Monstruous Functions

Speaker: Professor Ian Tice
Time: 5:30pm on Wednesday, October 25th, 2017
Location: Doherty Hall A302
Abstract: Until the 19th century it was widely believed among mathematicians that continuous functions of a real variable should in fact be differentiable at "most" points. It thus came as quite a surprise when Karl Weierstrass constructed a beast of a continuous function that is differentiable nowhere. The purpose of this talk is to construct such a monstrous function.

Have Yourself a Merry Little Christmas (Theorem)

Speaker: Professor Kate Thompson, Visiting Shelly Professor
Time: 5:30pm on Wednesday, October 11th, 2017
Location: Porter Hall 100
Abstract: One of the most classic results in number theory was originally stated by Fermat in a letter to Mersenne dated Christmas,1640. In "typical" Fermat fashion, the first proof didn't appear for over 100 years. Having said that, this result shows the breadth and beauty of number theory techniques. Time permitting, we will go through a (1) algebraic (2) geometric (3) analytic proof of this result. At the end, we will also discuss generalizations of this result, some proved as recently as 2013.
Talk Slides: Click here.

Constant Functions and Sobolev Spaces

Speaker: Michael Spoerl, Senior Math Major
Time: 5:30pm on Wednesday, September 27th, 2017
Location: Porter Hall 100
Abstract: In a 2002 paper, Brezis gave a (relatively) simple criterion for identifying constant functions. The proof, however, is not so simple - it requires a detour through Sobolev spaces. In this talk I will introduce the theory of Sobolev spaces and use the main theorem of Brezis' paper to prove his criterion. Only a conceptual understanding of calculus is needed.

Mathematical Contest in Modeling Interest Meeting

Time: 5:30pm on Wednesday, September 20th, 2017
Location: Porter Hall 100
Details: The Mathematical Contest in Modeling (MCM) is a yearly contest for undergraduates interested in applied mathematics- think of it like a cross between Putnam and a hack-a-thon for math. A team has 96 hours to put together a solution to a problem in math modeling. Some examples of problems from the past:
  • Estimate the global effects of a large asteroid impacting Antarctica
  • Study the hunting strategies of velociraptor dinosaurs based on fossil data
  • Develop a more efficient method of boarding passengers onto large commercial jets
If this sounds interesting to you, feel free to email Adrian Hagerty (Math Grad student, president of SIAM).

First General Body Meeting!

Time: 5:30pm on Wednesday, September 13th, 2017
Location: Porter Hall 100
Details: Welcome to Math Club! We'll introduce ourselves, give a brief overview of the type of things we do, show a tentative schedule of our events this weekend, and advertise our first event: Going to a Pirates Game on Friday, September 22nd! If we have time, we'll wrap up by watching the short film Flatland . There will be Pizza!

Viscosity Solutions

Speaker: Giovanni Leoni, Professor of Mathematics, CMU
Time: 5:30pm on Wednesday, April 12th, 2017
Location: Doherty Hall 2315
Abstract: In this talk I will try to motivate the notion of viscosity solutions for ordinary differential equations.

Star Numbers: from 17th-century oranges to delivery robots and beyond

Speaker: Josh Laison, Associate Professor of Mathematics, Willamette University
Time: 5:30pm on Monday, Mar. 27th, 2017
Location: Scaife Hall 125
Abstract: We discuss a new variation of a 400-year-old problem from Johannes Kepler and Isaac Newton about how densely geometric shapes can be packed together. Along the way we'll encounter Amazon delivery robots, geometric networks, an iPhone game, computers that prove theorems, 24-dimensional spheres, Tetris, cannonballs, and some exciting new theorems. Joint work with Andrew Bishop, Ben Gardiner, and David Livingston.

Counterintuitive Ideas in Analysis

Speakers: Joseph Zoller and Ani Sridhar, Juniors at CMU
Time: 5:30pm on Wednesday, Mar. 8th, 2017
Location: Margaret Morrison 103
Abstract: We will explore the basic notions in real analysis such as continuity, differentiability and integrability and completely annihilate any intuition you may have about them. We'll use and introduce basic tools from real analysis to construct strange functions. Memes will be provided.

Fermi Questions

Speaker: Liza Sulkin, Freshman Mathematics Major, CMU
Time: 5:30pm on Wednesday, Feb. 22nd, 2017
Location: Doherty Hall 2315
Abstract: Fermi Questions involve quickly determining the order of magnitude of absurd questions. We'll be answering several important questions such as:
  • How many Cathedral of Learnings would it take to reach Philadelphia from Pittsburgh?
  • How many wire cuts would it take to free the UC Stones?

Attenuated Tomography Old and New

Speaker: Nicholas Hoell, University of Toronto
Time: 4:30pm on Friday, Feb. 3rd, 2017
Location: Wean 7500
Abstract: We will explore the mathematics of the attenuated ray transform. The problem of recovering an unknown function based on measuring its weighted line integrals arose in medical imaging modalities and is connected to deep open questions in geometry. Some of the techniques used to address this problem involve ideas from harmonic analysis, representation theory, and gauge theory. No background knowledge of these fields or of inverse problems in medical imaging is assumed.

Some Applications of Differential Equations in the Social, Life, and Physical Sciences

Speaker: Jason Howell, College of Charleston
Time: 4:30pm on Friday, Jan. 27th, 2017
Location: Wean 7500
Abstract: In this talk we will survey several different applications of differential equations in a variety of settings. Witha primary focus on the modeling process and the influence of problem parameters on solutions, we will discuss how basic mathematical models of population growth and interaction can be extended to a multitude of areas, including love affairs, the spread of communicable diseases, warfare and combat, marketing, the spread of ideas, and even a zombie apocalypse. I will also discuss some ongoing undergraduate research projects that focus on popularity dynamics and targeted advertising. If time permits, a brief discussion of some applications of partial differential equations will be discussed.

How to Multiply Big Numbers

Speaker: Boris Bukh, Associate Professor of Mathematics, CMU
Time: 5:30pm on Wednesday, Nov. 16th, 2016
Location: Gates 4307
Abstract: All over the globe, young school children spend millions of hours multiplying numbers. I will demonstrate possible ways of multiplying numbers. While much improvement is possible, they offer no relief to the children. I will conclude the talk with cryptic remarks about college students multiplying matrices.

Triforce of Chaos Theory

Speaker: Zachary Singer, Junior Mathematics Major, CMU
Time: 5:30pm on Wednesday, Oct. 19th, 2016
Location: Wean 7500
Abstract: What does a butterfly flapping its wings have to do with the Triforce? In this talk we introduce a few ideas of chaos theory and fractal geometry, as well as how different notions of dimension can lead to a set having non-integral dimension. These ideas can be combined to help determine the behavior of how a biological virus spreads, which is one application I worked with over the summer.

Sequences, Series, and Subterfuge

Speaker: Misha Lavrov, Mathematics PhD Student, CMU
Time: 5:30pm on Wednesday, Oct. 5th, 2016
Location: Gates 4307
Abstract: Click here for a pdf version.
Talk Slides: Click for slides in a pdf version.

Partial Differential Equations

Speaker: Sam Zbarsky, Senior Mathematics Major, CMU
Time: 5:30pm on Wednesday, Sept. 21st, 2016
Location: Gates 4307
Abstract: This talk will be an introduction to two tools useful in studying partial differential equations: the Fourier transform and energy methods. I will introduce both of them, and show how they apply to several partial differential equations, including one I worked with over the summer.

Geometry, Combinatorics, Patterns and Fun

Speaker: Niraj Khare, Faculty Candidate for CMU Qatar
Time: 5:00pm on Wednesday, Mar.23rd, 2016
Location: DH 2315
Abstract: Click here for a pdf version
Talk Slides: slides in pdf version

Trigonometry Revisited or: If Fourier Had a Quantum Computer

Speaker: Zijian Diao, Faculty Candidate for CMU Qatar
Time: 5:00pm on Wednesday, March 2nd, 2016
Location: DH 2315
Abstract: Is sine of 1 degree rational? At a glance this question seems light years away from quantum computing, a cutting-edge research area where computer science meets quantum mechanics. Surprisingly, this question matters not only in the mathematical world, but also in the quantum realm. Its answer and many other rudimentary facts of trigonometry have found their way into the study of various research problems in quantum computing. In this talk we will explore these intriguing connections through the interplay of quantum algorithms and topics from number theory and classical analysis. Along the way, we will solve a long-standing puzzle in quantum search and provide a quantum approach to the centuries-old Basel problem.

Thue Theorems and Avoidable Patterns

Speaker: Martin Juras, Faculty Candidate for CMU Qatar
Time: 5:00pm on Wednesday, Feb.24th, 2016
Location: WEH 8220
Abstract: Click here for a pdf version

Infinite Dimension in Linear Algebra: A Case for Functional Analysis

Speaker: Adrian Hagerty, PhD student in Math Dept at CMU
Time: 5:30pm on Wednesday, Feb.17th, 2016
Location: DH 2315
Abstract: The study of Functional Analysis forms an important role in the modern fields of PDEs and Calculus of Variations. This involves taking the familiar notion of a vector space and bringing it to the realm of infinite dimensions. What new problems arise as we attempt to generalize our old tools to the infinite setting?

What is Structure?

Speaker: Clive Newstead, PhD student in Math Dept at CMU
Time: 5:30pm on Wednesday, Jan.27th, 2016
Location: DH 2315
Abstract: The idea of equipping a set with a structure is ubiquitous in mathematics. For example, graphs are sets equipped with an edge relation, groups are sets equipped with an algebraic structure, topological spaces are sets equipped with a notion of openness, posets are sets equipped with a notion of order... the list goes on forever and ever and ever. This talk will introduce a candidate for defining 'structure' in the abstract. With the tools we develop, we can prove some pretty cool stuff in enumerative combinatorics.

Probability and Intuition

Speaker: Peter Winkler, William Morrill Professor of Mathematics and Computer Science, Dartmouth
Time: 5:30pm on Wednesday, Dec.2nd, 2015
Location: DH 1112
Abstract: Supposedly our intuition about probability---even though humans invented the concept---is faulty. Lots of puzzles, naturally, are designed to lead us down the garden path to incorrect conclusions. But maybe our intuition is not as bad as we think. We'll take a fresh look at some paradoxes, old and new, to see whether we or our assumptions are at fault.
Poster: Poster

Fine Structure in Shape-memory Alloys

Speaker: Winston Yin, senior at CMU, math and physics double major
Time: 5:30pm on Wednesday, Nov.18th, 2015
Location: PH 100
Abstract: The peculiar properties of a shape-memory alloy are the result of the fine structure formed between its several crystalline forms. How do these fine structures arise from basic ideas about a crystal? In this talk, I will explain how infinitely fine structures are a natural consequence of integral minimization problems, as well as the project I worked on at Oxford this summer.

Variational Problems in Materials Science and Image Inpainting

Speaker: Irene Fonseca, Mellon College of Science University Professor of Mathematics, Director of Center for Nonlinear Analysis
Time: 5:30pm on Wednesday, Nov.11th, 2015
Location: PH 100
Abstract: A plethora of physical and technological applications ranging from analyzing instabilities in materials science to image analysis in computer vision are subject to rigorous mathematical understanding through recently developed methods and articulation of ideas in the calculus of variations, geometric measure theory, and nonlinear partial differential equations. In this talk, these techniques will be used for recolorization of damaged images.

The Projective Heat Map

Speaker: Richard Schwartz, Chancellor`s Professor of Mathematics, Brown University
Time: 11:30 on Friday, Oct.16th, 2015
Location: WEH 8200
Abstract: Abstract and Slides

Meet-and-Greet with Paul Raff

Speaker: Paul Raff, CMU alumus, now Principal Data Scientist at Microsoft
Time: 5:30 on Thursday, Oct.1st, 2015
Location: WEH 7500
Abstract: From Pure Math to Data Science (pptx)

Triples

Speaker: Po-Shen Loh, Math Professor at CMU, USA IMO Lead Coach
Time: 5:30 on Wednesday, Sept.30th, 2015
Location: PH 100

What's the longest sequence of triples (x1, y1, z1), (x2, y2, z2), ... that satisfies the following properties? (1) Each number is an integer between 1 and N inclusive. (2) For every j less than k, if we compare the triples (x_j, y_j, z_j) and (x_k, y_k, z_k), there are at least two coordinates in which the latter triple strictly exceeds the former triple. It turns out that this simple-sounding problem is equivalent to a question from Ramsey Theory, inspired by a question from k-majority tournaments, and related to deep question involving induced matchings and Szemeredi' s Regularity Lemma. The talk will be a tour of Combinatorics which introduces many of these concepts along the way.


Sticky Particles

Speaker: Adam Williams, senior math major at CMU
Time: 5:30 on Wednesday, Apr 29th, 2015
Location: Scaife 125

Imagine you have a finite set of particles, moving along the x axis, with fixed masses and initial velocities. When two particles collide, they *stick* together to form a new particle, preserving the mass and momentum of the original two. This is a simple set-up, but seems potentially difficult to work with mathematically. Adam shows you how you can use math to study this system in a way that makes it much more understandable.


Unexpected distribution phenomenon resulting from Cantor series expansions

Speaker: William Mance, PhD, CMU alumnus
Time: 5:00 on Wednesday, Apr 22nd, 2015
Location: Scaife 125

We explore in depth the theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones. This work was coauthored with Dylan Airey while he was still a high school student at the Texas Academy of Mathematics and Science. Our main theorem will touch on issues related to computability theory, ergodic theory, fractal geometry, number theory, and probability theory.


Why the IRD cares about the Riemann Zeta Function and Number Theory

Speaker: Steven Miller, Professor at Williams College
Time: 5:30 on Wednesday, Mar 25th, 2015
Location: Scaife 125

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers or of 2^n equals 1 about 30% of the time; the IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford s Law, was first noticed by observing which pages of log tables were most worn from age – it is a good thing there were no calculators 100 years ago! We will discuss the general theory and application, talk about some fun examples (ranging from the 3x1 problem to the Riemann zeta function), and if time permits discuss some joint results with my REU students.


The Birch and Swinnerton-Dyer conjecture

Speaker: Tomer Reiter, senior math major at CMU
Time: 5:30 on Wednesday, Mar 4th, 2015
Location: Scaife 125

The Birch and Swinnerton-Dyer conjecture is one of the Millennium Problems. The conjecture relates the group of rational points on an elliptic curve to information associated with the equation for the elliptic curve over finite fields. In the talk, we will see some of setup for the conjecture, the statement and some of the partial progress made. I will basically assume David's talk last time as a prerequisite, but I will briefly go over the facts about elliptic curves he mentioned that we will need.


The Congruent Number Problem and Elliptic Curves

Speaker: David Mehrle, senior math major at CMU
Time: 5:30 on Wednesday, Feb 18th, 2015
Location: Scaife 125

A positive integer N is called congruent if it is the area of a right triangle whose sides have rational lengths. Determining whether or not a given positive integer is congruent is a millennia-old problem that is still open today. I will explore this problem and several attempted solutions, including an interesting modern approach using elliptic curves.


Missing the Point

Speaker: Misha Lavrov, PhD student at CMU
Time: 5:30 on Wednesday, Jan 28th, 2015
Location: Scaife 125

If you are speaking with an alien from Mars who knows a lot of math and has never heard the word Euclid, explaining what geometry is can be frustrating. You will try very hard to come up with foolproof ways to axiomatize geometry, and the alien will persist in coming up with bizarre structures that look nothing like what you imagined. In 1899, Hilbert came up with a set of 16 alien-proof axioms for plane geometry. (That is not quite true. He had 17. But there’s been some progress in the intervening century.) In this talk, I will play the role of the alien in an effort to persuade you that the task of Hilbert was not an easy one.


Special Guest Speaker

Speaker: Noam Elkies, Professor of Mathematics, Harvard University
Time: 5:30 to 6:30 on Wednesday, December 3rd
Location: WEH 7500

TBA


The fundamental theorem of algebra

Speaker: Giovanni Leoni, Professor of Mathematics
Time: 5:30 to 6:30 on Wednesday, October 27th
Location: DH 1112

In this talk we will discuss the fundamental theorem of algebra and give an analytical proof.


You do not have to believe in transcendental numbers

Speaker: Boris Bukh, Assistant Professor of Mathematics
Time: 5:30 to 6:30 on Wednesday, October 8th
Location: DH 1112

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!
Pizza will be served.


When IMO meets Research

Speaker: Po Shen Loh, Assistant Professor of Mathematics
Time: 5:30 to 6:30 on Wednesday, September 24th
Location: MM 103

This year, problem 6 on the International Math Olympiad turned out to be a semi-open research problem. The originally proposed problem is solvable using basic methods, but it is still open to determine the best possible result. A set of lines in the plane is in general position if no two are parallel and no three are concurrent. A set of lines in general position cuts the plane into cells, some of which have finite area; we call these finite cells. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to color at least $\sqrt{n}$ of the lines blue in such a way that none of its finite cells has a completely blue boundary. What's the best bound that you can get? We'll improve the stated bound using techniques from probabilistic combinatorics and extremal hypergraph theory.


Assessing Artifacts: Using Calculus of Variations to Segment Damaged Images

Speaker: Nick Takaki, Undergraduate
Time: 5:30 to 6:30 on Wednesday, September 10th
Location: DH 1112

A biphase approximation of an image is a two-color piecewise-constant approximation of the original image which maximizes regional fidelity while minimizing complicated boundaries. When it comes to segmenting images, human intuition is very strong. We often have strong gut feelings about which pixels belong in which regions, where object boundaries should be, and what should be discounted as noise or damage. Variational image segmentation seeks to formalize these "gut feelings" by creating a functional which quantifies the error of approximation, and then defining correct segmentations as local minima of this functional. In this talk, I will discuss the intuition behind variational image approaches, popular region-based segmentation methods including the Chan-Vese model, and my group's summer research into a segmentation method which dynamically identifies artifacts and corrects shadows.


Chaos In Cellular Automata

Speaker: Carson Sestili, Undergraduate
Time: 5:30 on Thursday, May 1, 2014
Location: WEH 5415

An elementary cellular automaton is a sort of computational model. States are sequences of bits. Transitions between states are specified by a simple, deterministic rule that updates each bit according to a small neighborhood around that bit in the previous state. Even though the model is simple to describe, many cellular automata have surprisingly beautiful and complicated behavior. In some cases their behavior is so unpredictable that it might be called "random," even though there is no randomness in the system. In this talk I will show why several automata exhibit chaotic behavior, for a popular definition of chaos.


The Estimathon

Speaker: Andy Niedermaier, Jane Street Capital
Time: 5:30pm on Wednesday, March 26, 2014
Location: DH 2210

They're called Fermi problems...

Jane Street presents "The Estimathon." Attempt 13 estimation problems in 30 minutes, ranging from totally trivial to positively Putnamesque. Work in teams to come up with the best set of confidence intervals. Compete against your fellow students for fame* and fortune^!
(* fame is subjective)
(^ there will be prizes for the winners)


How Physics can help with the "Big Picture" in Math

Speaker: Ira Rothstein, Professor of Physics
Time: 5:30 on Wednesday, February 19, 2014
Location: DH 2315

Why are differential equations so ubiquitous? It seems that all physical systems are described by a set of differential equations. Perhaps we should stop and ask why? In this talk I will explain how a certain set of fundamental principle that apply to all physical systems automatically lead to differential equations (DE's) as equations of motion, as opposed to say integral equations. I will also discuss how the symmetries of physical systems, and their corresponding DE's, naturally lead to solutions via the use of group theory. Finally I will discuss how the principle of observer independence leads to the idea of differential geometry.