Past Talks

Fall 2024

• August 30, 2024. Speaker: Kevin Metzler. Title: Dynamical Systems as Applied to Neural Networks           Abstract: Using Dynamical Systems and some other fancy tools we have learned to invert certain types of neural networks. We will observe the relationship between sparse neural network layers and their invertibility.

Spring 2024

• April 5, 2024.  Speaker: Sara Amato . Title:   Differential Equations and Data-Driven Methods for Modeling Microglial Cells During Ischemic Stroke 
  Abstract: Neuroinflammation immediately follows the onset of ischemic stroke in the middle cerebral artery. During this process, microglial cells are activated in and recruited to the penumbra. Microglial cells can be activated into two different phenotypes: M1, which can worsen brain injury; or M2, which can aid in long-term recovery. In this thesis, we contribute a summary of experimental data on microglial cell counts in the penumbra following ischemic stroke induced by middle cerebral artery occlusion (MCAO) in mice and compile available data sets into a single set suitable for time series analysis. Further, we formulate data-inspired and data-driven models of microglial cells in the penumbra during ischemic stroke due to MCAO. Through the use of machine learning algorithms, differential equations, parameter estimation, sensitivity analysis, and uncertainty quantification, we computationally explore microglial cell dynamics in the short-term and long-term. Results emphasize an initial M2 dominance followed by a takeover of M1 cells, show the importance of microglial cell switching on model outputs, and suggest a lingering inflammatory response. 

• March 22, 2024.  Speaker: Derek Drumm . Title:   Ocean Debris Transport
  Abstract: Ocean debris poses a problem of great importance in our current era. Many researchers are currently studying how debris and pollution are transported within the ocean. In this presentation, I will briefly discuss some of the methods by which oceanic phenomena transport pollution, and one common technique to study this transport.

• March 15, 2024.  Speaker: Guillermo Nunez . Title:   Butson Matrices and Their Asymptotic Existence
  Abstract:  Butson Hadamard matrices are an interesting algebraic and combinatorial object, with applications in quantum information theory and coding theory. I will present several results regarding their asymptotic existence, and an improvement of my own.

• February 23, 2024.  Speaker: Abigail Lindner. Title:   Applications of Mathematical Optimization to Art 
  Abstract:  One can use mathematical optimization tools to plan delivery routes, improve process performance, and launch efficient humanitarian operations, all of which are useful and important applications in an increasingly connected and growing world. Worthy of discussion though these are, we will not concern ourselves here with these laudable research efforts. We will instead turn to less practical but still fun considerations of using mathematical optimization to create art. 

• February 16, 2024.  Speaker: Derek Drumm   . Title:   Mesh Adaptive Direct Search using Quasi-Newton Search Step with Applications to Fluid Problems  
  Abstract:  Gradient-based techniques for optimization, such as Newton's Method, are quite common in applications, due to their well-known fast convergence results. However, these types of optimization methods may not be well suited for certain optimization problems, such as shape optimization problems involving fluid flows. This is due to the difficulties of numerically approximating gradient information from flow simulation data. These difficulties can be avoided by using derivative free optimization (DFO) methods, such as the mesh adaptive direct search (MADS) algorithm. MADS, a type of direct search algorithm, can be quite flexible in its applications. This is thanks to its unique "search and poll" routine, which allows the implementation of heuristic search methods. The downside to MADS, like many direct search methods, is that it can be very slow. In this talk, I will present a modification to MADS which utilizes a Quasi-Newton approach to optimization. This approach attempts to balance the utility of avoiding costly derivative calculations while exploiting the fast local convergence of Newton's Method. I will then show how to apply this new algorithm to solve a common fluid dynamics problem: minimizing drag in a viscous fluid.

• January 26, 2024.  Speaker: Taorui Wang   . Title:   Statistics as measurement device 1: Historical Introduction of Normal Distribution 
  Abstract:  The statistics introduced in our courses seem to assume a worldview based on statistical models. Every quantity can be represented by some probability distribution and then we can apply some statistical model to it. This view is useful sometimes. But in many cases, it introduced unnecessary assumptions, and validating them is hard. In this seminar, I will introduce the history of "Normal" Distribution. How people derive it from Astronomy as an error curve of measurement and how they introduce this idea into social science. 

• January 19, 2024.  Speaker: Abigail Lindner. Title:   Some Models of Bacterial Fouling of Filtration Membranes 
  Abstract:  From research to clinical development and production, Mycoplasma are well-recognized and widespread contaminants in biopharmaceutical manufacturing. Their successful proliferation despite the ultrafiltration performed to eliminate them has been, in hypotheses tested via lab experimentation, attributed in part to the flexibility of the cell walls of Mycoplasma. Parsing out the mechanisms of this fouling is of interest to engineers working to improve filtration processes, and research in this area has sought to develop accurate mathematical models that integrate membrane, fluid, and other characteristics and account for transitions in foulant behavior. I will present some existing models of fouling in filtration systems.

Fall 2023

• December 8, 2023. Speaker: Colin Streck  . Title:  Post Quantum Cryptography
  Abstract: I will give a brief overview of some of the important problems used for post-quantum cryptography systems. 

• December 1, 2023. Speaker: Derek Drumm  . Title:  Using Derivative-Free and Black-box Optimization Techniques to Solve Shape Optimization Problems in Fluid Mechanics 
  Abstract: Shape optimization problems are common in engineering applications. These problems arise in fluid dynamics through the need to control internal and external flow properties. In the case of a Navier-Stokes fluid, performing numerical optimization can be computationally challenging. Difficulties arise when numerically approximating gradient information from flow simulation data. These difficulties can be avoided by using derivative-free optimization (DFO) methods, such as the mesh adaptive direct search (MADS) algorithm. These types of algorithms are quite flexible in their application and are well-suited to solving optimization problems that require accurate flow solutions at each iteration. In this document, I will briefly introduce the theory of shape optimization and the previously mentioned MADS algorithm. I will then apply the algorithm to solve a few test problems; in particular, I will look at finding the obstacle that minimizes drag in a Stokes fluid. Next, I will introduce the concept of Lagrangian coherent structures, which are described as the most attractive and repelling regions of a fluid flow. Finally, I will propose some ideas on how to use the developed theory to analytically and numerically solve shape optimization problems involving Lagrangian coherent structures. 

• November 10, 2023. Speaker: Dane Johnson. Title:  Adaptive Biological Flow Networks
  Abstract: I will discuss ideas for modeling fluid and solute transport in animal vasculature systems, leaf vein networks, and fungal mycelium.  

• November 3, 2023. Speaker: Sara Amato  . Title:  Computational modeling and parameter estimation of microglial cell dynamics
  Abstract: The penumbra is an area in the ischemic brain where damage is reversible if blood flow is restored in time. The neuroinflammatory process immediately follows the onset of ischemic stroke and begins with the activation and recruitment of microglial cells to the penumbra. Microglial cells can be activated into two distinct phenotypes, M1 and M2, which can have both harmful and beneficial effects. In this talk, we discuss a mathematical model of microglial cells in the penumbra. First, we derive a model based on assumed interactions and dynamics of the cells. Next, we use Generalized Sobol Sensitivity Analysis to identify influential parameters. Then we use MCMC estimation to fit model parameters to experimental data of M1 and M2 microglial cell counts in the penumbra. 

• October 6, 2023. Speaker: Duy Pham. Title:  An Introduction to PINN   
  Abstract: Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for approximating functions in various scientific and engineering domains. While many studies have shown that PINNs are highly effective at approximating functions within a given domain (interpolation), few have investigated how well the framework can recover missing portions of a function beyond the available data (extrapolation). In this study, we evaluate the performance of PINN for both interpolation and extrapolation using Legendre polynomials. Our results demonstrate that PINN significantly outperforms a purely data-based approach in multiple cases. 

• September 29, 2023. Speaker: Kevin Metzler  . Title:  Spatial transformations and their effect on image classification 
  Abstract: Using a combination of the Fourier transform and several wavelet transforms we will delve into the world of data science to better identify defects in images. 

• September 22, 2023. Speaker: Mason DiCicco. Title:  Intro to Nearest Neighbor Complexity
  Abstract: The nearest neighbor (NN) instance is a collection of positive and negative points, or anchors. Given a query point x, we classify it as positive if its nearest neighbor (among all anchors) is positive. The key question of NN complexity is: How many anchors are required to represent a given function? This question was (to my knowledge) asked for the first time in the 2022 paper of Hajnal, Liu, and Turan, Nearest neighbor representations of Boolean functions. In this talk, I will cover the main points of this paper and discuss some open problems. 

• September 15, 2023. Speaker: Abigail Lindner. Title: Did the Adoption and Safe Families Act Make a Difference?: Statistical Models to Answer the Question  
  Abstract: Assessing whether a policy had the desired impact is no small task for a variety of reasons, including lack of data or low data quality, the influence of confounding factors, and heterogeneity of implementation. In 1997, Congress passed the bipartisan-led Adoption and Safe Families Act (ASFA), which sought to address the phenomenon of "foster care drift" in the U.S. child welfare system and accelerate the movement of children into permanent placements. In this talk, we'll review the best evidence for whether ASFA "worked" and explore the statistical models that enable this analysis. 

• September 1, 2023. Speaker: Derek Drumm. Title: Derivative-Free Optimization: GPS and MADS
  Abstract: Derivative Free Optimization (DFO) is a class of optimization algorithms that, as the name suggests, avoid the calculation of derivatives. This type of optimization is particularly useful when optimizing functions whose derivative data is not available, or is difficult to calculate. I will give a brief introduction to two famous DFO algorithms, Generalized Pattern Search (GPS) and Mesh Adaptive Direct Search (MADS), as well as discuss how these algorithms are helping me in my current research area. 

Spring 2023

• April 21, 2023. Speaker: Forrest Miller. Title: Combining Deep Learning and Stochastic Differential Equations for the Pricing of American Options
  Abstract: In this work, we will investigate the theory and numerics of reflected backward stochastic differential equations (RBSDEs), motivated from the problem of option pricing. We will go through some key theoretical results of RBSDEs, and a deep learning approach to numerically approximating solutions. We will then present new work on the Buyer's and Seller's indifference price of American Put Options under a stochastic volatility model. 

• April 14, 2023. Speaker: Teodor Hellgren. Title: Legislative Elections and Apportionment
  Abstract: Apportionment methods and election theory play a crucial role in determining how resources, such as seats or representation, are allocated among different groups or parties. In this presentation, we will consider the math behind different apportionment methods and their pros/cons to understand the challenges involved in achieving fair allocation. We’ll consider how apportionment methods are used and their applications to some real world case studies, especially in minimizing error. Since no election or apportionment method can be perfect, the theme will be on picking our priorities to best reflect the wills of voters.

• April 7, 2023. Speaker: Charlotte Clarke. Title: Homotopy Type Theory of Sewn Quilts
  Abstract: The higher equality structures of homotopy type theory (HoTT) lend the field to geometric interpretations in visualizations. In this work, we take these interpretations to their logical extreme in the development of PieceWork, a programming language to represent sewn quilts. We leverage several layers of equality to represent quilts as geometric shapes which can be manipulated in several ways, such as cutting into smaller shapes or sewn together to form an intricate design. 

• March 31, 2023. Speaker: Sara Amato. Title: Dynamic Bayesian Network Approaches for Neural Inflammation
  Abstract: Microglial cells and cytokines are key components in neural inflammation. Knowledge of the time-dependent dynamics between these four components is crucial to understanding the neural inflammatory process. Additionally, computationally predicting information about microglial cells and cytokines at future times is of interest. Dynamic Bayesian Network (DBN) approaches address these problems. DBN analysis involves learning dependencies between variables (i.e. learning the parents of each node) and learning the values of parameters in the DBNs closed-form expression. It is assumed that the values of each random variable at any given time are independent, which allows for the parents of each node to be learned separately; however, in some real-world examples, this assumption may be too restrictive as the variables may interact with each other over time. Further, parameter identification is often ignored in this setting; however, performing this step is crucial to understand which relationships can be reliably uncovered and quantified. In this talk, we break the independence assumption and construct a closed-form expression for the DBN using a coupled system of regression equations with possible nonlinear terms. Additionally, we present a workflow to select identifiable parameters and estimate this subset using Markov Chain Monte Carlo (MCMC) sampling methods. 

• March 17, 2023 Speaker: Abigail Lindner. Title: Magic Squares
  Abstract: Magic squares have been an interest in recreational mathematics since antiquity. The normal magic square is an n × n array containing 1, 2, ..., n^2 arranged in such a way that each row, each column, and the main diagonals add to the same magic number. Over the centuries, variations of the normal magic square have been identified and named, including semimagic squares that fail the diagonal sum criterion of a normal magic square and associative squares that have every pair of numbers symmetrically opposite about the center summing to n^2 + 1. The last century has seen a deeper investigation of magic squares as they relate to concepts in linear algebra.

• March 3, 2023. Speaker: Kevin Metzler. Title: Using Machine Learning to Predict Movie Success, and Other Fun Side Projects
  Abstract: Having collected information on nearly 700 movies, it was time to put some regression models to the test in predicting both the Tomato Score of a movie as well as it's projected profit. I tested out three different models: Linear, K Nearest Neighbors, and a Neural Network, as well as Google's polynomial regression line that was generated in Google Sheets to compare. We can see some issues as well as the Curse of Dimensionality and some points to fix in the future. We can also explore some attempts at fun data visualization if we have time!

• February 24, 2023. Speaker: Derek Drumm. Title: Determining Flow Obstacles from Lagrangian Coherent Structures
  Abstract: One of the ways to understand a fluid flow is to analyze the structures which govern their behavior. One such set of structures, called Lagrangian Coherent Structures (LCS), are of particular interest. LCSs serve as a "skeleton" of a fluid flow, as they describe the regions of maximal attraction, repulsion, and shearing of a flow. Typically these structures are used to qualitatively analyze flow behavior. However, one may wish for these structures to appear in specific regions of a flow, as a means of flow control. Specifically, we are interested in the following problem: can we introduce a flow obstruction to guarantee a specified LCS. In this talk, we will discuss the feasibility of this problem, whether or not solutions to this problem exist, and how we may go about solving it both analytically and numerically.

• February 17, 2023. Speaker: Elisa Negrini (UCLA). Title: No-collision Transportation Maps in Image Processing and Data Science
  Abstract: We investigate applications of no-collision transportation maps introduced in [Nurbekyan et. al., 2020] in image processing and data science. Recently there has been a surge in applying transportation-based distances and features for data representing motion-like or deformation-like phenomena. Indeed, comparing intensities at fixed locations often doesn't reveal the data structure. No-collision maps and distances developed in [Nurbekyan et. al., 2020] are sensitive to geometric features similar to optimal transportation maps but much cheaper to compute due to the absence of optimization. We evaluate the performance of no-collision maps in various data science tasks, such as clustering and dimensionality reduction. Preliminary experiments show results comparable to other transportation-based techniques for a fraction of the computational time and using much sparser features. This is joint work with L. Nurbekyan.

• February 3, 2023. Speaker: Ben Gobler. Title: An Introduction to Game Theory
  Abstract: Everyone is familiar with games. But how can you determine the best strategy given a set of circumstances? This introduction to game theory will demonstrate the primary logical ideas that surround this study, combining probability and human behavior. We will explore prisoners' dilemmas, war games, police stops, commitment problems, Hotelling's game, and more. Game on!

• January 27, 2023. Speaker: Taorui Wang. Title: Linking PDE and Sampling
  Abstract: In undergraduate study, we might learn inverse transform sampling method. Unfortunately, we can not always use this method to generate samples. If we want to sample from a large family of probability density functions, especially in high dimension, we can use methods like Hamiltonian Monte Carlo Method in Metropolis-Hastings framework. When we view those method as stochastic differential equation, we can find its relationship with PDE.

• January 20, 2023. Speaker: Frederick "Forrest" Miller. Title: Modern Portfolio Theory From Scratch
  Abstract: In 1952, Harry Markowitz developed mean variance portfolio management, forming the bedrock of modern and post modern portfolio theory, as well as establish finance as a field to be studied mathematically. In this talk, we will build the Markowitz portfolio selection problem and discuss the techniques used to solve it. Then, we will discuss the efficient frontier and what happens when risk free assets complement the risky assets of the market. This theory provides mathematical thinking to statements such as "diversify your portfolio", "be careful when selecting stocks", "shorting can be very risky", and many more. Time permitting, we will conclude with a more general discussion on risk measures and the shortcomings of mean variance portfolio management.

• January 13, 2023. Speaker: Jakob Misbach. Title: How Many People Drew a Graph: A Survey of Graph Drawing Algorithms
  Abstract: Graph drawing is the idea of taking the vertices of a graph and embedding them in some plane to create a layout what is appealing based on some heuristic. The heuristics are generally constructed to create visually appealing layouts, and can include factors such as symmetry, ensuring all edge lengths are the same, and minimizing edge crossings (Eades 1984). Graph drawing is an NP-hard problem since it is NP-hard to create a layout with equal length edges, and finding symmetry is at least as hard as graph isomorphism (Eades 1984). Because of this, we will survey approximation algorithms which generate reasonable layouts in polynomial time.

Fall 2022

• December 16 , 2022. Speaker: Matteo Pintonello (University of the Basque Country). Title: Word Problems in Groups
  Abstract: In 1902, Burnside asked whether a finitely generated group all whose elements have finite order is necessarily finite, the so-called Burnside problem. This question was answered in the negative only in 1964 by Golod and Shafarevich. This problem can be included in the framework of word problems in groups. We will give an overview of many types of these questions, from the General and Restricted Burnside conjectures to some problems regarding commutators, like the Ore conjecture. We will finish with finiteness properties, like conciseness.

• December 9, 2022. Speaker: Ethan Washock. Title: 3-cube-free constructions in the integers modulo 2^n
  Abstract: Classical theorems in extremal combinatorics due to Sperner, Erdős, Kleitman, and Samotij state that families minimizing the amount of chains in a Boolean lattice are restricted to a "layered" construction. These theorems translate from the Boolean lattice to the integers modulo 2^n when k-chains are replaced with projective cubes of dimension 2^(k-1) in the case of k being a power of two. This case was proven by Long and Wagner in 2018. Conjectured constructions of largest k-cube-free for any k are also conjectured in their paper, which also have a specific layered construction. However, these bounds on the size of a k-cube-free set aren’t proven. In this presentation, I will investigate the exact structure of 3-cube free subsets of the integers modulo 2^n and discuss strategies that could be used for bounding the "fullness" of layers in a 3-cube-free construction. I will also discuss methods for finding the largest possible 3-cube-free sets using computer software. 

• December 2, 2022. Speaker: Guillermo N. Ponasso. Title: Hadamard matrices, Latin squares and Projective planes
  Abstract: In this talk I will introduce projective planes from the point of view of synthetic (or axiomatic) geometry. This outlook on geometry traces back at least to ancient Greek mathematics and it is perhaps much more relevant now as it was when it was first conceived. More recently with the development of combinatorics many relationships have been established between projective planes and other objects such as Hadamard matrices and Latin squares. I will survey these connections and highlight the role played by the dual interpretation of the regular representation of a group as both a linear representation and a permutation representation. 

• November 18, 2022. Speaker: Ben Gobler. Title: An Introduction to Visual Group Theory
  Abstract: Would you teach someone to play chess without telling them about chessboards? Of course not; the chessboard is essential to the way we understand and exhibit games of chess. Algebra courses deserve the same treatment— how can we expect students to grasp the core ideas of abstract algebra without providing them the fundamental tools for interpretation? In this talk, we will explore the magic of Cayley diagrams and the strong intuition they offer to the study of group theory.

• November 11, 2022. Speaker: Abigail Lindner. Title: Stability Analysis in Mathematical Ecology
  Abstract: The interplay between complexity and stability has been a major topic of discussion in theoretical ecology since the early 1970s, when Robert May published his paradoxical and well-known paper suggesting that the probability of instability is greater in complex communities than in simple communities. Much research in the 40+ years since has had central the desire to better understand how natural communities maintain their internal stability against natural and artificial fluctuations in their environments. Here, we will be considering May’s original work compared to current mathematical models that support the opposite hypothesis and a few of the mechanisms that contribute to stability in ecological communities. 

• November 4, 2022. Speaker: Leonardo Saud Maia Leite (KTH Stockholm). Title: The Lattice of Flats of a Matroid and Real-rootedness
  Abstract: The chain polynomial of a finite lattice L is given by p_L = Σ_{k ≥ 0} c_k (L) x^k, where c_k (L) is the number of chains of length k in L starting at 0^ and ending at 1^. There is a conjecture which states that, if L is a geometric lattice, then its chain polynomial p_L is real-rooted. In particular, it is log-concave. Here, we will consider a finite matroid M, define its lattice of flats L(M), and study the polynomial p_L(M). We verified that the conjecture is true for paving matroids and for generalized paving matroids, a new class of matroids introduced during this study. This is an ongoing and joint work with Petter Brändén.

• October 28, 2022. Speaker: Charlotte Clark. Title: Blinkers, Heat, and Windshield Wipers: Another Perspective on Hybrid Systems for Vehicle Safety
  Abstract: Cyber-physical systems are typically computerized control mechanisms for interacting with and controlling physical machines and systems. Often, in cases such as power plants and self-driving cars, these computerized controllers manage systems where human life is on the line. In these cases, we want to be certain that they are programmed correctly. Using the KeYmaeraX theorem prover, we can create models of cyber-physical systems that we can prove are safe. While autonomous vehicles are always a popular modelling subject, in this talk we will be modelling something critical to all vehicles on the road today: internal electronic components such as turn signals and climate control, and how they impact vehicle safety.

• October 14, 2022. Speaker: Forrest Miller. Title: Optimizing the Benefit to Cost Ratio for Public Sector Decision Making
  Abstract: Runaway and homeless youth (RHY) face a high risk of being trafficked. While an existing shelter system provides necessary RHY services such as shelter, mental health and financial support, there is a great need for system-wide, yet costly, shelter capacity expansion. We undertake an RHY and service provider-informed, systematic, and data-driven approach that considers the benefits to society obtained from rolling out new capacity against associated costs. We propose a mixed integer linear fractional program (MILFP) that maximizes the benefit to cost ratio of capacity expansion for the New York City shelter system. We employ Dinkelbach’s algorithm to convert the MILFP to a series of linearized versions, improving tractability. Our results provide data-informed recommendations for NYC shelter expansion opportunities to better serve RHY.

• October 7, 2022. Speaker: Riuji Sato. Title: An introduction to regularity theory for PDEs
  Abstract: When looking for solutions to PDEs, oftentimes an explicit solution is available only in special cases. By enlarging the space of functions to look for solutions, one can then show the existence of weak solutions. These are often proved using standard estimates and tools from functional analysis. We are then interested in knowing whether such solutions are in fact classical solutions by showing that they indeed have the necessary regularity that the original PDE requires. To do this requires a more careful analysis of some intricate estimates. In this talk, we revisit the notion of weak solutions for PDEs in divergence form and motivate the importance of studying the regularity of these solutions by looking at some famous open problems in PDEs. We then consider an elliptic PDE in a bounded domain and sketch the proof of the interior H2-regularity of its weak solution. We emphasize that at the heart of the proof are important algebraic assumptions on the PDE that allows us to obtain the necessary analytic estimates to prove higher regularity. We end with a discussion on the other directions of the theory, e.g., boundary regularity, theory for parabolic PDEs, Cα-regularity, etc. 

• September 30, 2022. Speaker: Mason DiCicco. Title: Expected Length of the Longest Common Subsequence
  Abstract: Given two random binary sequences of length n, what is L(n), the expected length of their longest common subsequence? Chvátal and Sankoff (1975) proved that L(n) is asymptotically equal to γn for some constant γ (which we call the Chvátal–Sankoff constant). However, the exact value of γ is still unknown. (The best bounds are 0.788 < γ < 0.826 due to Lueker (2009).) In this talk I will describe the general method used to calculate upper bounds for γ.

• September 23, 2022. Speaker: Derek Drumm. Title: Lagrangian Coherent Structures
  Abstract: One of the ways to understand a fluid flow is to analyze the structures which govern their behavior. One such set of structures, called Lagrangian Coherent Structures (LCS), are of particular interest. LCSs serve as a "skeleton" of a fluid flow, as they describe the regions of maximal attraction, repulsion, and shearing of a flow (analogous to stable and unstable manifolds of a dynamical system). I will discuss how we derive these structures, and some interesting applications of them, as well as how they are related to my area of research.

• September 16, 2022. Speaker: Guillermo N. Ponasso. Title: Quaternions and Sums of Four Squares
  Abstract: In 1770 Lagrange showed that every natural number can be written as the sum of four squares. Several years later, in 1843, Hamilton discovered quaternions in a flash of insight, and carved his discovery on a rock of Broom bridge, Dublin. In our talk we will see how Lagrange's theorem is related to the Hurwitz quaternions. To do so we will first discuss sums of two squares and how they relate to Gaussian integers.

Spring 2022

• April 15, 2022. Speaker: Sara Amato. Title: Data-driven and mechanistic modeling approaches with applications to neuroscience
  Abstract: Ischemic stroke is a leading cause of death and disability, with limited treatment options. The neuroinflammatory process immediately follows ischemia and plays a large role in determining the clinical outcome of stroke. However, interactions between cellular components involved in the neuroinflammatory process are not fully understood. In this talk, we will discuss how Dynamic Bayesian Networks (DBNs) can be utilized to learn these interactions and to inform ODE modeling in this area.

• April 8, 2022. Speaker: Tony Vuolo. Title: I've Got a Blank Space AB: Completing One's Understanding of Linear Algebra
  Abstract: One teaching strategy common to math classes is providing students with formulae they can assume without proving. Identities and operations on matrices and vectors in linear algebra are subject to this kind of treatment despite being provable within the realm of the material taught in that class. In this discussion we explore a few such identities in linear algebra, how to prove them, and what their truth implies about generalizations of other concepts in geometry. These concepts are the dot and cross product of vectors and the orthogonality of vectors and linear spaces in the Cartesian plane.

• April 1, 2022. Speaker: Mason DiCicco. Title: Communication Complexity of Subsequence Detection
  Abstract: I will prove some bounds on the communication complexity of subsequence detection.

• March 25, 2022. Speaker: Taorui Wang. Title: Halting Problem and Chaitin's Number
  Abstract: Halting Problem is a famous problem in computer science. it is,  given a program and an input to the program, whether the program will eventually halt when run with that input.​ I will talk about how halting problem relates to uncomputable real and the construction of Chaitin's number.  

• February 25, 2022. Speaker: Guillermo N. Ponasso. Title: Can you hear the shape of a drum?
  Abstract: The topic is mostly explained by the title! If you hear the sound of a drum, i.e. a vibrating membrane, can you recover its shape? We will answer this question, and look at some generalizations (can you hear the shape of a vibrating object in several dimensions?). This has interesting connections to quadratic forms and modular forms!

• February 18, 2022. Speaker: Jessica Wang. Title: Tiling of Prime and Composite Kirchhoff Graphs
  Abstract: A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. We present an algorithm that constructs all Kirchhoff graphs up to a fixed edge multiplicity. We explore the tiling of prime Kirchhoff graphs, specifically, the number of possible prime Kirchhoff graphs given a set of initial fundamental Kirchhoff graphs.

• February 11, 2022. Speaker: Riuji Sato. Title: An Introduction to Viscosity Solutions
  Abstract: Ideally, one is interested in explicit solutions to PDEs. However, in most situations this can be very difficult, if not impossible. What one can do is to show that a solution exists instead, and to help with this, one can look for solutions in a larger class of functions. In this talk, we review the notion of weak solutions for elliptic PDEs in divergence form and discuss how it is reliant on an integration-by-parts step that might not be possible for other PDEs of interest. We introduce the notion of a viscosity solution and give examples for which the theory of viscosity solutions can be applied. We then state the maximum principle for viscosity solutions and sketch how using Perron’s method, we can show that viscosity solutions for the Dirichlet problem exist. We then sketch a proof of the maximum principle. We end by discussing extensions of the theory to other boundary conditions, parabolic equations, and systems.

• February 4, 2022. Speaker: Forrest Miller. Title: A brief survey of Optimization and Operations Research
  Abstract: Operations Research (OR) is a powerful branch of mathematics that focuses on applying optimization to solve problems in resource allocation and many other areas of applied mathematics. In this talk, I will discuss the theoretical underpinnings of optimization, and how it is applied in operations research. Convex and fractional optimization problems will be discussed, as well as algorithms to solve types of optimization problems. Finally, I will present some work I am doing applying optimization techniques to a resettlement problem.

• January 28, 2022. Speaker: Ben Gobler. Title: Listing the Rationals using Continued Fractions
  Abstract: It is well known that the rational numbers are countable. From a list which includes each rational number exactly once, we may ask, "What is the 200th rational in the list?" or "Where does 22/7 appear in the list?" To answer these questions, we will explore an original method which uses continued fractions to evaluate and locate terms in the Calkin-Wilf sequence, one such list of the rationals.

Fall 2021

• December 3, 2021. Speaker: Nikolaos Kalampalikis. Title: A graph theoretic proof for the Sensitivity Conjecture
  Abstract: The Sensitivity conjecture has been important problem in theoretical computer science. In this presentation, we will go over Huang's proof of showing that every (2^(n-1) + 1)-vertex induced subgraph of the n-dimensional cube graph has maximum degree of at least sqrt(n). As a direct consequence we will show that sensitivity and degree of a boolean function are polynomially related.

• November 19, 2021. Speaker: Elisa Negrini. Title: A Neural Network Ensemble Approach to System Identification
  Abstract: We present a new algorithm for learning unknown governing equations from trajectory data, using and ensemble of neural networks. Given samples of solutions $x(t)$ to an unknown dynamical system ẋ(t) = f(t, x(t)), we approximate the function f using an ensemble of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for f. This procedure yields M-1 time-independent networks, where M is the number of time steps at which x(t) is observed. Finally, we obtain a single function f(t, x(t)) by neural network interpolation. Unlike our earlier work, where we numerically computed the derivatives of data, and used them as target in a Lipschitz regularized neural network to approximate f, our new method avoids numerical differentiations, which are unstable in presence of noise.  We test the new algorithm on multiple examples both with and without noise in the data. We empirically show that generalization and recovery of the governing equation improve by adding a Lipschitz regularization term in our loss function and that this method improves our previous one especially in presence of noise, when numerical differentiation provides low quality target data. Finally, we compare our proposed method with other algorithms for system identification.

• November 5, 2021. Speaker: Sean Fraser. Title: Geometric Measure Theory: An Introduction
  Abstract: In this talk, we discuss some basic ways to use measure theory to analyze geometric properties of certain subsets of R^n. We begin with a refresher on measure spaces. We then discuss Hausdorff measure and the Hausdorff dimension of a set. We find that Hausdorff measure is like "surface area" like how Lebesgue measure is like "volume," and state the isoperimetric inequality. We then conclude the talk by improving the bound on the isoperimetric inequality, by using calculus of variations to define the reduced boundary of a set.

• October 8, 2021. Speaker: Guillermo N. Ponasso. Title: Basics of Diophantine Approximation
  Abstract: Diophantine approximation is concerned with the approximation of irrational numbers by rational numbers. I will go through some of the basic results and also discuss some problems on cyclotomy such as non-vanishing sums of roots of unity.