Wednesday Colloquium
UPCOMING EVENTS
Date: 5:40-6:40pm on Wednesday 4/17
Location: PH 100
Speaker: John Mackey
Title: Computer Assisted Mathematical Discovery
Abstract: Chat GPT, AlphaGo, and AlphaGeometry have attracted a lot of attention recently. Computers are getting better and better at problem solving. But what of problem posing? Can computers help us arrive at interesting conjectures and state more nuanced theorems?
In this talk, I'll introduce SAT solvers and talk about their increasing utility in solving combinatorial problems. In particular, we can ask for the minimum number of convex pentagons when n points are placed in the plane in general position. SAT can solve the problem for large enough sets of points that allow us to conjecture the minimum for all n.
We'll discuss other open problems and see how SAT may be applied.
PAST EVENTS
Date: 5:40-7:00pm on Wednesday 4/3
Location: PH 100
Title: Math Club's Annual Meme Competition
Description: It's the most wonderful time of year, Math Club's annual math meme competition! I can't believe it's been three years since I created this tradition and if you would like to continue this tradition you should consider running for math club exec!
This year there will be two categories: originally created memes and memes that you have found from the internet that you think is the best in the entire set of math memes that already exist. These memes can be either an image, gif, or video. The winner of the competitions will be decided on Wednesday, April 3rd, in Porter 100 at 5:40pm where we will have a wonderful dinner and vote on the best meme together (PLEASE NOTE RSVP IS REQUIRED FOR DINNER).
This is the link for the meme submission for the competition, the deadline is April 3rd by noon. Link: https://forms.gle/XKwoiiCDXdCgVcRk6
This is the link for the Meme night Dinner RSVP for the competition, the deadline is April 3rd by noon. Link: https://forms.gle/HiV3bEtuYNQmVG7n9
Date: 5:40-6:40pm on Wednesday 3/27
Location: PH 100
Speaker: Patrick Massot
Title: Why explain mathematics to computers?
Abstract: A growing number of mathematicians explain mathematics to computers. This process is called formalization. In this talk, I will show what formalization looks like, explain what kind of things it teaches us, and how it could become very useful.
Date: 5:40-7:00pm on Wednesday 3/19
Location: PH 100
Speaker: Lillian Dukes
Title: Q&A with Lillian Dukes
Description: This week's speaker/guest will be Lillian Dukes, a CMU alumna (math and electrical engineering) who has worked in many aviation/aerospace related jobs, most recently as the VP of Technical Operations for Atlas Air. Rather than a standard talk, she is interested in a more discussion-based format, and thus the event will likely unfold more as a Q&A/interview than a talk as usual, but it should be very interesting! We encourage you to come if you are interested in hearing of her 30 years of experience! As usual, there will be pizza, but if you are interested please fill out THIS RSVP FORM so we know how much to order. We hope to see you there!
Date: 5:40-6:40pm on Wednesday 2/28
Location: PH 100
Speaker: Bernardo Subercaseaux
Title: We made a computer solve my favorite graph coloring problem, and so can you!
Abstract: In this talk I will present the basics of SAT-solving, and how it can be used to solve problems in discrete mathematics. In particular, I will show how Marijn Heule and I used SAT-solving to determine the packing-chromatic number of the infinite square grid, a problem in graph coloring that was open since 2002. The final "proof" consists in a single file of over 30 terabytes which implies the packing-chromatic number to be 15. The main technical ingredients of our solution are an efficient SAT encoding discovered by reverse engineering, a cube-and-conquer approach designed to leverage parallel computation, and symmetry-breaking techniques.
Date: 5:40-6:40pm on Wednesday 2/21
Location: PH 100
Title: "Which Math Concentration is Right for Me?"
Description: If you are interested in learning about the different math concentrations to declare your own (or switch out of your current one), then join us as we discuss the 5 math concentrations and hear from panel members about each of them! We will discuss the requirements for each, the differences between them, and how each of the panel members decided on their concentration. Come with questions about the concentrations or your future plans and we'll help you decide with concentration is right for you! As usual, there will be pizza, but please fill out THIS RSVP FORM if you would like some! (And please only fill out the form if you are planning on coming to and staying for the event.)
Date: 5:40-6:40pm on Wednesday 2/14
Location: PH 100
Speaker: David Offner
Title: Packing and decomposition problems on hypercube graphs
Abstract: Given graphs $G$ and $H$, a graph packing problem asks how densely vertex- or edge-disjoint copies of $H$ can be packed in $G$. A decomposition problem asks whether the vertex or edge set of $G$ can be partitioned by vertex- or edge-disjoint copies of $H$. This talk concerns packing and decomposition problems when $G$ is the $n$-dimensional hypercube graph $Q_n$. The history of such problems date to a 1955 theorem of Ringel that if $n$ is a power of 2, $Q_n$ has an edge decomposition into Hamiltonian cycles, and despite a number of recent advances, there are still many open problems. For example, in 2014 Erde conjectured that if $n$ is even, $k < 2^n$, and $k$ divides the number of edges of $Q_n$, then $Q_n$ has an edge decomposition into paths of length $k$. We survey what is known about packing and decomposition problems on the hypercube, including new progress on Erde's conjecture.
Date: 5:40-6:40pm on Wednesday 2/7
Location: PH 100
Title: Lunar New Year Dinner Celebration
Description: Please RSVP no later than Monday, 2/5, by 12:00 pm. Lunar New Year is celebrated in many Asian countries and it marks the beginning of the new year based on the lunisolar calendar. For many people who celebrate Lunar New Year, we often will eat a nice dinner with our family/friends and give each other wishes for the New Year (and in my family we exchange Lì Xì) . However, since all of us are away at school we hope that you'll join us on Wednesday to celebrate with the Math Community here at CMU!
*Note: Everyone is encouraged to join even if you don't celebrate the Lunar New Year personally!
Date: 5:40-6:40pm on Wednesday 1/31
Location: PH 100
Speaker: Justin Hsieh
Title: Regular Expressions and Friends
Abstract: What do (0|1(01*0)*1)* and (0|-?[1-9]\d*)(\.\d+)?|-0\.\d+ mean? Regular expressions essentially allow us to "find and replace" many different strings at the same time. We will use the theory of regular languages and finite automata to implement regular expressions, and then we will find out how to surpass this power.
This talk will not require any background knowledge about theoretical computer science.
Date: 5:40-6:40pm on Wednesday 1/17
Location: PH 100
Speaker: Robert Trosten
Title: Way Too Many Applications of the Compactness Theorem
Abstract: Model theory lies at the intersection of set theory, combinatorics, universal algebra, and logic, and it allows us to play God - to create and destroy universes at will. A particularly important result is that of the compactness theorem, which allows us to extend from finite knowledge to infinite knowledge - that is, if we desire to craft a universe with X properties, it suffices to build a universe with Y properties for each finite subset Y of X. In this talk, we will take you on a guided tour to build the machinery of propositional logic, the methods of first-order logic, and use compactness of each to get results in all sorts of areas! By creating an appropriate universe, we can easily color infinite graphs, create algebraic closures, make rigorous infinitesimal analysis, discover how your typical set of natural numbers is ordered, and so much more both outside and inside of model theory itself. In total, I will present eight applications of the compactness theorem over the course of the hour. (#7 will shock you.)
We will try to assume minimal prerequisites: Concepts and an open mind should be sufficient, as we’ll define, prove, and/or handwave any technical details we need along the way in our exploration into model-theoretic methods. Linear algebra may be helpful to build intuition, and a teeny bit of cardinal arithmetic may be useful. The applications will be all over the place, so we hope there’s something for everyone! I find this stuff really cool, and think it's a damn shame that someone could go their entire undergrad without even seeing it once.
Date: 5:40-6:40pm on Wednesday 12/6
Location: PH 100
Speaker: Brenda Chen
Title: Bounding Lifts of Markoff Triples mod p
Abstract: In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo p solutions to the equation x^2+y^2+z^2=3xyz are covered by the integer points for most primes p. In this talk, we show how the algorithm given in the paper of Bourgain, Gamburd, and Sarnak can be used to obtain upper bounds on lifts of Markoff triples modulo p. We provide numerical evidence that these bounds can be improved on average and with high probability, and present an implementation of the BGS algorithm. This is joint work with Elisa Bellah, Elena Fuchs and Lynnelle Ye.
Date: 5:40-6:40pm on Wednesday 11/15
Location: PH 100
Speaker: Krishan Canzius
Title: How to slay hydras and count past infinity
Abstract: The Hydra is a many-headed monster from Greek mythology. Every time a hero cuts a head off the Hydra several more will grow back in its place. In this talk we'll model the Hydra as a certain kind of graph, and we'll model a battle with the Hydra as a single-player mathematical game. The main question we'll answer is: Can we win the hydra game, and if so which strategies will guarantee a win? In order to answer this question, we'll need to introduce the ordinals: a number system which extends the natural numbers and allows us to count past infinity.
Summer Experiences Panel on Friday
Time: Wednesday, 11/08/23, 5:40-7:00 pm
Location: PH 1000
Description: At this panel, we'll be discussing what you can do over the summer as a math major! We'll have a panel of upperclassmen who have done a variety of things over the summers talking about their experiences, too, and giving advice on how you can do what they did. If you have any questions you want the panel/event to answer, fill out this RSVP form !
Date: 5:40-6:40pm on Wednesday 11/1
Location: PH 100
Speaker: Udita Katugampola
Title: Non-traditional and faster approach to find eigenvectors
Abstract: For decades, if not for centuries, we have been overlooking the use of columns of a matrix when it comes to producing eigenvectors even though the vectors are sitting in the column space of a matrix. The traditional Gaussian elimination approach uses rows instead of columns. The old school says that "we cannot use columns to find eigenvectors." However, to the contrary, the opposite also happens to be true.
The idea of an eigenvector came as a brainchild of Daniel Bernoulli’s work on vibrating strings in 1732, despite there being no explicit reference to it in his writings. Since then, the idea of an eigenvector has continued to evolve over several centuries.
To our surprise, we still use the same old idea when it comes to finding eigenvectors. The traditional approach requires two steps: first find the eigenvalues and then find the eigenvectors. In this talk we show that we do not need the second step since they already appear as nonzero columns of a certain matrix called an eigenmatrix, a term introduced to mathematics in our work. To exaggerate a bit, we can find an eigenvector in few seconds, while the traditional method takes several minutes. This then gives a faster approach to diagonalize a given matrix, if it has ‘enough’ linearly independent eigenvectors.
Date: 5:40-6:40pm on Wednesday 10/25
Location: Zoom
Title: Course Selection Meeting
Description: We'll be going through the list of math courses offered next semester and upperclassmen will give their thoughts! Drop by to learn which classes to take, which professors to avoid, and many, many other fun things! At the appropriate time, you can join at this link . We hope to see you there!
Date: 5:40-6:40pm on Wednesday 10/11
Location: PH 100
Speaker: Emma Hayes
Title: Surrogate Modelling of PDEs and Applications to Inverse Problems
Abstract: Neural networks have become a powerful tool to provide numerical solutions for scientific problems with increased computational efficiency. This efficiency can be advantageous for numerically challenging problems where time to solution is important or when evaluation of many similar analysis scenarios is required. In my research, we considered solutions of the 2D acoustic wave equation under the influence of a driving force at a source point. We trained neural networks on data generated using both single and multiple source locations, where our ground truth was generated on a course grid using a Discontinuous Galerkin Method. Our neural networks are able to produce an accurate solution over a square domain over a fine grid roughly 300 times faster than the Discontinuous Galerkin method on a single source. During this talk, I will discuss the mathematical foundations for this research, the construction of Physics Informed Neural Networks, the results thus far, and the continued work underway.
Date: 5:40-6:40pm on Wednesday 10/04
Location: PH 100
Speaker: Joshua Siktar
Title: Grad School Journeys: Nonlocal Edition
Abstract: This talk will be divided into two parts. The first will tell the story of how my involvement in Carnegie Mellon's Mathematics department as an undergraduate sparked my interest in conducting research and teaching, which ultimately led to me pursuing a PhD in mathematics. Along the way I will give some advice on deciding whether to go to graduate school, choosing where to apply, and managing the workload as a graduate student.
The second part of the talk will be an overview of my research that will be presented in my dissertation. I've studied the optimal control of non-local, or integral equation, models in peridynamics, which is a contemporary approach to modeling displacement of materials that does not rely on a continuity assumption of the material. This gives peridynamics models a distinct advantage over continuum mechanics models, which inherently assume continuity; thus these models can be used to study spontaneous crack formation and fracture. I will discuss some of the tools used to show that such optimal control problems have unique solutions, and then talk about using finite element approximations to discretize these problems. Consequently, both theoretical and numerical aspects of the problems will be covered
SEMS Symposium
Time: Wednesday, 9/27/23, 5:00-7:00 pm
Location: PH 100
Description: he SEMS Symposium is similar to our standard talks, but instead of a single talk, four groups of students who participated in CMU's Summer Experiences in Mathematical Sciences (SEMS) will talk about the research that they did this past summer. If you have never participated in research at CMU and are interested as to what that may entail, you are especially encouraged to come! Of course, everyone is welcome. In addition to the standard pizza, Irina will be ordering extra food for people who stay the whole time. If you would like either of this, please RSVP .
Date: 5:40-6:40pm on Wednesday 9/20
Location: PH 100
Speaker: William Mance
Title: Normal Numbers
Abstract: Informally, a real number is normal in base b if in its b-ary expansion all digits and blocks of digits occur as often as one would expect them to, uniformly at random. Borel introduced normal numbers in 1909 and proved that Lebesgue-almost every real number is normal in all bases b ≥ 2. Even though this shows that, in some sense, normal numbers are "typical," no example of a number normal in all bases was given until 1939 by Turing. In the last 100 years, the study of normal numbers has spread over a wide breadth of seemingly unrelated disciplines. Normality is closely related to number theory, ergodic theory, theoretical computer science, probability theory, fractal geometry, descriptive set theory, and other areas of math. We will explore the basic properties of normal numbers and surprising connections they have, depending on the interest of the audience.
Date: 5:40-6:40pm on Wednesday 9/13
Location: PH 100
Speaker: Connor Gordon
Title: NIM!
Abstract: Nim is a simple mathematical game about picking stones from piles. Despite this simplicity, there's a lot of rich mathematics hiding in its strategies! In this talk we will discuss the optimal way to play Nim, along the way exploring non-standard inductions, a perhaps unexpected appearance of binary arithmetic, and (questionably qualified) commentary on the nature of mathematical discovery.
Date: 5:40-6:40pm on Wednesday 4/26
Location: PH 100
Speaker: Irene Fonseca
Title: Mathematics and Imaging Science
Abstract: In this talk we will address the mathematical treatment of image processing, including inpainting, recolorization, denoising, and machine learning schemes.
Date: 5:40-6:40pm on Wednesday 4/19
Location: PH 100
Speaker: Sangmin Park
Title: Identifying boundary points of point clouds
Abstract: Given i.i.d. sample points from an unknown distribution supported on a bounded domain, how can we efficiently and reliably identify points close to the boundary of the domain? Knowledge of the boundary points is crucial to solving PDEs on point clouds/graphs, which have found applications in semi-supervised learning and data-depth assignment. In this talk, we will use elementary calculus and basic probability to understand and analyze a boundary estimation algorithm, and see its application to assigning data depth on real data sets.
Date: 5:40-6:40pm on Wednesday 4/12
Location: PH 100
Speaker: John Mackey
Title: Codes, Tilings, and Hadamard Matrices
Abstract: This will be a chalk talk with little direct preparation. It will mostly be a recollection of some mathematics that I've seen. Hopefully people will find it interesting and/or humorous. It is mostly an excuse to see old friends and make new friends.
Date: 5:40-6:40pm on Wednesday 3/22
Location: PH 100
Speaker: Jeremy Avigad
Title: A formal verification of the Cantor-Schröder-Bernstein theorem
Abstract: Computational proof assistants now make it possible to write mathematical definitions, theorems, and proofs in stylized languages, like programming languages, in such a way that the details can be fully checked by computer. The technology has been around for decades, but only recently has it had significant uptake in the mathematics community. I'll talk about the technology and illustrate it by working through the formalization of the Cantor-Schröder-Bernstein, which says that if A and B are any two sets such that there is an injection from A to B and vice-versa, then A and B are in one-to-one correspondence.
Date: 5:40-6:40pm on Wednesday 3/1
Location: PH 100
Speaker: Amzi Jeffs
Title: How to realize a convex code
Abstract: We will give an overview of the study of "convex codes," a vein of research that seeks to classify how convex sets can be arranged in d-dimensional space. This task is quite difficult, even in the plane, and gives rise to a number of striking examples and neat geometric theorems. We will share some choice techniques and results, along with plenty of pictures.
Date: 5:40-6:40pm on Wednesday 2/22
Location: PH 100
Speaker: James Cummings
Title: Fun with randomness
Abstract: Probabilistic algorithms and constructions turn out to be surprisingly powerful. I will discuss some examples in number theory and combinatorics.
Date: 5:40-6:40pm on Wednesday 2/15
Location: PH 100
Speaker: Sherry Sarkar
Title: Algorithms with Uncertainty: Online Bipartite Matching
Abstract: "Algorithms with uncertainty" is an exciting new regime of research taking the theory CS world by storm. In this talk, I'll begin by giving a brief exposition of the types of problems and settings often considered in these areas. Then I'll delve into a popular problem in the field -- online bipartite matching. We'll investigate different varieties of online bipartite matching, ranging from the classic maximization problem, random arrival, and recourse. The goal of my talk will be to expose the audience to the rich variety of relatively new settings inspired by "uncertainty", with which we can re-examine the classical combinatorial problems.
Date: 5:40-6:40pm on Wednesday 2/8
Location: PH 100
Speaker: Ryan O'Donnell
Title: High-dimensional expanders, and graphs of constant link, and the Classification of Finite Simple Groups
Abstract: High-dimensional expanders" are a relatively new generalization of expander graphs, and they have recently been used to solve major open problems in theoretical computer science (e.g., the design of good quantum error-correcting codes). Only trouble is, it's really really complicated to construct them; known methods use a lot of complicated group and number theory. One way you could construct them is to find big graphs where the neighborhood of every vertex induces the same graph L, and L itself is a little expander graph. But how easy is it to find graphs like that? I thought it was going to be easy! But... it wasn't, and I ended up having to use some complicated group theory myself. But at least this motivated me to learn about some new things about group theory. Anyway, I'll tell you about some of this stuff.
Date: 6:05 PM - 7:05 PM on Wednesday 2/1
Location: PH 100
Speaker: Florian Frick
Title: Topology For Discrete Problems
Abstract: Topology is the study of continuous maps and properties that are preserved under continuous deformations. It might be surprising that topology has found numerous applications in combinatorics, discrete geometry, economics, and data science -- areas that are discrete as opposed to continuous. I will give a few examples of such applications and explain how to spot problems, where topology might be useful. No prior knowledge of topology is needed.
Date: 5:40-6:40pm on Wednesday 1/25
Location: PH 100
Speaker: Connor Gordon
Title: From ZFC to Forcing: A Speedrun
Abstract: Cantor's famous diagonal proof showed that the infinite size of the real numbers is in some sense "genuinely larger" than the infinite size of the natural numbers. This raises a natural question: is there some "intermediate infinity" between the two? This question is known as the Continuum Hypothesis (CH), and as it turns out, we don't know the answer; in fact, we can't know!
This talk will aim to take us from the axioms of set theory, ZFC, to a (sketch of a) proof of the independence of the CH from said axioms. Along the way, we'll establish the relevant set theory background and answer questions of what it even means prove something of this nature. The only prerequisite required is concepts-level understanding of logic, sets, and functions. Everything else will be defined as needed.
Date: 5:40-6:40pm on Wednesday 11/30
Location: PH 100
Speaker: Vlad Oleksenko
Title: The obstacle problem
Abstract: In this talk, we will examine the one-dimensional obstacle problem — a classic motivating example in the field of variational calculus. Using only elementary tools and methods, we will prove several interesting results from my research on this topic. Additionally, we will discuss concepts like weak derivatives, Sobolev spaces, energy functionals, and inner variations. Overall, this presentation will provide a gentle introduction to motifs commonly encountered in calculus of variations, without going too much into technical details.
Date: 5:40-6:40pm on Wednesday 11/9
Location: PH 100
Speaker: Clinton Conley
Title: Continued fractions
Abstract: The standard way of representing real numbers as a decimal expansion by natural numbers ultimately boils down to counting to ten on our fingers (and summing some geometric series). But what would we have done without fingers? We discuss an alternate method of representing real numbers by sequences of naturals, and analyze some ways in which it is superior to the standard decimal expansion. Time permitting, we also mention some connections between these "continued fraction" representations and descriptive set theory and dynamical systems.
Date: 5:40-6:40pm on Wednesday 11/2
Location: PH 100
Speaker: Elchanan Haas
Title: The Math Behind DALL-E 2
Abstract: DALL-E, Stable Diffusion and Imagen are able to generate images based off of any prompt. They have revolutionized AI content generation. How do they do that? Learn the math behind diffusion models and the connections to thermodynamics in this talk.
Date: 5:40-6:40pm on Wednesday 10/26
Location: PH 100
Speaker: Prof. Giovanni Leoni
Title: Ingenious short proofs
Abstract: In this talk, I will present some recent short proofs in analysis that I like.
Date: 5:40-6:40pm on Wednesday 10/12
Location: PH 100
Speaker: Prof. Tom Bohman
Title: Lonely runners and coprime matchings
Abstract: Suppose n runners are running on a circular track of circumference 1, with all runners starting at the same time and place. It is not a race. Each runner proceeds at their own constant speed. We say that a runner is lonely at some point in time if the distance around the track to the nearest other runner is at least 1/n at that point in time. For example, if there two runners then there will come a moment when they are at opposite points on the track, and at this moment both runners are lonely. The lonely runner conjecture asserts that for every runner there is a point in time when that runner is lonely. A coprime matching of two sets of integers is a matching that pairs every element of one set with a coprime element of the other set. In this talk we will discuss a recent partial result on the lonely runner conjecture. Coprime matchings of intervals of integers play a central role in the proof of this result.
Joint work with Fei Peng
Date: 5:40-6:40pm on Wednesday 10/5
Location: PH 100
Speaker: Tomas Tkocz
Title: Extremal-Volume Sections of the Cube
Abstract: How small can the volume of a slice of the unit cube be? This question, asked by Good in the 70s in the context of its application in geometry of numbers has turned out to be influential, prompting development of several important methods, as well as spurring further problems and research directions of independent interest in convex geometry, with strong ties to probability and harmonic analysis. We shall present sharp bounds for hyperplanes and discuss some of its spectacular applications.
Date: 5:30-6:30pm on Thursday 9/29
Location: DH 2210
Speaker: David Vaughn, Chief Investment Officer, Clarivest Assset Management LLC
Title: Quantitative Equity Portfolio Management and its Challenges
Abstract: David Vaughn holds a B.S. degree in Economics from California Institute of Technology and an M.S. degree in Computatonal Finance from Carnegie Mellon University. Using a combination of mathematics and economics, Mr. Vaughn founded the ClairVest Asset Management company in 2000 and built the company to the point that it now manages a portfolio valued at $5.3 billion. In this talk, the Mr. Vaughn will discuss his experience and key lessons learned in the field of quantitative equity portfolio management. In addition, he will cover some of the current and upcoming challenges faced in the industry.
Mr. Vaughn values and promotes mathematics education. In particular, he coaches elementary and middle school math teams. In 2014 Mr. Vaughn coached California's state MATHCOUNTS team to victory at the national competition.
Date: 5:40-6:40pm on Wednesday 9/21
Location: PH 100
Speaker: Subhasish Mukherjee
Title: A Brief Trek through Abstract Harmonic Analysis
Abstract: How do techniques for solving many classes of differential equations and the decomposition of symmetries of vector spaces relate to each other? They are both glimpses into the ideas of harmonic analysis. We'll take a brief tour through some elements of the field and see some amazing properties we can glean from Fourier analysis on R and then see generalizations to much wider arrays of groups.
Date: 5:40-6:40pm on Wednesday 9/14
Location: PH 100
Speaker: Zoe Wellner
Title: Counting things with bad maps
Abstract: There is a particularly fun example of proof by contradiction which says that the existence of a combinatorial object is equivalent to a map behaving in weird ways. Then we use topological facts, like Brouwer's fixed point theorem, or the Borsuk--Ulam theorem to say that those maps were in fact not possible. Aside from my own bias towards this proof style, this is a useful technique for looking at questions of fair division of rent, splitting necklaces, graph colorings, and many more. Come to this talk to hear about this surprising connection between topology and combinatorics!
Date: 5:40-6:40pm on Wednesday 9/7
Location: WEH 7500
Speaker: Thomas Lam
Bio: Thomas is an undergrad at CMU and VP External in the Math Club.
Title: The Number Rotation Puzzle
Abstract: Have you ever solved a 15 puzzle? What about a Rubik's Cube? Combination puzzles are a lot of fun to solve, and unsurprisingly, there's quite a bit of math behind them too. In this talk, we will explore a puzzle that works like a "2D Rubik's Cube": You're given a scrambled grid of numbers, and you can take a square subgrid of fixed size and rotate it 90 degrees. When can you unscramble the grid using a sequence of such rotations, and how? Our journey to answering this will involve many surprising connections to very different fields of math
Date: 5:40-6:40pm on Wednesday 4/20
Location: HOA 160
Speaker: Tess Anderson
Bio: Professor Tess Anderson is currently at Purdue University, where she studies the interplay of harmonic analysis and number theory. This fall, she will be joining us at CMU.
Title: Let's count things: Arithmetic statistics meets Fourier analysis
Abstract: Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves. Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields. How often does a random polynomial fail to have full Galois group? How many number fields of a given degree and bounded discriminant are there? In nontechnical terms: let's count things together!
Date: 5:40-6:40pm on Wednesday 4/13
Location: HOA 160
Speaker: Varun Gudibanda
Bio: Varun is a PhD student at UW Madison and is an alum of CMU's math department. His interests are in mathematical physics and PDEs. He also started the "Today I did x" email intro to math club emails!
Title: It's Getting Hot In Here Let's Study Heat Convection
Abstract: Have you ever boiled water? If so, you’re familiar with the concept of convection. But did you know that one single number involving convection has divided a research community? Come to this week's talk to find out about the world of convection and this one number that has been debated for decades.
Date: 5:40-6:40pm on Wednesday 3/23
Location: HOA 160
Speaker: Robert Pego
Bio: Dr. Pego is a professor here at CMU interested in analysis, dynamics, and PDEs. His work is often motivated by fluids, physics, and even population ecology!
Title: How to count fish using mathematics
Abstract: In 2003 the Japanese fisheries scientist H.-S. Niwa published a remarkable study of the distribution of sizes of schools of fish in the mid-ocean. Niwa's ideas led to studying an infinite set of kinetic equations for the merging and splitting of animal groups. We explain the non-Gaussian nature of the equilibrium school-size distribution using complex-function theory for Bernstein functions (related to Laplace transforms). For this we make use of a new double-transform theorem, which touches on a diverse collection of topics across mathematics: generating functions for Hausdorff moment sequences, Fuss-Catalan numbers, convolution semigroups, and a striking integral formula for binomial coefficients.
Date: Wednesday, March 16, 2022
Location: 5:40-6:40pm HOA 160
Speaker: Ian Tice
Bio: Prof. Tice is a professor here at CMU interested in analysis and PDES. He's also involved with the honors program here!
Abstract: Banach spaces are essential tools in infinite dimensional analysis because they serve as useful container spaces in which to construct various interesting mathematical objects (solutions to differential or functional equations, pathological functions, integrals, generalized limits, notions of fractional derivatives, etc). Due to their ubiquity in modern analysis, it is often convenient to have tools for constructing new Banach spaces that interpolate between a pair of given spaces. In this talk we will give a completely elementary but nontrivial example of how one might go about constructing such interpolation spaces. We will use nothing more than one dimensional calculus (continuity and the fundamental theorem of calculus) and basic linear algebra.
Date: TUESDAY, March 15, 2022
Location: 4:30 P.M., WeH 7218 & Zoom
Speaker: Terrance Pendleton, Drake University
Title: The Camassa-Holm Equation: Analysis, Numerics, Generalizations and Applications
Abstract: In this talk, we study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation and its generalizations. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a variety of numerical methods to study both the analytical and physical properties of the Camassa-Holm equation and show its potential for modeling the propagation of tsunami waves. In particular, we provide global existence and uniqueness results for the Camassa-Holm Equation by establishing convergence results for the particle method applied to these equations, and then use this same method to numerically quantify the nonlinear interaction among the peakon solutions. We conclude the talk by proposing new invariant-preserving finite difference schemes for a generalization of the Camassa-Holm equation as a potential model for the propagation of tsunami waves.
Date: 5:40-6:40pm on Wednesday, 3/2
Location: HOA 160
Speaker: Simran Khunger
Bio: Simran Khunger is the math club president. She's currently a senior preparing to go off to graduate school!
Title: So you want to go to grad school?
Abstract: Our math club president Simran Khunger will be giving a talk "So you want to go to grad school?". She'll explain the math grad admissions process, what opportunities CMU math students have for each of their undergrad years, and her experience going through the graduate process herself. There will also be a part at the end discussing post-grad school careers and applying to non-math grad programs. This is a great opportunity for underclassmen to hear about what grad school is like, how to get there, and whether or not it's right for you!
Date: Wednesday, 2/23, 5:40-6:40pm
Location: Porter Hall A18B
Speaker: Andrew Newman
Bio: Andrew is a postdoc at CMU who's interested in the intersection of probability, topology, and combinatorics. He might be your 21-325 professor!
Title: An introduction to stochastic topology!
Abstract: Over the last 50 years, mathematicians have been able to study random graphs to answer questions about what graphs typically look like. A graph though, is just 1-dimensional; it's built out of lines and points. In this talk we'll discuss what random higher-dimensional spaces are, why they are worth studying, and overview the basic ideas from topology that allows us to understand and, sometimes, answer questions like: What is the expected number of 10-dimensional holes in a random 11-dimensional object, or what is the probability that a random 12-dimensional shape can be drawn in 24-dimensional space? The only background knowledge that will be assumed is a basic familiarity with matrices.
Date: Tuesday, 2/15, 5:40-6:40pm
Location: Zoom
Speaker: Zoe Wellner
Bio: Zoe is a 4th year grad student in the math department's ACO program who has a very cute cat (see the flyer)! The title and abstract are also attached below!
Title: Adventures in Weird Geometries
Abstract: Looking at drawings, Pac-man’s universe, or even the ground we walk on, many things appear “flat”. Looking at space locally only tells us so much about the fundamental properties of the space itself. It can be extremely valuable to consider problems in spaces where they naturally reside. Mathematically, if we have a problem that loops in on itself we might want to consider solving that problem on a torus. If we need to differentiate the parity of an object, maybe it lives in a Möbius strip? If we are looking to accurately represent what we see around us in a drawing on a piece of paper, we are using the properties of projective space! We will discuss how these differ mathematically and the different objects these connect with.
Date: Wednesday, 2/2, 5:40-5:50pm
Location: Zoom
Speaker: Alec Sun
Bio: Alec is a grad student in the math department interested in combinatorics and theoretical CS. He is also a TA for Probability and Computing this semester!
Title: Sperner's Lemma
Abstract: Sperner's Lemma is a combinatorial analog of Brouwer's Fixed Point Theorem. Given a triangle ABC with triangulation T, suppose that the set of vertices of T is colored in 3 colors such that A, B, C are colored 1, 2, and 3, and the color of each vertex on an edge of ABC is the same as one of its endpoints. Sperner's Lemma states that there exists a triangle from T whose vertices are colored with 1, 2, and 3. After sketching a proof, we will see how Sperner's Lemma can be applied to resolve fair division problems as well as establish the Brouwer's Fixed Point Theorem.
Date: Thursday, 1/27/22, 5:40-6:40PM
Location: Zoom
Speaker: Misha Lavrov
Bio: Misha Lavrov is formerly a graduate student at CMU, and currently a math professor at Kennesaw State University in Georgia who occasionally writes about himself in the third person. His interests mathematical and nonmathematical include graph theory, baking cookies, probability, playing board games, and very large numbers
Title: The Mathematics of Sproutball
Abstract: Sproutball is a variant of dodgeball which is exceptionally good at entertaining small children for almost arbitrarily long periods of time. But for how long exactly? We will find out. Along the way, we'll take an excursion into the theory of excursions, a tool that lets us find quick solutions to some probability problems that would otherwise require solving terrifying systems of equations.
Date: Wednesday, 1/19/22, 5:40-6:40PM
Location: Zoom
Speaker: John Voight
Bio: John Voight is a professor of mathematics at Dartmouth College, where he specializes in arithmetic algebraic geometry and number theory.
Title: Quaternions
Abstract: Quaternion algebras arose from Sir William Rowan Hamilton's attempts to understand rotations in three dimensional space, but they turn up naturally across mathematical domains. Indeed, wherever you see matrices of size two by two, quaternions are lurking too! In this talk, we will give a broad survey of quaternions, including some applications. You can read his book on quaternions here.
Date: Wednesday, 12/8/21, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Vivian Kuperberg
Bio: Vivian is currently a fifth year graduate student at Stanford University studying analytic number theory under Soundararajan.
Title: On pseudo-polynomials
Abstract: Every polynomial f with integer coefficients has the following useful property: for every integers n and k, f(n+k) is congruent to f(k) mod n. To say this a different way, f is well-defined mod n for every n. This talk will be about pseudo-polynomials, which are arbitrary functions on the integers that satisfy this same property. All polynomials are pseudo-polynomials, but it turns out there are many more: for example, floor(e*n!) is a pseudo-polynomial. We'll discuss the history of pseudo-polynomials, outstanding open questions about pseudo-polynomials, and ways to construct pseudo-polynomials that are very poorly behaved.
Date: Wednesday, 12/1/21, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Irene Fonseca
Bio: Prof. Fonseca is a University Professor at CMU and is also the director of our very own Center for Nonlinear Analysis. Her research in pure analysis, specifically the calculus of variations, has been applied in a variety of fields, such as materials science and image processing.
Title: Mathematics and Imaging Science
Abstract: Have you ever thought that mathematical analysis and computer science are two very disjoint fields? What about topology and machine learning? All of these fields are more interconnected than you might think. In this talk we will address the mathematical treatment of image processing, including inpainting, recolorization, denoising, and machine learning schemes.
Date: Wednesday, 11/17/2021, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Professor Adam Bjorndahl
Title: An Introduction to Knowledge Representation and Reasoning
Abstract: Formal mathematical models of knowing, learning, and communicating are useful in a variety of fields, including computer science (e.g., for reasoning about distributed systems), economics (e.g., for characterizing game-theoretic notions of strategic equilibria), linguistics (e.g., for representing semantic context and the presuppositions under which discussions take place), and philosophy (e.g., for formalizing the relationship between knowledge, belief, and justification and assessing the force of skeptical arguments). In this talk we'll motivate and explore a simple but versatile class of models for representing knowledge in both single- and multi-agent settings, paying special attention to the concept of "common knowledge". Time permitting we'll also discuss some extensions of these basic models to include dynamic representations of learning and evidence. No prior experience with these concepts/models is assumed.
Date: Wednesday, 11/3/2021, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Kayla Wright
Bio: Kayla is a Ph.D. candidate at the University of Minnesota interested in algebraic combinatorics and representation theory!
Title: An Introduction to Schubert Calculus
Abstract: I will be giving a brief introduction to a subject called Schubert Calculus!
How many lines pass through two points in the plane? How many points do a line and a conic intersect in? And given four lines drawn randomly in three-dimensional space, how many lines intersect all four of them?
These are classical problems in enumerative geometry, the study of counting the intersection points of geometric objects. We will talk about classical ways these were thought of and also introduce modern-day ways of thinking about these problems using combinatorics of tableaux.
Date: Wednesday, 10/27/2021, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Professor Boris Bukh
Title: You do not have to believe in transcendental numbers
Abstract: The transcendental numbers are the atoms of the number line --- they are everywhere, but few have seen one. No longer you will have to believe in the existence of these half-mythical numbers! I will arm you with the knowledge that will enable you to write a number down, and prove to your cat that it is transcendental!
Date: Wednesday, 10/20/2021, 5:40-6:40PM
Location: Porter Hall 100 or Zoom
Speaker: Professor Robin Neumayer
Title: An Introduction to Shape Optimization Problems
Abstract: In this talk, we introduce two classical shape optimization problems. The first is the isoperimetric inequality, which says that balls have the smallest perimeter among sets of a given volume. This is a mathematical reason why soap bubbles are round, as they take the shape of minimal surface area. Next, we will discuss the Faber-Krahn inequality, which says that among all drum heads of a fixed area, a circular drum produces the vibration of lowest frequency. Finally, we will discuss connections between these two shape optimization problems.
Date: Wednesday, 10/13/2021, 5:30-6:30PM
Location: Porter Hall 100 or Zoom
Speaker: Dr. Eric Stubley
Title: Solving Equations with Origami
Abstract: I'll show you how to use origami to calculate roots of any (rational) cubic polynomial! To understand why this procedure works we'll only need to know about similar triangles, but along the way we'll encounter some principles from abstract algebra and algebraic geometry that help illuminate what's really going on. I'll weave in some of the history of the mathematics of paperfolding as well. A talk about origami wouldn't be complete without some actual paper folding, so bring a blank piece of paper and a writing implement for some short activities.
Date: Wednesday, 9/29/2021, 5:30-6:30PM
Location: Porter Hall 100 or Zoom
Speaker: Dr. Tomasz Tkocz
Title: Extremal-volume projections of the cross-polytope
Abstract: To give a flavour of certain problems in high dimensional convex geometry, I shall discuss extremal-volume projections of the cross-polytope onto hyperplanes and present a beautiful solution using probabilistic and analytic ideas such as decoupling. Only knowledge about R^n is required.
Date: Wednesday, 9/23/2021, 5:30-6:30PM
Location: Porter Hall 100 or Zoom
Speaker: Dr. John Mackey
Title: I am in disarray, also Ramsey Theory
Abstract: This is mostly for new(er) members, I'll talk about a subject called Ramsey Theory, which is the art of finding structure that must exist even within the most random objects. It will just be fun.
Date: 5:40-6:40pm on Wednesday 4/6
Location: HOA 160
Speaker: Francesca Zaffora Blando
Bio: Francesca is a logician in CMU's Philosophy department, whose research intersects probability theory, information theory, and computability theory.
Title: Algorithmic randomness and Bayesian epistemology
Abstract: In this talk, I will provide an informal introduction to the theory of algorithmic randomness—a branch of computability theory that specifies what it means for an individual mathematical object (such as a real number or a binary sequence) not to display any algorithmically detectable regularities. Then, I will argue that algorithmic randomness has fruitful philosophical applications. In particular, I will show that algorithmic randomness can be used to shed light on Bayesian convergence to the truth, a phenomenon of fundamental importance to Bayesian epistemology (a probabilistic theory of rational learning).
Date: 5:40-6:40pm on Wednesday 4/27
Location: HOA 160
Speaker: Edward Hou
Bio: Edward is an undergrad at CMU interested in mathematical logic and its applications!
Title: Proofs without words, Banach–Tarski, and Circle Squaring
Abstract: We investigate the problem of given two arbitrary shapes, can one find a finite partition of one shape, so as to move around the pieces and recombine into a partition of the other shape. A celebrated 2022 result by Máthé–Noel–Pikhurko says that one can "Circle the Square" with geometrically very simple pieces. We decompose their proof into several fundamental components, revealing key insights from algebra, combinatorics, and analysis. Along the way we find old friends such as the Pythagorean theorem and the Banach–Tarski paradox.