PROMYS Guest Lectures
Title: What are Ramsey numbers?
Speaker: Rehana Patel (Wesleyan University, Middletown, USA)
Date: Fri, 14 Jun 2024
Time: 11:45 am
Venue: LH-5, Mathematics Department
Finite Ramsey theory is the study of structure that becomes unavoidable in large finite objects. In this talk, we will provide a brief taste of this rich and beautiful subject. We will start with the following question: In any group of six people, can we always find three who know one another or three who don’t? A far-reaching generalization of this question was first answered in a paper on logic by Frank Ramsey in 1928. Our approach to it will involve graph theory and combinatorics, with a dash of probability. No prerequisites will be needed to understand the talk.
Title: Sums of n squares
Speaker: Raman Parimala (Emory University, Atlanta, USA)
Date: Mon, 03 Jun 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
We discuss representation of integers as a sum of $n$ squares. We explain the quaternionic composition law for sums of four squares and a proof of a theorem of Lagrange on which positive integers can be expressed as a sum of four squares. We outline general connections to the theory of quadratic forms.
Title: Counting potato print patterns
Speaker: Tulsi Srinivasan (Azim Premji University, Bengaluru)
Date: Fri, 31 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
You might have carved a piece of potato to create a stamp to print with. How many patterns can we get from a single potato stamp? One approach to answering this question sheds light on the rich connections between objects and their symmetries, and leads us to a more general counting strategy.
Title: Rounding sequences and counting by congruence classes
Speaker: Caleb Shor (Western New England University, Springfield, USA)
Date: Tue, 21 May 2024
Time: 11:45 am
Venue: LH-1, Mathematics Department
For $n$ a natural number, consider the sequence of $n$ rational numbers $n/1, n/2, n/3, \dots, n/n$. Round each to the nearest integer to obtain sequence of $n$ integers. How many are odd?
In this talk we will see how knowledge of sums of squares and a result of Gauss will help to lead us to a somewhat surprising result. Time permitting, we will discuss similar results.
Title: Who cut my cheese?
Speaker: Purvi Gupta (IISc Mathematics)
Date: Fri, 17 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
Given a box packed with identical cubes of cheese, what is the maximum damage one can cause with a single straight cut through the box? This seemingly simple puzzle represents an old but recurrent mathematical theme that slices through numerous fields such as number theory, functional analysis, probability theory, and computational complexity theory. The cross-sections of convex bodies hold many mysteries, some of which continue to puzzle mathematicians today. We will focus on the deceptively simple case of the cube to demonstrate some of these ideas and open questions. No cheese will be harmed in the making of this talk.
Title: Are we there yet?
Speaker: Jaclyn Lang (Temple University, Philadelphia, USA)
Date: Fri, 10 May 2024
Time: 2:30 pm
Venue: LH-1, Mathematics Department
How far is it from Bangalore to Chennai? Is there a single correct answer to this question? In this talk we will explore different notions of distance as well as why you might choose one over another depending on the context. This will take us on a brief sight seeing tour through geometry, graph theory, and number theory.
Title: Prime numbers
Speaker: Sujatha Ramdorai (University of British Columbia, Vancouver, Canada)
Date: Wed, 07 Jun 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
Prime numbers have been studied by Humankind for centuries and have applications in Internet Cryptography. We will outline this connection and also talk about how prime numbers give rise to different number systems.
Title: Divisibility tests, recurring decimals, and Artin's conjecture
Speaker: Apoorva Khare (IISc Mathematics)
Date: Wed, 31 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
If we had two extra thumbs, how would we check if “2024” is divisible by eleven? Or by “11”? We will see a simple test in any base $B$, i.e. usable by species having any number of fingers (whether shaped like hot-dogs or not); and for any divisor $d$. That is, the test works for everything ($d$), everywhere ($B$), all at once.
We will then move to recurring decimals. Note that 1/3 = 0.3333… and 1/3x3 = 0.1111… have the same number of digits - one - in their recurring parts. (Is 3 the only prime with this property in base 10?) More generally, we will see how many digits $1/d$ has in its recurring “decimal” expansion, for us or for any species as above.
Finally, for a species with a given number of fingers (= digits!), are there infinitely many primes $p$ for which the recurring part of $1/p$ has $p-1$ digits? (E.g. for us, 1/7 has the decimal recurring string (142857).) And what does this have to do with Gauss, Fermat, and one of the Bernoullis? Or with Artin and a decimal number starting with 0.3739558136… ? I will end by mentioning why this infinitude of primes holds for at least one species among humans (10), emus (6), ichthyostega (14), and computers (2) - but, we don’t know which one!
Title: Zeta functions and Euler products
Speaker: Alina Bucur (University of California, San Diego, USA)
Date: Wed, 24 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
We will define one of the most famous functions in all of mathematics, the Riemann zeta function, whose properties are the subject of one of the Millenium Problems. We will also look at some of its analogues for other objects.
Title: Cyclotomic polynomials and roots of unity
Speaker: Kiran S. Kedlaya (University of California, San Diego, USA)
Date: Mon, 22 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
We introduce an important family of polynomials, the cyclotomic polynomials, whose roots are the roots of unity of a fixed order. We explore the structure of these polynomials and the number fields that they generate, including a brief look at Gauss sums.
Title: Could Euclid trisect every angle? How about us?
Speaker: Parthanil Roy (ISI, Bangalore)
Date: Wed, 17 May 2023
Time: 5 pm
Venue: LH-1, Mathematics Department
This talk will be a lucid introduction to the formal mathematics behind Euclidean Constructions, which we all learn in our middle school curriculum. The rules, regulations and restrictions of this type of construction will be discussed in detail. An alternative will also be suggested. We shall also find out how a completely geometric question can be answered using purely algebraic techniques giving rise to an elegant theory introduced in the nineteenth century by a famous French mathematician named Évariste Galois.