SPRING 2024
Dewan Chowdhury — Exploring Geometric Flows: An Introduction to Curve-Shortening Flow
Speaker: Dewan Chowdhury
Abstract: Geometric flows represent a fascinating area of study in mathematics, blending geometry and analysis to understand the evolution of geometric objects over time. One of the most intuitive flows is the curve-shortening flow (CSF), in which a curve evolves by moving in the direction of its normal vector, with speed proportional to its curvature. In this talk, we will examine many of the concepts central to the theory of geometric flows through an exploration of the CSF. Only calculus is assumed.
Feng Luo — The story of circle packing
Speaker: Feng Luo
Abstract: A circle packing in the plane is a collection of round disks with disjoint interior. I will tell you some of the fascinating theorems concerning circle packing and their relationship to modern 2-dimensional conformal geometry (e.g., the Riemann mapping theorem and the uniformization theorem). These results relate combinatorics (planar graphs) with geometry. The contributions of Koebe, Thurston, Andreev, and He-Schramm will be discussed.
Alex Kontorovich — Applications of Thin Orbits
Speaker: Alex Kontorovich
Abstract: We will discuss some recent advances in our understanding of various problems at the intersection of dynamics, geometry, and number theory.
Pi Day
Pi day is coming up over spring break, so we're celebrating a week early! There will be pie and tea!
Zheng-Chao Han — A glimpse into some mathematics behind soap films and soap bubbles
Speaker: Zheng-Chao Han
Abstract: Soap films and bubbles appear abundantly in daily life. We will explore some mathematics used to describe these commonly encountered yet fascinating geometric objects. What are the underlying laws in nature that govern their formation? How are they studied mathematically? What kind of mathematical tools can be brought to bear on these problems? Some of the questions have relatively easy answers, but these are challenging and not so easy questions. In this talk I will try to provide a glimpse into a small collection of such problems and discuss some of the mathematical background for tackling such problems.
Sam Spiro — Solving Math Problems with Anime
Speaker: Sam Spiro
Abstract: The minimal superpermutation problem asks how many symbols a string must have in order to contains all n! permutations of {1,…,n} as substrings. The Haruhi problem asks for the most efficient way to watch a certain anime. Remarkably, these two problems turn out to be equivalent. Even more remarkably, the anime community has made substantially more progress than the mathematicians! In this talk, we discuss an improved lower bound to the minimal superpermutation problem discovered on 4chan, as well as the anime theoretic motivation for this problem. As time permits, we also discuss a problem involving Rock, Paper, Scissors which was also inspired by anime. The talk will be entirely self contained and assume no prior knowledge of anime.
Kristen Hendricks — A first look at knots and symmetries
Speaker: Kristen Hendricks
Abstract: A mathematical knot is a simple object — take a piece of string, tie it up however you like, and glue the ends together so you can't untie it. But these deceptively easy objects to describe and fiddle with are key to understanding deep geometric questions, many not nearly so accessible. We'll introduce knots and consider some possible measures of how complicated a knot is, before turning our attention to one of my favorite topics, possible symmetries of knots. In the end, we'll see how different types of symmetry have wildly different relationships with how "complicated" the knots involved are.
Kimberly Weston — Using probability and differential equations to solve social and economic problems
Speaker: Kimberly Weston
Abstract: What models make good models for asset prices? Financial equilibrium theory provides one such answer — good models are those that are formed by equating supply and demand. Studying financial equilibria requires the use of several areas of math, and this talk will introduce you to some of the related areas and terminology. Financial equilibria can be characterized by solutions of systems of backward stochastic differential equations (BSDEs). We will discuss what BSDEs are, the structure of relevant BSDE systems, and how BSDE solutions can be useful in solving social and economic problems through financial equilibria.
Simon Thomas — Large Cardinals and The Axiom of Determinancy
Speaker: Simon Thomas
Abstract: Let 𝒩 be the space of infinite sequences of natural numbers (a₀, a₁, …, aₙ, …); and for each subset X ⊆ 𝒩, let G(X) be the two player game in which:
- Players I and II alternately choose natural numbers bₙ, cₙ respectively;
- Player I wins if and only if (b₀, c₀, b₁, c₁, …, bₙ, cₙ, …) ∈ X.
Tom Benhamou — Filters and Ultrafilters - a mathematical approach to smallness and largeness
Speaker: Tom Benhamou
Abstract: In this talk, we will present the notion of filters and ideals along with several key examples. These notions provide an abstract framework to different notions of "small" and "large" across mathematics. Then we will present the notion of ultrafilters, some of the most fundamental results in the area, and classical applications, such as the ultrapower constructions. If time permits, we will also show some applications to infinite combinatorics.
FALL 2023
Paul Ellis — Hyperbinary Numbers and the Calkin-Wilf Tree
Speaker: Paul Ellis
Abstract: Hyperbinary is the same as binary except you are allowed to use the digit '2'. So for example, 21 and 101 both represent the number five. It turns out that if you make a sequence where f(n) is the number of ways to represent the number n in "hyperbinary" form, then the sequence f(n)/f(n+1) will list out every positive rational number with no repeats! Come see why!
Estimathon
Brought to you by RUMA and AWM
This is a team based contest that combines trivia, game theory, and mathematical thinking. Teams have 30 minutes to work on a set of 13 estimation problems, the winning team being the one with the best set of estimates. Come join us for a fun time and food!
Dinner will be provided, and (limited) T-shirts will be available!
Benjamin Gunby — Generating Functions
Speaker: Benjamin Gunby
Abstract: Encoding a sequence as a power series seems like a strange thing to do on the surface. However, it turns out to be widely applicable in counting, approximating, and comparing sequences. We will talk about how to count things like the Fibonacci sequence, zigzag permutations, Catalan paths, and more using this powerful tool.
Dima Sinapova — Infinity and the Powerset Function
Speaker: Dima Sinapova
Abstract: Set theory is the study of infinite objects. After Cantor discovered that the cardinality of the reals is strictly bigger than the cardinality of the natural numbers, a natural question emerged: is there anything in between? The continuum hypothesis (CH) states the answer is no, and it became the famous Hilbert's first problem. CH was shown to be independent of the usual axioms of mathematics (ZFC) by works of Gödel (in 1940) and Cohen (in 1964). That means that neither CH, nor its negation, can be proven from the usual axioms. Since then, modern set theory investigates ZFC constraints (i.e. "what is necessary") versus consistency results (i.e. "what is possible"). I will survey some classical results and then discuss recent developments in the study of infinite sets.
Doron Zeilberger — Gambling on Gambling
Speaker: Doron Zeilberger
Abstract: The mathematics of gambling is not always easy, but I will show that it is a good gamble to study the mathematics of gambling.
Robert Dougherty-Bliss — Proofs by Example
Speaker: Robert Dougherty-Bliss
Abstract: You have been told that mathematical statements require "rigor" and "proof." This is an overstated belief. Broad classes of combinatorial, algebraic, and even analytic statements can be established purely based on empirical evidence. These techniques are not as well-known as they should be. We will share in the joy of proof by example with a series of problems about sums, integrals, and recurrences.
Michael Kiessling — Optimal Social Distancing, and such: a Mathematical Perspective
Speaker: Michael Kiessling
Abstract: At the height of the pandemic it was advised to practice social distancing. The rule was to leave a certain distance of closest approach between you and the next person. But how many people then fit into a room of given square footage, and shape? Mathematicians have studied such optimization problems and many of their variations for a very long time, for various reasons and under various names. They are surprisingly rich, interesting, and they are difficult to solve, many of them of the type "NP hard". One of these problems gave rise to Stephen Smale's 7th problem worth the attention of mathematicians in the 21st century.
In my talk I will introduce you to some of the mathematical questions that are being asked, some of the surprising answers that have been found, and to the challenges one faces.
Math Major Q&A Session
Abstract: Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have a professor, a current math graduate student, a graduated math major in tech, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come.
SPRING 2023
William Chen — Groups, numbers, and moduli stacks of elliptic curves
Speaker: William Chen
Abstract: Let G be a finite group. Given a generating pair (x,y) of G, consider the three transformations sending (x,y) to (y,x), (x,y-1), and (x,xy). Two generating pairs of G are said to be Nielsen equivalent if one can be obtained from the other by applying a sequence of such transformations (or their inverses). A natural problem, dating back to the 50's, asks for a description of the Nielsen equivalence classes of G. While simple to state, the problem is surprisingly deep, and touches on topics ranging from number theory to topology and algebraic geometry. An interesting case is when G = SL(2,p), in which case Nielsen equivalence classes are related to Markoff numbers, and the moduli of SL(2,p)-covers of elliptic curves. In this talk we will try to sketch some of these relationships.
Short Undergraduate Special (SUS) Lectures
Abstract: Why is the Earth round? How do we stop climate change? What even is an RU Screw? In honor of April Fools' Day, we will have attendees and board members alike answer the deep questions plaguing the world in the most serious way possible — through mathematics!
Dr. Corrine Yap — Intersections of Combinatorics and Statistical Physics
Speaker: Dr. Corrine Yap
Abstract: This talk will introduce how statistical physics (also known as statistical mechanics) can provide a useful perspective on problems in combinatorics and computer science, and vice versa. In recent years, mathematicians have tackled questions about phase transitions in particle systems by analyzing graph structures, and produced counting and sampling algorithms by using physics-inspired polymer models. We'll start with an introduction to some combinatorially relevant models such as the Ising, Potts, and hardcore models and explore a variety of questions and tools that lie in the intersection of combinatorics and statistical physics.
Paul Ellis — The Four Numbers Game
Speaker: Paul Ellis
Abstract: Write any four numbers at the corners of a square. Next, write the difference of each pair at the midpoint of the corresponding side. Connect these midpoints to make a smaller square inscribed in the original one. Repeat the process. What happens? Always? Can you prove it?
Michael Beals — Generalized Functions and Generalized Derivatives
Speaker: Michael Beals
Abstract: How do we make sense of the derivative of a discontinuous function? How do we make sense of the sum of a non-converging series of functions? How do we take half of a derivative of a function? And why would we want to do any of these things? The answer to that last question is, perhaps, because the physicists and mathematicians of the nineteenth century needed the results. And the resulting theory of generalized functions changed analysis in the twentieth century. We will consider how it all works.
FALL 2022
Priyadip Mondal — Tetrahedral decomposition of hyperbolic knot and link complements
Speaker: Priyadip Mondal
Abstract: The study of geometry and topology largely rests on the study of geometric structures of manifolds. In this talk, we will concentrate on hyperbolic structures of 3-manifolds. A hyperbolic 3-manifold is a geometric object which locally looks like the hyperbolic 3-space (i.e., the upper-half space), and consequently, they inherit a hyperbolic metric. One property that significantly distinguishes the (hyperbolic) geometry in the hyperbolic 3-space from the standard Euclidean geometry is that, unlike the standard Euclidean geometry, the shortest distance between points is not always given by straight line segments between them. The hyperbolic 3-manifolds that we will primarily see in our talk are the hyperbolic knot and link complements. The hyperbolic structure of a hyperbolic knot or link complement is often best understood in terms of its decomposition into ideal tetrahedra (if it has one). This talk will be devoted to exploring hyperbolic knot and link complements through such tetrahedral decompositions. If time permits, we will also talk about some 3-manifold softwares that people use to understand such tetrahedral decompositions. (Since the talk will be geometric in nature, it will be accompanied by a lot of pictures, and no prior knowledge of hyperbolic geometry at any level will be assumed.)
Lars Ruthotto — Neural Network Approaches for High-Dimensional Optimal Control
Speaker: Lars Ruthotto
Abstract: This talk presents recent advances in neural network approaches for approximating the value function of high-dimensional control problems. A core challenge of the training process is that the value function estimate and the relevant parts of the state space (those likely to be visited by optimal policies) need to be discovered. We show how insights from optimal control theory and – in the stochastic case - the fundamental relation between semi-linear parabolic partial differential equations and forward-backward stochastic differential equations can be leveraged to achieve these goals. To focus the sampling on relevant states during neural network training, we use the Pontryagin maximum principle (PMP) to obtain the optimal controls for the current value function estimate. Our approaches can handle both stochastic and deterministic control problems. Our training loss consists of a weighted sum of the objective functional of the control problem and penalty terms that enforce the HJB equations along the sampled trajectories. Importantly, training is unsupervised in that it does not require solutions of the control problem.
We will present several numerical experiments for deterministic and stochastic problems with state dimensions of about 100 and compare our methods to nonlinear optimization approaches and existing approaches.
Robert Dougherty-Bliss — When does a sum have an answer?
Speaker: Robert Dougherty-Bliss
Abstract: Many sums have an answer, like 1+2+…+n. How do we find these answers? Do all sums have answers? For example, does 1!+2!+…+n! have an answer? What is an "answer" anyway?
These questions are analogous to what you learn to ask in Calculus II: which integrals have answers, and how do you find them? Fortunately, the state of affairs for summation is much nicer. Unfortunately, it is less known to most people! I will try to rectify that by showing us some summation techniques and answering a few of the above questions.
Eilidh McKemmie — The Galois Group of a Random Polynomial
Speaker: Eilidh McKemmie
Abstract: A Galois group is an algebraic structure which we can assign to a polynomial and gives us information about the polynomial. For example, Galois theory explains why we have a quadratic equation to find roots of quadratic polynomials, but no similar equation to find roots of polynomials of degree 5 or more. We will consider some different ways of picking a random polynomial and look at what the Galois group of a random polynomial is likely to be.
Yi-Zhi Huang — Representation Theory of Geometric Objects and Quantum Theory
Speaker: Yi-Zhi Huang
Abstract: Quantum mechanics and quantum field theory play a fundamental role in physics. Mathematically, they can be viewed as the representation theory of geometric objects, that is, the theory about how to study geometric objects using vector spaces and linear transformations. I will give an introduction to this mathematical approach to quantum theories.
Paul Ellis — Modular Origami and the Platonic Solids
Speaker: Paul Ellis
Abstract: We will be working mostly with the Sonobe units. By folding a bunch of these units and fitting them together, we can construct some interesting shapes, including all the platonic solids. If time permits, we will also talk about why there are only five of these. I will bring plenty of other interesting origami models to share, in case we have more time.
Ian Jauslin — Statistical Mechanics
Speaker: Ian Jauslin
Abstract: Many of us were taught that the world around us is made of invisible, microscopic particles such as molecules, electrons or atoms, and that the physical laws that bind them explain the observations we make on a human scale: for instance, temperature is explained as molecules moving, rotating and vibrating; electrical currents are accounted for as electrons zipping through a metal. But how much do we really understand about the relationship between the invisible microscopic world and the observable macroscopic one? Can we formulate natural mathematical models for the microscopic world, for which we can prove macroscopic physical laws?
Dennis Kriventsov — From optimization to free boundaries
Speaker: Dennis Kriventsov
Abstract: Calculus of variations is an area of analysis concerned with minimizing energies, generally over large collections of objects like functions or sets. I will describe how with some ideas from Calculus I (and maybe some hidden machinery) you can successfully find minimizers to such energies and then try to understand what they look like. Depending on the energy, this leads directly to partial differential equations and to free boundary problems (which are like partial differential equations satisfied by set boundaries). These subjects contain the most successful analysis developments of the past century--but also some basic questions which we just do not know how to answer.
Math Major Q&A Session
Speakers: Michael Beals, Stephen Hu, Adam Jamil, Brittany Gelb
Abstract: Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have Professor Beals, a current Rutgers graduate student, a math major now in tech, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come. Don't miss out on the first event of the semester!
SPRING 2022
Natasa Sesum — Ancient Solutions in Geometric Flows
Speaker: Natasa Sesum
Abstract: We will discuss the Ricci flow, importance of ancient solutions for the flow and classification results of ancient solutions in certain cases.
Recording password: WswS?.F1
Colin Fan — Holomorphic Nullstellensatz
Speaker: Colin Fan
Abstract: Hilbert proved his Nullstellensatz (German for "theorem of zeros") in 1893, establishing a duality (1-1 correspondence) between geometry and algebra in the setting of polynomials. In particular, the Nullstellensatz allows us to use algebra (easier) to study geometry (harder).
Being more ambitious, we can ask whether or not we can establish a duality between geometry and algebra in the setting of holomorphic functions. This question is intrinsically interesting, and is further motivated by the need to study complex manifolds. Thankfully, Rückert, in 1931 established his analytic Nullstellensatz, being the first one to introduce algebraic tools to complex analytic geometry.
In this talk, I hope to keep the prerequisites at a minimum with little to no proofs and highlight the beauty of both the Hilbert and the Rückert Nullstellensatz, as well as the difficulties one may face in the study of holomorphic functions of several variables.
James Propp — Tiling problems, old and new
Speaker: James Propp, UMass-Lowell
Abstract: A classic puzzle is to show that an 8-by-8 square cannot be tiled by 1-by-2 and 2-by-1 rectangles if two opposite 1-by-1 corner-squares are removed. A less famous puzzle is to count how many tilings there are if you DON’T remove those corners. This is an example of a dimer problem, first considered by physicists, at the intersection of graph theory and enumerative combinatorics. I’ll review some highlights in the theory of dimers, displaying elegant formulas for the number of tilings and striking pictures of the kinds of long-range order that can arise spontaneously.
Then I’ll discuss beautiful work of Conway, Lagarias, and Thurston applying combinatorial group theory to the study of tiling problems, yielding criteria for distinguishing between doable and non-doable tiling problems. I’ll apply the method to obtain a surprising result about a class of tiling problems that are ALMOST never doable.
Finally, bringing the two strands together, I’ll discuss my recent empirical work counting tilings related to trimer models. None of the methods that I know of for obtaining exact formulas for dimers apply here, yet the data show a wealth of patterns suggesting that there are deep theorems to be proved. Especially mysterious are some patterns involving the 2-adic numbers.
This talk should be understandable to undergraduates. No background in graph theory, enumerative combinatorics, combinatorial group theory, or p-adic analysis will be required, and you won’t need to know what a dimer or a trimer is when you arrive (though you will when you leave).
Sangjun Ko — A topological proof of Euclid's Theorem
Speaker: Sangjun Ko
Abstract: Euclid's Theorem, which states that there are infinitely many primes, is a classic theorem whose proof is studied by every student who is beginning to learn about proofs. As an undergraduate, Abel Prize winner Hillel Furstenberg came up with a different proof which takes ideas from point-set topology. We will give a quick introduction to basic point-set topology and present Furstenberg's proof, and, if time permits, discuss some properties of the so-called "evenly spaced integer topology" as well as other similar constructions.
Konstantin Mischaikow — We have Data and Computers, why do we need Math?
Speaker: Konstantin Mischaikow
Abstract: With today technology we can collect massive high dimensional sets of data from experiments and generate massive high dimensional sets of data numerically, but at the end of the day it is still a finite set of points. Assume that we are trying to understand a continuous process that generated the data. For example a process that can be modeled by a differential equation. In this case we probably want to be able to extract a continuous function, which raises the question how can one go from finite data to continuum objects with some sense of certainty.
To be a bit more concrete this talk will focus on how to go from finite data to the identification of a periodic orbit using algebraic topology.
Prerequisites for this talk: Calculus and Linear Algebra
Brian Pinsky — Weakening the Axiom of Choice
Speaker: Brian Pinsky
Abstract: You and your countably many friends are being held captive by an evil set theorist. Tomorrow, they will give each of you a hat that is one of countably many colors (multiple people may have the same color). Then you will all simultaneously try to guess your hat's color. You can see everyone else's hat, but not your own. You lose if infinitely many of your friends guess wrong.
Assuming the axiom of choice, your friends have a winning strategy. However, assuming the axiom of choice lets the evil set theorist cheat and banach-tarski the finite set of friends who guess wrong into infinitely many friends, and you still lose.
Is there an axiom that is strong enough for your friends to win, but too weak for the set theorist to cheat? In this talk, we'll explore strictly weaker versions of the axiom of choice, and hopefully find an answer.
Ian Coley — What is homotopy theory?
Speaker: Ian Coley
Abstract: Multivariable calculus naturally leads to the study of manifolds, spaces that look like our first friend ℝn. But what if we went the other way, where we stopped thinking about rigid spaces and starting thinking more floppily? I'll introduce you to homotopy theory: its origins, some cool little tools, and how it ties in with both algebra and geometry.
FALL 2021
Yunus E. Zeytuncu — Spectral Geometry of CR manifolds
Speaker: Yunus E. Zeytuncu (University of Michigan - Dearborn)
Abstract: In this talk, we will look at the relation between the spectrum of differential operators and the geometry of underlying manifolds. In particular, we will talk if one can hear the shape of CR manifolds. No prior knowledge of differential equations and geometry is necessary; we plan to construct everything from scratch.
Recording password: *$c1Ysx=
Olena Kozhushkina — On Everywhere Surjective Functions
Speaker: Olena Kozhushkina (Ursinus College)
Abstract: Analysis is full of examples verifying some kind of pathological property — continuous nowhere differentiable functions; linear discontinuous functions; differentiable nowhere monotone functions, and others. In this talk, we will focus on another interesting example — everywhere surjective functions.
A function f: R to R is called everywhere surjective, if for every non-empty interval (a, b), the image f((a, b)) is the entire R. Such function is doomed to be discontinuous at every real number, and in particular, provides a “strong” counterexample to the converse of the Intermediate Value Theorem. Namely, f satisfies the Intermediate Value Property — for any two numbers x1<x2 in R, if y is a real number between f(x_1) and f(x_2), then there exists a real number c in (x_1, x_2) such that f(c)=y — but f is not even close to being continuous.
The existence of everywhere surjective functions was first noticed by H. Lebesgue. In fact, there are plenty of such functions. The set S of everywhere surjective functions is 2^c-lineable; that is, SU{0} contains a vector subspace of the largest possible dimension, 2^c.
Nicholas Scoville — Discrete Morse theory as an introduction to topology
Speaker: Nicholas Scoville (Ursinus College)
Abstract: Like algebra, geometry, and number theory, topology is one of the main branches of mathematics today. Yet unlike algebra, geometry, and number theory, most people are not aware of what topology is. Because of its abstract nature, explaining the main concepts found in topology can be challenging. In this talk, we will discuss some of the main ideas in topology via an introduction to discrete Morse theory. Discrete Morse theory is a tool used in topology, but viewing this tool from the right point of view can shed great light on the important ideas in topology. We will look at two original research problems in discrete Morse theory that I have worked on with undergraduate students with many pictures, animations, and no equations.
Recording password: BP5cwg&%
Daniel Krashen — Algebraic structures and the algebraic structures on which they live
Speaker: Daniel Krashen
Abstract: Mathematical objects often have two lives. On the one hand, they are something like creatures living in complex ecosystems, but on the other hand, they can also be like the planets on which these creatures live. Sometimes these "planetary" algebraic structures can be given additional geometric or topological features, giving a range of new tools from other mathematical disciplines.
In this talk I will try to push this metaphor, and use it as a lens through which to view some open problems in the theory of quadratic forms and division algebras. In particular, we will see how these problems connect to ideas in geometry, such as the (in)famous "hearing the shape of the drum" problem.
Purdue Mathematics Society intercollegiate math social
The event will include:
- Games/Icebreakers
- Research Talks
- Problem Solving Sets (where people can present solutions in groups)
- Presentation of Solutions
Louis Gaudet — The circle method in analytic number theory
Speaker: Louis Gaudet
Abstract: Analytic number theory is a part of mathematics that studies properties of the integers and the prime numbers via tools from analysis (like calculus). The circle method, first developed by Hardy and Ramanujan in 1918, is an amazing tool in this field for studying “Diophantine problems” (i.e. solving equations with whole numbers). We’ll take a look at the basic ideas of the method in the context of some famous examples, like the Goldbach problem about sums of two primes.
Harlin Lee — Topics from Spectral Graph Theory
Speaker: Harlin Lee
Abstract: Spectral graph theory is the merging of linear algebra and graph theory to provide a novel understanding of graph structure in relationship to various matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. In this lecture, we will develop the mathematical machinery of SGT and explore some of its most spectacular results.
Math Major Q&A Session
Speakers: Michael Beals, John Kerrigan, Brooke Ogrodnik, Aaron Scheiner
Abstract: Q&A session for current, incoming, or prospective math majors. Come with any questions you have about the math major, graduate school, teaching in mathematics, and more. We will have Professor Beals, Professor Kerrigan, a former graduate student now in industry, and a representative from RUMA on the panel. Those interested in different major tracks are encouraged to come. Don’t miss out on the first event of the semester!
SPRING 2021
Matthew Issac — A Tour of Universal Properties
Speaker: Matthew Issac
Abstract: In this lecture, we will examine "universal properties," a concept which lurks in the background of many math classes. Moreover, these special properties can be used as definitions, once we have shown what we have defined indeed exists. We will begin with familiar examples (the empty set) and move onto more exotic constructions (the tensor product). Moreover, we will give some formal notion to what exactly it means to satisfy a "universal property," and what are the consequences of this.
Fioralba Cakoni — Given the answer how do you find the question?
Speaker: Fioralba Cakoni
Abstract: Inverse Problems are problems where causes for a desired or an observed effect are to be determined. They arise in many real life applications. Mathematical methods for solving inverse problems are rich and challenging. In this presentation we will discuss some toy examples and some more realistic ones to get a feeling for mathematics behind inverse problems.
Brandon Gomes — An Introduction to Logic and Formal Reasoning
Speaker: Brandon Gomes
Abstract: Logic is the foundation on which all of mathematics, the sciences, and philosophy are built. In this talk we explore what logic is and discuss how to study logic as its own branch of mathematics. We will cover historical and modern theories of logic as well as the intersection of logic with computer science and computability theory.
Eric Ling — Spacetime Singularities in Black Holes
Speaker: Eric Ling, Department of Mathematics
Abstract: In 2020 Sir Roger Penrose (a mathematician) won the Nobel Prize for his theoretical work on spacetime singularities in black holes. In this talk we give an introduction to the differential geometry techniques used to prove his theorem.
Recording password: !p16tM4b
Mariusz Mirek — Fourier Analysis in Additive Combinatorics
Speaker: Mariusz Mirek, Department of Mathematics
Abstract: We will discuss the proof of famous Roth's theorem asserting that every subset of integers with non-vanishing density contains infinitely many arithmetic progressions of length at least three.
Doron Zeilberger — Asking as Few Questions as Possible
Speaker: Doron Zeilberger, Department of Mathematics
Abstract: Any fool can figure out the answer if he or she (or it) is allowed to ask many questions, even if the answer to each question is Yes or No. But figuring out how to ALWAYS get the right answer in as few questions as possible, is not so easy.
Recording password: 5nm%BWsG
Nicholas McConnell — On the Matrix Exponential
Speaker: Nicholas McConnell, RU '21
Abstract: The exponential function is an important function in mathematics. Among other reasons, y = e^(at) solves the differential equation dy/dt = ay, y(0) = 1. However, using both differential equations and linear algebra, one can take an n x n matrix A, and establish a matrix y = exp(tA) that depends on a parameter t. Then the differential equation dx/dt = Ax, x(0) = x_0 (where x is in R^n) has solution x = exp(tA)x_0.
Linear operators (which are not typically thought of using matrices) can also be exponentiated, and whenever T is a linear operator, exp(tT)x_0 is the solution to dx/dt = Tx, x(0) = x_0. The main example of this is, if D = d/dx then exp(tD) f(x) = f(x+t). We will see the definition, basic properties and use of exponentiation of linear operators and matrices.
A Lie group is a smooth manifold which has a group structure where the multiplication and inverse are smooth maps (e.g., R^n under addition; T^n (where T = {z in C: |z| = 1}) under multiplication; GL(n, R) under multiplication; GL(n, C); SL(n, R); SL(n, C); O(n); SO(n); U(n); SU(n)). If G is a Lie group, the tangent space g at the identity is a Lie algebra, as left-invariant vector fields are closed under the Lie bracket and correspond bijectively to vectors at the identity. There is an exponential map exp : g -> G such that for each X in g, the map t -> exp(tX) is the unique smooth homomorphism f : R -> G such that f'(0) = X. For GL(n, R), GL(n, C), and their subgroups, this is merely the usual matrix exponential.
Recording password: .PmmOdV9
Roberta Shapiro — Mugs and Donuts: An Introduction to Surface Topology
Speaker: Roberta Shapiro, Graduate Student
Abstract: What is a surface? What are all* the types of surfaces that exist in the wild? How do we tell them apart? We will answer these questions and more** as we take a first dive into the surface topology.
*not all
**BYOQ: bring your own questions
Recording password: XXL!h.32
Board Game Night with RUCP
Description: Join us for Among Us, Skribbl.io, and Codenames. The event will be hosted over discord.
FALL 2020
RUMA Q&A Session
Speakers:
Robert Beals, Chair of the Undergraduate Honors
John Kerrigan, Part Time Lecturer
Brooke Logan, Department of Mathematics, graduate student
Adam Jamil, RU ’21, Mathematics and Computer Science
About: This will be a talk all about the math major and the different paths it can lead you on. With perspectives from a wide variety of mathematical perspectives, come prepared with questions!
Nick Backes — p-adic Numbers
Speaker: Nick Backes, Graduate Student
Abstract: The rational numbers are fantastic; however, they have gaps. You can make sequences of rational numbers which should converge, but they do not converge to a rational number. Thus, we introduce the real numbers as a completion of the rational numbers. However, convergence of a Cauchy sequence depends on a chosen norm (or absolute value, if you prefer). As it turns out, there is more than norm on the rationals, and if the rationals are completed with respect to a different norm, then we obtain the p-adic numbers. The p-adics are somehow a different way to fill the gaps between rational numbers. The p-adics have unexpected and beautiful geometric, analytic, and number-theoretic properties.
Recording password: ya0x^Q8i
Konstantin Matveev — A Surprising Journey of Longest Increasing Subsequences
Speaker: Konstantin Matveev, Department of Mathematics
Abstract: I will talk about an eventful story of longest increasing subsequences. It will touch on a combinatorial geometry problem that has led to a 68 years long marriage, on computer experiments run by one of the key participants of the Manhattan project, on representation theory of symmetric groups, on random matrices, and on statistical physics.
Recording password: k5d?&p8%
Simon Thomas — The Axiom of Choice
Speaker: Simon Thomas, Department of Mathematics
Abstract: In this talk, I will discuss the Axiom of Choice and its striking consequences.
Recording password: MK4Md^*H
Kristen Hendricks — Classical and Modern Invariants of Knots
Speaker: Kristen Hendricks, Department of Mathematics
Abstract: We’ll give a brief introduction to what knot theory is and why you might be interested in it, and talk about some classical invariants of knots and what they detect. We’ll then introduce a modern invariant called Heegaard-Floer knot homology from the early 2000s, and talk about its properties and its relationship to classical invariants.
Recording password: t!aeE9p!
