors. As a result, arithmetic combinatorics has gradually drawn much attention recently.

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Title: Domain decomposition-based coupling of Operator Inference Reduced Order Models via the Schwarz alternating method

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Title: Counting small cycles in the superspecial Abelian surface graphs

Abstract: Unlike isogeny graphs for elliptic curves, which are locally tree-like, the isogeny graphs for superspecial Abelian surfaces typically have small cycles. These cycles correspond to hash function failures in the setting of cryptography. In this talk, we will introduce the cryptography background, the related problem in the superspecial Abelian surface graph, and the equivalent lattice path enumeration problem. We will give a formula for the number of short cycles in genus-2 isogeny graphs and give a bijective for it.