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Events
IMPRS Combo V

White text on a blue background: "IMPRS Combo V"

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

09.03.26 11.03.26

IMPRS Combo V

Please register

conference

09.03.26 11.03.26

IMPRS Combo V

The IMPRS Combo V is an interdisciplinary and introductory workshop for PhD students and PostDocs in their early career. It aims to bring IMPRS students together and to encourage them to learn about a wider spectrum of mathematics. This year’s Combo features two mini-courses along with two additional guest lecturers. We draw connections not only between several fields of mathematics and its history, but also to physics, computing science, and neurobiology.

Our ﬁrst mini-course by Karin Baur is an introduction to the field of cluster algebras, which are relevant to areas including combinatorics, algebraic geometry, hyperbolic geometry, and dynamical systems. Our second mini-course by Alessandra Caraceni focuses on stochastic mixing and includes connections to probability, physics, and computing science. To round out the program, we have lectures by Jürgen Jost on the history of modern geometry, as well as a lecture by Nico Scherf on geometry encountered in the context of neurobiology. He is from the MPI for Cognitive and Behavioral Sciences, providing an opportunity for participants to network with other scientists from Leipzig.

Registration open till February 23, 2026.

Speakers

Karin Baur

Alessandra Caraceni

Jürgen Jost

Dani Kaufman

Nico Scherf

Program

09:30 - 10:00	Welcome Coffee and Registration
10:00 - 11:00	Dani Kaufman Intro Session on Cluster Algebra
11:15 - 12:15	Karin Baur Combinatorial surface models for cluster algebras and cluster categories - Lecture 1 Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this course, I will show how triangulations give rise to cluster algebras and to cluster categories. We will also study the combinatorial approach to Grassmannian cluster algebras and cluster categories. Time permitted, we will also discuss how tilings of surfaces give gentle algebras and module categories.
12:15 - 13:15	Lunch Break
13:15 - 14:15	Intro Session on Stochastic Mixing
14:30 - 15:30	Alessandra Caraceni How long does it take to shufﬂe a deck of cards? An introduction to mixing - Lecture 1 Many familiar random processes can be modelled as Markov chains. For example, repeatedly shufﬂing a deck of $n$ cards using a simple random rule deﬁnes a Markov chain whose states are the $n!$ possible orderings of the deck, and whose transition probabilities can be arranged into an $n!$ by $n!$ stochastic matrix $P$. A natural question is: how many shufﬂes are needed before the deck is well mixed?The answer is given by the theory of mixing times, which quantify how quickly a random process approaches its long-term behaviour. Notably, the mixing time of many natural card shufﬂes will turn out to be polynomial in $n$, allowing to roughly uniformly sample one of a super-exponential number of possible orderings via a small number of simple operations. In this minicourse, we shall introduce the concept of mixing time via concrete examples and gradually develop key tools for estimating it, such as couplings, spectral methods and conductance. We shall see how this rich theory plays a central role in probability, theoretical computer science and statistical physics, and discuss recent developments as well as several beautiful open problems.
15:30 - 16:00	Coffee Break
16:00 - 17:00	Jürgen Jost The development of modern geometry: From Gauss and Riemann to the present - Lecture 1 Bernhard Riemann's seminal work that founded Riemannian geometry was inspired by Gauss' insights on the geometry of surfaces. Riemann introduced what is now called a Riemannian manifold and showed that the sectional curvatures derived from the Riemann curvature tensor yield a complete set of local invariants of a Riemannian metric. Riemannian geometry became fundamental for Einstein's theory of general relativity theory as well as modern quantum field theory. Currently, it also provides the basis for novel methods in machine learning. From a philosophical perspective, it offers new insight into the nature of space.

10:00 - 11:00	Karin Baur Combinatorial surface models for cluster algebras and cluster categories - Lecture 2 Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this course, I will show how triangulations give rise to cluster algebras and to cluster categories. We will also study the combinatorial approach to Grassmannian cluster algebras and cluster categories. Time permitted, we will also discuss how tilings of surfaces give gentle algebras and module categories.
11:15 - 12:15	Alessandra Caraceni How long does it take to shufﬂe a deck of cards? An introduction to mixing - Lecture 2 Many familiar random processes can be modelled as Markov chains. For example, repeatedly shufﬂing a deck of $n$ cards using a simple random rule deﬁnes a Markov chain whose states are the $n!$ possible orderings of the deck, and whose transition probabilities can be arranged into an $n!$ by $n!$ stochastic matrix $P$. A natural question is: how many shufﬂes are needed before the deck is well mixed?The answer is given by the theory of mixing times, which quantify how quickly a random process approaches its long-term behaviour. Notably, the mixing time of many natural card shufﬂes will turn out to be polynomial in $n$, allowing to roughly uniformly sample one of a super-exponential number of possible orderings via a small number of simple operations. In this minicourse, we shall introduce the concept of mixing time via concrete examples and gradually develop key tools for estimating it, such as couplings, spectral methods and conductance. We shall see how this rich theory plays a central role in probability, theoretical computer science and statistical physics, and discuss recent developments as well as several beautiful open problems.
12:15 - 13:15	Lunch Break
13:15 - 15:30	Nico Scherf The Geometry of Thought: Perspectives on Neural and Artiﬁcial Representations Understanding how complex systems represent, transform, and act on information is a shared challenge across neuroscience, machine learning, and cognitive science. In both biological and artiﬁcial systems, internal representations exhibit structure that goes beyond individual variables and can often be fruitfully described using mathematical concepts such as geometry, dynamics, and optimization. In this lecture, I will present a selection of ideas from our recent work that explore how mathematical structures emerge in neural and artiﬁcial representations, and how they can be used to compare, interpret, and constrain models of cognition and learning. The examples will be drawn from empirical neural data, deep learning models, and learning agents interacting with complex environments. Rather than focusing on a single method or result, the talk will emphasize conceptual connections. The goal is to offer an intuitive entry point for mathematicians into current questions at the interface of data-driven neuroscience and (Machine) learning theory, highlight fruitful applications of mathematical thinking to real systems, and outline open problems where further mathematical insight could play a decisive role.
15:30 - 16:00	Coffee Break
16:00 - 17:00	Discussion/Talk
18:00 -	Conference Dinner - Individual Payment

10:00 - 11:00	Karin Baur Combinatorial surface models for cluster algebras and cluster categories - Lecture 3 Surface combinatorics have been instrumental in describing algebraic structures such as cluster algebras and cluster categories, gentle algebras, etc. In this course, I will show how triangulations give rise to cluster algebras and to cluster categories. We will also study the combinatorial approach to Grassmannian cluster algebras and cluster categories. Time permitted, we will also discuss how tilings of surfaces give gentle algebras and module categories.
11:15 - 12:15	Karin Baur Exercise Session on Cluster Algebra
12:15 - 13:15	Lunch Break
13:15 - 14:15	Alessandra Caraceni How long does it take to shufﬂe a deck of cards? An introduction to mixing - Lecture 3 Many familiar random processes can be modelled as Markov chains. For example, repeatedly shufﬂing a deck of $n$ cards using a simple random rule deﬁnes a Markov chain whose states are the $n!$ possible orderings of the deck, and whose transition probabilities can be arranged into an $n!$ by $n!$ stochastic matrix $P$. A natural question is: how many shufﬂes are needed before the deck is well mixed? The answer is given by the theory of mixing times, which quantify how quickly a random process approaches its long-term behaviour. Notably, the mixing time of many natural card shufﬂes will turn out to be polynomial in $n$, allowing to roughly uniformly sample one of a super-exponential number of possible orderings via a small number of simple operations. In this minicourse, we shall introduce the concept of mixing time via concrete examples and gradually develop key tools for estimating it, such as couplings, spectral methods and conductance. We shall see how this rich theory plays a central role in probability, theoretical computer science and statistical physics, and discuss recent developments as well as several beautiful open problems.
14:30 - 15:30	Alessandra Caraceni Exercise Session on Stochastic Mixing
15:30 - 16:00	Coffee Break
16:00 - 17:00	Jürgen Jost The development of modern geometry: From Gauss and Riemann to the present - Lecture 2 Bernhard Riemann's seminal work that founded Riemannian geometry was inspired by Gauss' insights on the geometry of surfaces. Riemann introduced what is now called a Riemannian manifold and showed that the sectional curvatures derived from the Riemann curvature tensor yield a complete set of local invariants of a Riemannian metric. Riemannian geometry became fundamental for Einstein's theory of general relativity theory as well as modern quantum field theory. Currently, it also provides the basis for novel methods in machine learning. From a philosophical perspective, it offers new insight into the nature of space.