The Guide to Inverse Problems for the Perplexed

Speaker: 

Nicholas Hoell

Institution: 

University of Toronto

Time: 

Thursday, February 14, 2019 - 12:00pm to 1:00pm

Location: 

RH 440R

We discuss the philosophy, mathematics, and varied approaches to inverse problems, particularly geometric inverse problems with applications to medical imaging using tomography as a case study. The main mathematical topics we touch on are differential equations, harmonic analysis, and some basic representation theory. No background in medical imaging or sophisticated mathematics is required.

A surprising application of mathematics: How to name a color

Speaker: 

Nicole Fider

Institution: 

University of California, Irvine

Time: 

Friday, February 8, 2019 - 12:00pm to 1:00pm

Location: 

RH 340P

Your brain likes patterns and categories; by grouping related ideas together, it can store and recall information quickly.  Real-life continuous domains (like time, temperature, and taste) are inherently composed of infinitely many points of information, which our brains segment into finitely many categories for convenience (such as morning/afternoon/evening/ night, or sweet/sour/salty/bitter).  This phenomenon is well-documented and is a topic of interest in the behavioral, cognitive, and social sciences.

The set of colors is another example of a continuous domain, which in English is segmented into categories called “blue,” “red,” “green,” etc.  In this talk, I discuss how we apply mathematics —including linear algebra, geometry, and probability—to real-world data to study the occurrence of different categorizations schemes of the color space.  I then outline several related open projects, which could be pursued as part of a 199 Reading/Research course in the Spring or Summer sessions.

Factorizations in Numerical Monoids and Quasi-polynomial Growth

Speaker: 

Roberto Pelayo

Institution: 

University of Hawaii at Hilo

Time: 

Monday, February 4, 2019 - 12:00pm to 1:00pm

Location: 

RH 340P

With its multiplicative structure, the monoid of natural numbers is a unique factorization domain.  In contrast with this, co-finite, additive submonoids of the natural numbers, called numerical monoids, have elements with plural factorizations into irreducible elements.  Understanding factorizations within numerical monoids has led to the development of several factorization invariants, including the delta set and omega-primality.  These invariants, which generally measure how far an element is from being uniquely factorable, have garnered significant scholarly interest.  Most notably, many recent results demonstrate that these invariants experience eventual quasi-polynomial growth.  While this growth is evident in essentially every factorization invariant, we also provide examples of quasi-polynomial growth of invariants of families of monoids, including the Frobenius number and the Apery set.  

 

In this talk, we will present several of our results and their connections to questions in commutative algebra. As many of these results were generated in the Pacific Undergraduate Research Experience in Mathematics (PURE Math), we discuss why these problems are particularly accessible to undergraduates.

Surely Joking with Mr. Feynman: Can You Compute Integrals?

Speaker: 

Shuhao Cao

Institution: 

University of California, Irvine

Time: 

Tuesday, January 29, 2019 - 12:00pm to 1:00pm

Location: 

RH 440R

In his memoir-like short story book, "Surely You're Joking, Mr. Feynman!”,  Nobel-winning physicist Richard Feynman taunted an integration trick his high school physics teacher Mr. Bader taught him. According to Mr. Feynman, he used this trick to compute certain definite integrals that "guys at MIT or Princeton had trouble doing", and thus earned a great reputation for doing obscurely hard integrals, which often appear in Quantum Field Theory. In this short excursion, we will study this trick, then put it into action to evaluate the Gaussian integral, which is arguably the most useful definite integral in Probability and Data Science. Lastly we will solve one of the most difficult integral question in the 2005 Putnam competition.

Meandering Around

Speaker: 

Lucia Simonelli

Institution: 

International Center for Theoretical Physics

Time: 

Friday, January 25, 2019 - 12:00pm to 1:00pm

Location: 

RH 340P

Meander

1. (noun) a winding curve

2. (verb) to wander or ramble casually 

We will discuss mathematical properties of some special winding curves and then ramble casually about different experiences studying mathematics.

Mathematics in Cryptography

Speaker: 

Christopher Jankiwski

Institution: 

New York University

Time: 

Tuesday, January 20, 2015 - 12:00pm

Location: 

Rowland Hall 306

We often see mathematical material in an abstract manner, but behind formal theorems and rigorous proofs, we can find significant applications to the real world. One example which has become crucial to our daily lives is cryptography. When we access a bank account online or purchase an item electronically, we rely on some method of cryptography to ensure that our information is secure. In this talk, we present the RSA model for cryptography as built from concepts involving prime numbers, factorization, and other fundamental topics in number theory.

Big numbers in number theory

Speaker: 

Christopher Davis

Institution: 

University of Copenhagen

Time: 

Wednesday, January 7, 2015 - 4:00pm to 5:00pm

Location: 

RH 306

There is a joke that when one counts something in linear algebra, the only possible results are 0, 1, and infinity.  What about in number theory?  Well, if we count the primes, we get infinity.  To get a finite number, we could try to count primes less than some fixed N.  It was proven in 1955 that something interesting happens sometime before N = 10^(10^(10^1000)).  Huh?!  What if we count gaps between primes?  Just in 2013, it was proven that something interesting happens for gaps less than 70,000,000.  What if we try to write down a really big prime number?  Right now, nobody can write down a prime bigger than 10^(10^8).  In this talk, we'll introduce some of these big numbers in number theory, we'll carefully say what they mean, and we'll describe progress on making the first two big numbers smaller and on making the third big number bigger.

We're Going to the Moon: The N-body problem

Speaker: 

Jeremy Pecharich

Institution: 

Pomona College

Time: 

Monday, December 8, 2014 - 4:00pm to 5:00pm

Location: 

NS2, 1201

It is May 25, 1961, President Kennedy announced that the United States would land a person on the moon. But, in 1961 nobody actually knew how to get to the moon! It wasn't until 1963 that a viable flight path to the moon was found to exist. We will discuss the mathematics that made this path possible and some of the long history of the N-body problem.

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