IMPRS minicourse: Sieving in algebra and geometry, I
IMPRS lecture course "Sieving in algebra and geometry"
A course on modern sieve theory:
"Sieving in algebra and geometry"
This course explores recent advances in the application of classical methods from analytic number theory—such as Brun’s sieve, the large sieve, and the Hardy–Littlewood circle method—to problems in algebraic number theory and arithmetic geometry.
Emphasis will be placed on the interplay between analytic techniques and geometric structures, with applications including the Cohen–Lenstra heuristics in the statistical study of class groups and the Batyrev–Manin conjecture on the distribution of rational points on Fano varieties. No prior familiarity with sieves will be assumed.
MPIM seminar room on:
Monday 12 May, time 16.30-17.30
Tuesday 13 May, time 16.30-17.30
Wednesday 14 May, time 16.30-17.15
Thursday 15 May, time 16.30-17.30
Vorlesung Selected Topics in Algebraic Geometry: Habiro Cohomology
Description: https://people.mpim-bonn.mpg.de/scholze/ss2025_habiro.pdf
Vorlesung: Selected Topics in Algebra: The Habiro Ring of a Number Field
Details & abstract:
https://people.mpim-bonn.mpg.de/scholze/veranstaltungen.html
https://people.mpim-bonn.mpg.de/scholze/ws202425_habiro.pdf
Video recordings:
Lecture 10: Dualizable categories and localizing motives, V
I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.
Lecture 9: Dualizable categories and localizing motives, IV
I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.
Lecture 8: Dualizable categories and localizing motives, III
I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.
Lecture 7: Introduction to dualisable categories and their $K$-theory, V
In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories.
Lecture 6: Dualizable categories and localizing motives, II
I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.
Lecture 5: Introduction to dualisable categories and their $K$-theory, IV
In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories.
Lecture 4: Dualizable categories and localizing motives, I
I will give an introduction to localizing invariants of dualizable categories. I will start with the general theory of dualizable categories, and explain some non-trivial results, such as equivalence between dualizability and flatness for a presentable stable category. Then we will compute the localizing invariants of various dualizable categories which come from topology and non-Archimedean analysis. These include sheaves on locally compact Hausdorff spaces and categories of nuclear modules on formal schemes. I will also explain a deep connection between the algebra of dualizable categories and the category of localizing motives -- the target of the universal finitary localizing invariant (of categories over some base). We will see that the category of localizing motives (considered as a symmetric monoidal category) is in fact rigid in the sense of Gaitsgory and Rozenblyum, and sketch some applications of this result.
Lecture 3: Introduction to dualisable categories and their $K$-theory, III
In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories.
Lecture 2: Introduction to dualisable categories and their $K$-theory, II
In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories
Lecture 1: Introduction to dualisable categories and their $K$-theory, I
In this series of lectures, I will give an introduction to some fundamental concepts that will be relevant for this workshop:
localizing invariants of stable $\infty$-categories, Waldhausen $K$-theory, noncommutative motives, and presentable and dualisable categories
Lecture course "Analytic Stacks" by Dustin Clausen and Peter Scholze
The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:
1. Light condensed abelian groups.
2. Analytic rings.
3. Analytic stacks.
4. Examples.
Lectures will be given by Dustin Clausen at IHES and Peter Scholze at MPI, and broadcast live at the other location.
We also plan to make the lectures accessible by Zoom, and record them.
Mi, 10(c.t.) - 12 Uhr, und Fr, 10(c.t.) - 12 Uhr, MPI-Hörsaal
First Lecture: October 18, 2023
Recordings can be found on Youtube.
Vorlesung "Selected Topics in Differential Geometry - The classical Plateau Problem"
Felix Klein Lectures 2022 with Jacob Lurie
Felix Klein Lectures 2022
organised by the HCM. Please find the official website of the event here.
A Riemann-Hilbert Correspondence in p-adic Geometry
Jacob Lurie (Institute for Advanced Studies, Princeton)
Dates:
Lecture 1: MPI, Tuesday, Nov 15, 12:00--13:00;
Lecture 2: MPI, Thursday, Nov 17, 14:00--15:00;
Lecture 3: MPI, Tuesday, Nov 22, 16:30--17:30;
Lecture 4: MPI, Thursday, Nov 24, 14:00--15:00;
Lecture 5: MPI, Tuesday, Nov 29, 16:30--17:30;
Lecture 6: MPI, Thursday, Dec 1, 14:00--15:00.
Venue: MPIM lecture hall
Members of the Bonn mathematics community do not have to apply for participation via this platform. Access to the lecture hall will be granted to them on a first come first served basis until all available seats are taken. The registration for external people is already closed.
Abstract:
At the start of the 20th century, David Hilbert asked which representations can arise by studying the monodromy of Fuchsian equations. This question was the starting point for a beautiful circle of ideas relating the topology of a complex algebraic variety X to the study of algebraic differential equations. A central result is the celebrated Riemann-Hilbert correspondence of Kashiwara and Mebkhout, which supplies a fully faithful embedding from the category of perverse sheaves on $X$ to the category of algebraic $\mathfrak{D}_X$-modules. This embedding is transcendental in nature: that is, it depends essentially on the (archimedean) topology of the field of complex numbers. It is natural to ask if there is some counterpart of the Riemann-Hilbert correspondence over nonarchimedean fields, such as the field $\mathbf{Q}_p$ of $p$-adic rational numbers. In this series of lectures, I will survey some of what is known about this question and describe some recent progress, using tools from the theory of prismatic cohomology (joint work with Bhargav Bhatt).Image source: https://commons.wikimedia.org/wiki/File:Felix_Klein.jpeg
Lectures on minimal 3-manifolds
Lectures on "Topics in 4-manifolds"
Contact: Peter Teichner, Rob Schneiderman
Lecture course by Don Zagier on "Standard and less standard asymptotic methods"
Starting February 15, Don Zagier will give a lecture course entitled "Standard and less standard asymptotic methods". This course will be given in collaboration with the IGAP (Institute of Geometry and Physics, a new joint venture between SISSA and ICTP in Trieste).
The course will be streamed from Trieste twice a week (Tu/Th 4-5:30 for the first four weeks and Mo/We 2-3:30 for the last two weeks).
Abstract:
In every branch of mathematics, one is sometimes confronted with the problem of evaluating an infinite sum numerically and trying to guess its exact value, or of recognizing the precise asymptotic law of formation of a sequence of numbers ${A_n}$ of which one knows, for instance, the first couple of hundred values. The course will tell a number of ways to study both problems, some relatively standard (like the Euler-Maclaurin formula and its variants) and some much less so, with lots of examples. Here are three typical examples: 1. The slowly convergent sum $\sum_{j=0}^\infty (\binom{j+4/3}{j})^{-4/3}$ arose in the work of a colleague. Evaluate it to 250 decimal digits. 2. Expand the infinite sum $\sum_{n=0}^\infty (1-q)(1-q^2)...(1-q^n)$ as $\sum A_n (1-q)^n$, with first coefficients 1, 1, 2, 5, 15, 53, ... Show numerically that $A_n$ is asymptotic to $n! * a * n^b * c$ for some real constants $a$, $b$ and $c$, evaluate all three to high precision, and recognize their exact values. 3. The infinite series $H(x) = \sum_{k=1}^\infty sin(x/k)/k$ converges for every complex number $x$. Compute this series to high accuracy when $x$ is a large real number, so that the series is highly oscillatory.
The courses are scheduled as follows:
1 Tue 15-Feb 16.00 - 17-30
2 Thu 17-Feb 16.00 - 17-30
3 Tue 22-Feb 16.00 - 17-30
4 Thu 24-Feb 16.00 - 17-30
5 Tue 01-Mar 16.00 - 17-30
6 Thu 03-Mar 16.00 - 17-30
7 Tue 08-Mar 16.00 - 17-30
8 Thu 10-Mar 16.00 - 17-30
9 Mon 14-Mar 14.00 - 15.30
10 Wed 16-Mar 14.00 - 15.30
11 Mon 21-Mar 14.00 - 15.30
12 Wed 23-Mar 14.00 - 15.30
These will be hybrid courses. All are very welcome to join either online or in person (if provided with a green pass). Venue: Budinich Lecture Hall (ICTP Leonardo Da Vinci Building), for those wishing to attend in person. Zoom: https://unesco-org.zoom.us/j/91274263707
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