Mailing List: https://forms.gle/seUsJK5rufxwiXGTA
When: Tuesday 4:00 - 5:00 pm
Where: Vincent 311
Organizers: Andy (hale0198 AT umn.edu), Sam (coyle158 AT umn.edu), Walton (and09562 AT umn.edu), Ramanuja (telek002 AT umn.edu), Quan (tran1178 AT umn.edu).
The Undergraduate Math Research Seminar (UMRS) is a seminar aimed to create a welcoming undergraduate research community at UMN. We will have opportunities for students to give low stakes talks on the research and reading they’ve done as well as hold sessions devoted to learning what research is like, the different areas of math research and how to create research presentations. If you’d like to be updated on the seminar, see the mailing list above. See here for the 2023-2024 schedule.
Jan 28: Social Hour
Come and learn more about the seminar! There will be food provided by the hosts.
Feb 4: Andrew Hale. "The Method of Brackets Applied to Definite Integrals."
We introduce the Method of Brackets, a tool for evaluating certain definite integrals using the Ramanujan Master Theorem. This method works whenever the integrand can be expanded with a Taylor series. We will demonstrate how the method works via the formation and evaluation of bracket series, and how the method can efficiently evaluate hard integrals. A generalization of the method to multiple dimensions is introduced for evaluating iterated integrals. Finally, we use the method to prove a formula in the Table of Integrals, Series, and Products.
Feb 11: Cameron Beckman. "Explaining the Wiener-Ikehara Tauberian Theorem."
The Wiener-Ikehara Tauberian Theorem is a theorem based in Fourier Analysis that can be used to prove the prime number theorem without advanced complex analysis. In this talk, we will compare this theorem to the standard proof and will explain how it can be used to prove theorems on arithmetic progressions.
Feb 18: Annie Warren. "Bubbles and Fissures: Gap Formation in a Two-Dimensional Swarm Model."
We investigate large systems of particles interacting through an attractive-repulsive potential, both for finite system size and in the McKean-Vlasov continuum limit. We build on work that examines the formation of vacuum regions in one-dimensional systems to investigate the size and topology of such regions in two dimensions, finding (for instance) bubbles or fissures. Our main results establish the existence and scaling laws for small bubbles and fissures, assuming square or hexagonal lattice symmetries. Interestingly, the transition between a uniform density distribution and finite-size vacuum bubbles does not exhibit hysteresis in the limit of large system size. Numerically, we investigate both finite-size effects on the transition and the stability of bubbles and fissures.
Feb 25: Archisman Bandyopadhyay. "Shrinkage Estimation: Why Learning from Others is Always Useful."
We are all familiar with the problem of estimating the mean of a distribution by independent samples generated from the same distribution. The classical (and very useful) solution is to use the sample mean as the estimate. However when we have independent groups of samples with each group being generated by distributions with different means, can we do better? The answer is yes, and perhaps surprisingly, in shrinkage estimation, we use the information from each group to influence our estimates for the others. In this talk, I will outline the idea of shrinkage in statistical estimation, the different types of shrinkage estimators and also go over a proof that the Stein shrinkage estimator is always guaranteed to be better than the classical mean estimator if there are at least 3 groups.
Mar 4: Ramanuja Telekicherla Kandalam. "A Short Talk on the Ising Model."
The Ising model is a model originating from statistical physics which describes the magnetism of metals. Despite its simplicity, the Ising model gives rise to some very deep mathematics. In this talk, we discuss some basic results about the Ising model such as its solvability and the uniqueness of the Gibbs measure along with the fascinating mathematics.
Mar 11: Spring Break!
Mar 18: Andrew Hale. "Homotopy Path Algebras."
Homotopy Path Algebras (HPAs) were introduced to relatie structures in algebraic geometry to topological spaces. This connection is a part of mirror symmetry and the mathematics of high energy physics (string theory). In this presentation, we provide background about HPAs and describe some of their applications. Our project explores 3-dimensional Calabi-Yau HPAs which have additional algebraic structure. We show that the additional structure provides an algorithm for writing an explicit cellular resolution of the HPA, which is useful for computations in mirror symmetry.
Mar 25: Nathan Mihm. "Computational Methods on Optimal Addition-Subtraction Chains."
This talk explores the formalization of addition-subtraction chains as an extension to the well-researched addition chains. An addition chain for a positive integer n is a sequence
1 = a_0, a_1, ..., a_l = n
in which each a_k = a_i + a_j for some i, j < k, whereas an addition-subtraction chain allows a_k = a_i ± a_j. The talk explores a graph-theoretic method to compute the length of a shortest addition-subtraction chain reaching n, similar to the methods known for addition chains. We present an algorithm for efficiently computing such chains and some preliminary computation results. This research may prove practical in applications where this arithmetic represents computationally intensive tasks including elliptic curves. Other generalizations of additions chains are also explored.
Apr 1: Joint Panel with Math Club!
Apr 8: Samuel Coyle. "A Theoretical Background of Persistent Homology and Vietoris-Rips Complexes for Topological Data Analysis."
Across many scientific disciplines, modern research is often interested in large and high-dimensional datasets. Very powerful tools from linear algebra and cluster analysis have been successfully used to study such datasets; however, these methods have a weakness in studying from highly complex and/or continuous datasets. A relatively new approach to modeling these complex datasets is Topological Data Analysis (TDA). In this talk, we will begin by describing one of the most prevalent tools in TDA, persistent homology, which assumes that the dataset comes with a filtration. From the results of persistent homology, we will be able to define a pseudo-metric which, under sufficiently nice conditions, provides stability in the distance of data sets. We will also discuss how to construct filtrations which satisfies these sufficient conditions for datasets which are not equipped with a filtration beforehand. If time permits, we will discuss some open problems in TDA relating to interpreting results from persistent homology and about the homotopy type of Vietoris-Rips complexes including my current research for the case of embedded lattice points of the torus.
Apr 15: Riley Lundgren. "Seiberg-Witten Theory and Applications."
Seiberg-Witten gauge theory describes the behaviour of magnetic monopoles in compact oriented smooth 4-manifolds via the Seiberg-Witten monopole equation. The space of solutions to the monopole equation modulo gauge equivalence is a compact smooth manifold. We obtain the Seiberg-Witten invariants by considering a characteristic number of this space of solutions. In this talk, we provide an introduction to spin geometry and the Seiberg-Witten invariants, and describe some alternative methods for using the monopole equation which have arisen since its discovery.
Apr 22: Daniel Kumm. "The Monoid of Biset Products for Dynkin Categories"
We examine so called 'bisets' associated to the Dynkin category A_n. These can be represented as n x n matrices of 0s and 1s of a certain shape, with an unusual multiplication. These bisets, together with the zero matrix, form a monoid. We describe the idempotents, ideals, and the J-classes and L-classes of this monoid. Then, we develop our results by presenting conjectures about the sizes of the J-classes and L-classes and the connection with the principal series. We conclude by discussing the representation of the matrices of A_n as a union of rectangles and how it describes the ideals.
Apr 29: DRP Presentations
Sep 19: Presentation Workshop
Sep 26: Dinh-Quan Tran. "Viscosity Solutions in Optimal Control Theory and Zero-sum Differential Games."
In this talk, we present the method of viscosity solutions for first-order Hamilton-Jacobi equations. Developed in the 80s, viscosity solutions are meant to deal with PDEs that do not have classical, continuously differentiable solutions. It provides us with a way to maintain the well-posedness of the problem with weak solutions and has proven to be extremely flexible and applicable to a broad range of problems.
We will also review two primary applications of this method in optimal control theory and zero-sum differential games. Through identities such as the Dynamic Programming Principle, we can identify corresponding equations that provide the value function as solutions, thereby adding a PDE dimension to problems in dynamical systems.
Oct 3: Samuel Coyle. "Explicit Forms of Digital Sums and Their Relation to the Takagi Function."
In 1901, Teiji Takagi described a continuous yet nowhere differentiable fractal function. Since then, Takagi's function has many applications to various problems in mathematics, but we will be analyzing the explicit expression for Digital sums, that is, sums related to base expansions of integers. Through a certain class of regular sequences and Salem's description of Lebesgue's singular function, we will find a very natural relationship between Takagi's function and explicit forms for Digital sums.
Oct 10: Riley Lundgren. "Puiseux and Hahn series with Applications to Tropical Geometry."
We introduce the study of formal series by generalizing the ring of formal power series. We will define Laurent, Puiseux, and Hahn series over a field, as well as the relationships between these series. A key result is the algebraic closure of both the field of formal Puiseux series and the field of formal Hahn series when the base fields are algebraically closed. We will also use Tropical Geometry to study the algebraic geometry of the fields of Puiseux and Hahn series.
Oct 17: Undergraduate Mathematics Research Panel
Oct 24: Andrew Hale. "The Method of Brackets Applied to Definite Integrals."
We introduce the Method of Brackets, a tool for evaluating certain definite integrals using the Ramanujan Master Theorem. This method works whenever the integrand can be expanded with a Taylor series. We will demonstrate how the method works via the formation and evaluation of bracket series, and how the method can efficiently evaluate hard integrals. A generalization of the method to multiple dimensions is introduced for evaluating iterated integrals. Finally, we use the method to prove a formula in the Table of Integrals, Series, and Products.
Oct 31: Ramanuja Telekicherla Kandalam. "Basic Results in Social Choice Theory."
Social choice theory was initially developed by the Marquis de Condorcet, who applied probability theory to analyze decisions made by majority vote. Following Condorcet, other researchers were able to study other decision-making models (board voting, dictators, etc...). The purpose of this talk is to give a brief overview of some important results from social choice theory, along with the background needed to understand them.
Nov 7: Mixer
Nov 14: Estimathon!
The Estimathon is a team-based contest that combines trivia, game theory, and mathematical thinking. Teams have 30 minutes to work on a set of 10 estimation problems, the winning team being the one with the best set of estimates.
Nov 21: Andrew Hale. "Optimal Design Problems with Cost Functions: Regularity of Minimizers and Free Boundary Analysis."
We describe an important Partial Differential Equation known as Laplace's equation, and how it applies to several optimal design problems. We then describe a generalized Alt-Cafferelli problem in terms of this operator, then show the existence of solutions with a technique from singular perturbation theory. Finally, we examine the properties of these solutions, which include regularity estimates, a free boundary condition, and sufficient conditions for uniqueness.
Nov 28: Thanksgiving break!
Dec 5: DRP Presentation