"Mathematics is a continuing creative endeavor with aesthetic values similar to those found in art and music. In particular, mathematics is a living subject, not one that has long since been embalmed in textbooks." - From 1955 College Board Mathematics Commission's report.

Part of the Contemporary Mathematics series, Lectures on Tight Closure and Its Applications, this volume consists of edited lecture notes from the online International Graduate course on Tight Closure and Its Applications from the ICTP and School on Commutative Algebra and Algebraic Geometry in Prime Characteristics. The lecture notes also come with exercises and solutions. Access via your institution: Buy the volume: Don’t have access? Ask your library to contact cust-serv@ams.org to subscribe!

"Logic is the youth of mathematics and mathematics is the manhood of logic." On May 18, 154 years ago, Bertrand Russell was born. This British logician and mathematician discovered Russell’s Paradox in 1901. The paradox considers “the set of all sets that do not contain themselves” and exposed a contradiction in naive set theory. This led mathematicians to re-examine the foundations of set theory. Russell then collaborated with Alfred North Whitehead on Principia Mathematica, published in three volumes from 1910 to 1913. The work attempted to derive all of mathematics from pure logic using a system of type theory. It became one of the major projects in the history of logic and mathematics.

The man who literally rebuilt modern mathematics from the ground up: Alexander Grothendieck (1928–2014). He invented schemes, toposes, and étale cohomology, turning algebraic geometry into the most powerful language in all of math. What Hilbert started, Grothendieck finished. The most influential mathematician of the 20th century.

One fascinating lesser-known fact in mathematics is Kaprekar's constant: 6174. Indian mathematician D.R. Kaprekar discovered it in 1949 through patient experimentation. Take any four-digit number where not all digits are the same (e.g., 3141, but not 1111). Rearrange its digits to form the largest and smallest possible numbers, then subtract the smaller from the larger. Repeat the process with the result. Within at most 7 steps, you'll always reach 6174; and once there, it stays at 6174 forever (it's a fixed point). Quick example with 3141: 4311 − 1134 = 3177 7731 − 1377 = 6354 6543 − 3456 = 3087 8730 − 0378 = 8352 8532 − 2358 = 6174 7641 − 1467 = 6174 (and it loops here) This works for every qualifying four-digit number due to the finite set of possibilities and the structure of the operation in base 10.

A visual history of π approximations From ancient civilizations to modern mathematics, this infographic showcases key milestones in our quest to understand π. See how approximations evolved; starting with simple fractions and polygon methods by Archimedes, through brilliant contributions like Zu Chongzhi’s fraction, infinite products, series, and fast-converging formulas developed over centuries.

They called it one of the most powerful tools in mathematics—a way to represent complex functions as an infinite sum of simpler terms. It was developed by an English mathematician named Brooke Taylor. Born in 1685, Taylor studied at Cambridge, earning a law degree, but devoted himself to pure mathematics. By his twenties, he had contributed to geometry, mechanics, and the mathematics of vibrating strings, helping lay the foundations of wave theory. In 1715, he introduced what we now call the Taylor series, an expansion that expresses a function as the sum of its derivatives at a single point, with each term multiplied by powers of the input offset. This allowed mathematicians to approximate functions with polynomials, making them easier to analyze, compute, and apply. The Taylor series became indispensable in physics, engineering, and mathematical analysis—from orbital mechanics and optics to thermodynamics, quantum theory, and numerical algorithms. Brook Taylor died in 1731 at just 46, but his method remains a cornerstone of mathematical analysis, shaping how we model the world.