Summer Graduate School

One of the hottest topics in analytic number theory involves the use of statistics and probability in understanding different aspects of algebraic and analytic number theory, through various new lenses. This is reflected in some of the most exciting number theory research of the last few years (for example, of Bhargava, of Ellenberg and Venkatesh, of Alexander Smith, of Sawin and Wood, of Adam Harper, of Koukoulopoulos and Maynard, of Helfgott and Radziwill, of Pilatte,....). As a consequence the CRM will host a thematic semester Mar 2-July 3, 2026 on these topics involving some of the world leaders in the subject. Since this new area can roughly be split in two into Algebraic and Analytic, we will focus for two months on each, with the SMS school placed in the middle. The 2026 SMS will introduce junior mathematicians to important trends in number theory. 

School Structure

The first week will have a focus on arithmetic statistics but will include some of the more basic probabilistic number theory lectures.  The second week will have a focus on probabilistic number theory, but will include some of the more advanced arithmetic statistics lectures.

There will be three or four hous of lecture courses per day, accompanied by several problem sessions and working groups.

Prerequisites

Students should be familiar with first courses in algebraic number theory and in analytic number theory.

Preparatory Reading for certain mini-courses. Each of the following are about 30 pages long:

-- Melanie Wood (Harvard): "Probability theory for random groups arising in number theory" https://ems.press/books/standalone/278/5565

-- Ashvin Swaminathan (Harvard): "Counting Cubic Number Fields" by Niven Achenjang https://www.mit.edu/~NivenT/assets/pdf/Counting_Cubic_Number_Fields.pdf.

-- Adam Harper "Moments of random multiplicative functions, III: A short review" https://arxiv.org/abs/2410.11523, & the introduction of "On the limiting distribution of sums of random multiplicative functions"  https://arxiv.org/abs/2508.12956

-- Alexandra Florea (UC Irvine): "Traces of high powers of the Frobenius class in the hyperelliptic ensemble" by Zeev Rudnick, Acta Arithmetic, 143.1 (2010), 81-99, obatin from this LINK

-- Alex Smith: "The Selmer group, the Shafarevich-Tate group, and the weak Mordell Weil theorem" by Bjorn Poonen https://math.mit.edu/~poonen/f01/weakmw.pdf

-- Tim Browning (IST Austria): "Beginners guide  to the circle method" by Andrew Granville https://dms.umontreal.ca/~andrew/CircleMethodNotes.pdf

Application Procedure

SLMath is only able to support a limited number of students to attend this school.  Therefore, it is likely that only one student per institution will be funded by SLMath.

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Venue

The summer school will take place at Centre de recherches mathématiques (CRM), Montréal, Canada.

Additional Links

For additional information about the thematic program, check out this webpage: https://www.crmath.ca/en/activities/#/type/activity/id/3952
For additional information about the summer school, check out this webpage: https://www.crmath.ca/en/activities/#/type/activity/id/4055