Embedded Files
UPCOMING EVENTS
Date: 5:30-6:50pm on Wednesday 11/5
Location: Porter 100
Title: Mathematical Modeling in Medicine: From Virtual Populations to Predictive Confidence
Abstract: What can mathematics tell us about biology when experiments are costly, slow, or impossible? In this talk, I’ll show how mechanistic models and virtual cohorts can help explain why some cancer treatments fail, and how surrogate modeling is enabling model-based inference in computationally intensive, high-dimensional models representing complex biological systems.
PAST EVENTS
Date: 5:15-6:50pm on Wednesday 10/29
Location: Porter 100
Title: SEMS Symposium
Abstract: This summer the Department of Mathematical Sciences ran several undergraduate research programs, including the Summer Experiences in Mathematical Sciences (SEMS). Students who participated in SEMS worked on projects in teams. They will be presenting their work at the 2025 SEMS Symposium, which will take place on Wednesday, October 29, from 5:15 to 6:50 pm in Porter Hall 100. This meeting will coincide with the weekly Math Club meeting (except that the symposium will be a bit longer). Each team will present for 15-20 min followed by 5 min for questions. The schedule of their presentations is attached.
For those who want to see what undergraduate research is about and those who want to support the SEMS participants, we invite you to attend our event. It should be fun! There will be pizza and dessert (Prantl's burnt almond cake and vegan choices).
Date: 5:30-6:40pm on Wednesday 10/22
Location: Porter 100
Speaker: Nicholas Yoder
Title: AI Safety and the Geometry of High-Dimensional Spaces
Abstract: This talk explores recent advances in Artificial Intelligence with a focus on the geometry of high-dimensional embedding spaces (exceeding 10,000 dimensions) and how their unintuitive properties influence both the capabilities and vulnerabilities of modern AI systems. We’ll examine why understanding these mathematical spaces is essential for developing safer and more transparent models, as well as future careers in Artificial Intelligence.
Date: 5:30-6:40pm on Wednesday 10/8
Location: Porter 100
Speaker: Dylan Altschuler
Title: I'll Give You My Car If You Can Draw a Kite: Graph Embeddings and Dimension Reduction
Abstract: Many computational tasks that were "solved" decades ago, such as matrix multiplication or linear programming, have been reopened in the age of big data. Google famously encodes the advertisement profiles in billion-dimensional feature space; matrices of this size cannot even be loaded into computers, let alone computed on. This motivates the study of dimension reduction, a topic which arises at the intersection of combinatorics, geometry, probability, information theory, machine learning, and theoretical CS. We will discuss some basic definitions and examples.
Date: 5:30-6:40pm on Wednesday 10/1
Location: Porter 100
Speaker: Matteo Gravinese
Title: Calculating Values of the Riemann Zeta function
Abstract: The Riemann Zeta function is extremely relevant in many areas of mathematics such as complex analysis, analytic number theory, and the study of prime numbers. Calculating exact values of this function gives us a deeper understanding of this function, and thereby helps us understand these related fields. At this meeting, we will analyze a seemingly unrelated infinite summation in order to derive a formula that will allow us to compute values of this function at positive even integer values.
Date: 5:30-6:40pm on Wednesday 9/17
Location: Porter 100
Speaker: Ian Tice
Title: An elementary invitation to the abstract interpolation of Banach spaces
Abstract: Suppose we have two mathematical structures of the same type (e.g. vector space, metric space, group, field, etc) X and Y that nest via the subset inclusion $X \subset Y$. A natural mathematical question (and one that is asked in many contexts!) is whether there exist more such structures sitting between X and Y, say Z with $X \subset Z \subset Y$. We might think of such a Z as "interpolating" between X and Y. The purpose of this talk to give an elementary (1D calculus and a bit of linear algebra) introduction to this topic by introducing a general mechanism for producing infinitely many spaces interpolating between the space of continuous functions and the space of continuously differentiable functions.
Date: 5:30-6:40pm on Wednesday 9/10
Location: Porter 100
Speaker: Advait Nene
Title: Generating Functions
Abstract: Counting feels discrete. Calculus feels continuous. But when we bring the two ideas together, we discover one of the most overpowered ways to count: generating functions. We will explore the combinatorial and analytic properties of generating functions, in order to count things that are really hard to count.
Date: 5:30-6:40pm on Wednesday 9/3
Location: Posner 151
Speaker: Leo Goldmakher
Title: Artin's conjecture and patterns in long division
Abstract: Using long division, we find that 1/7 = 0.142857142857142857..., 1/11 = 0.0909090909..., 1/13 = 0.076923076923076923..., etc.. These decimals may look random, but they contain a lot of secret structure---some proved, some observed but still open. We'll discuss some of this structure and connect these decimals to the structure of the multiplicative group of integers (mod p). This will lead us to Artin's conjecture on primitive roots, a major open problem in number theory / group theory. I will conclude by discussing some very recent results refining Artin's conjecture, discovered jointly with Greg Martin and Paul Péringuey (both at UBC).
Date: 5:30-6:40pm on Wednesday 1/22/25
Location: PH 100
Speaker: Robin Neumayer
Title: Shape Optimization: From Soap Films to Acoustics
Abstract: The task of optimizing an object's shape to make it the most efficient, least costly, or most streamlined arises in nature as well as across engineering and design fields. For instance, due to surface tension, soap bubbles take the shape with the least surface area among all possible shapes enclosing a fixed amount of air — a sphere! We will introduce the mathematical study of so-called “shape optimization problems” and survey some classical and contemporary results in the field.
Date: 5:30-6:40pm on Wednesday 10/23/25
Location: PH 100
Speaker: Dave Levermore
Title: History of the Equations of Fluid Dynamics
Abstract: The equations of fluid dynamics are associated with Euler, Navier and Stokes. However many more of the leading mathematicians and scientists were involved. The
story spans over 200 years and intertwines with the those of the law of energy conservation, of thermodynamics, of the kinetic theory of heat, of statistical physics, and of the theory of partial differential equations. It is hoped that knowing this story will help clarify your
understanding of the various systems of these equations.
Date: 5:30-6:40pm on Wednesday 10/9
Location: PH 100
Speaker: Bernardo Subercaseaux
Title: A Demi-god's Number for the Rubik's Cube.
Abstract: It is well-known that any state of the 3x3x3 Rubik's Cube can be solved in at most 20 moves, a result known as "God's Number". However, this result took Rokicki et al. ~35 CPU years to prove and is therefore challenging to reproduce. I will show a simple approach to obtain a worse upper bound of 36 moves, that is justified probabilistically. The main idea generalizes to bounding the diameter of other vertex-transitive graphs by at most twice its true value, hence the name "demi-god number''. This talk will include the group-theoretic definition of the Rubik's cube, illustrating some nice elementary ideas of abstract algebra, and also a summary of the "beginner's method" to solve the cube. I will not assume prior knowledge of how to solve Rubik's cubes (although if you already know how to solve one that will be helpful!).
Date: 5:30-6:40pm on Wednesday 10/2
Location: PH 100
Speaker: Clinton Conley
Title: Acyclic graphs can be hard to color?
Abstract: A (proper) coloring of a graph is a way to paint its vertices so that adjacent vertices get different colors. A basic fact in graph theory is that acyclic graphs can be colored with two colors of paint. Unbothered by this fact, we discuss how a very special acyclic graph called G_0 requires uncountably many colors of paint (in some models of ZF).
Date: 4:50-6:50pm on Wednesday 9/25
Location: PH 100
Speaker: SEMS Participants
The SEMS Symposium is similar to our standard talks, but instead of a single talk, 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. As always, please RSVP! There will be pizza and dessert provided.
Date: 5:30-6:40pm on Wednesday 9/18
Location: PH 100
Speaker: 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 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: 5:40-6:40pm on Wednesday 9/11
Location: PH 100
Speaker: Chris Eur
Title: Why would a sequence be unimodal?
Abstract: Numerical sequences abound in combinatorics, for example arising from the number of forests in a graph, the number of faces in a polytope, the number of partitions of a natural number, the Stirling numbers of the first kind, etc. --- the list goes on and on. Among the most basic questions we can ask about them concerns "unimodality" --- does the sequence monotonically increase and then decrease? We discuss how ideas from algebraic geometry and topology have led to successful resolutions of some of these combinatorial problems.
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.
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
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