Special Course by MATH+ Visiting Scholar Guido Montufar (UCLA) (20.07 – 24.07)

Mathematical Foundations of Modern Machine Learning:
Parameter Spaces, Function Spaces, and Optimization Dynamics

This mini-course explores a mathematical perspective on modern machine learning centered around three fundamental objects: the parameter
spaces of neural networks, the realized function spaces, and the optimization dynamics that connect the two. Each lecture will be accompanied by representative research papers and pointers to current research directions.

A more detailed description is included below. 

There will be one lecture per day (10:30 a.m.–12:00 p.m.) in room IMoS 3003:

If you want to attend, please send an email to This email address is being protected from spambots. You need JavaScript enabled to view it. with the subject line "Attending MATH+ Lecture'', including the following information: Name; Program (e.g., BSc, MSc, PhD); Year (e.g., 1st year, 2nd year); Email address. 

Lecture 1. Parameter Spaces of Neural Networks
* Identifiability
* Parameter symmetries
* Fibers of the realization map

Lecture 2. Function Spaces and Realization Spaces
* The realization map and realization spaces
* Examples from ReLU networks and attention architectures

Lecture 3. Expressivity and Complexity
* Linear regions in ReLU, Maxout, Max-Pooling, and Graph Neural Networks
* Expected number of linear regions

Lecture 4. Optimization Dynamics: Implicit Bias and Feature Learning
* Implicit bias of parameter optimization in the kernel regime
* Feature learning and effective width collapse in the mean-field regime

Lecture 5. Verification of Neural Networks
* Mathematical verification and certified robustness
* Current challenges and open problems

Workshop: AI for Math (28.07 – 29.07)

Further information, the program, and the registration form are available here:
https://www.ai4math.de/events/other/ai-workshop-ai4math-2026-07/
Please register by 21 July 2026.

MiniCourse by Alan Stapledon (17.08 - 21.08)

The first two days will be devoted to background material
like simplicial complexes and their face rings or lattice
polytopes and Ehrhart theory.
The course brings students up to speed to work in groups
on research projects during the following two months.
Please register by August 3: [https://tally.so/r/dWpped]

Block Lecture: Computational Integer Optimization (24.08 - 28.08)

Integer optimization lies at the core of many real-world decision problems in logistics, finance, energy systems, and production planning. Solving large-scale mixed-integer programs efficiently requires sophisticated algorithms, numerical rigor, and a deep understanding of the mathematics behind the applications.

This block course focuses on the computational and mathematical foundations of modern integer optimization solvers. In dedicated lectures, we study the algorithmic building blocks that make state-of-the-art solvers effective in practice, as well as the mathematical principles underlying their correctness and performance.

Topics include different classes of cutting planes and techniques to prove their correctness; the mathematical foundations of presolving (with excursions to number theory and graph theory); primal heuristics for finding high-quality feasible solutions; logical deduction mechanisms in propagation and infeasibility analysis; and the integration of machine learning techniques into optimization algorithms. Further emphasis is placed on numerics in limited-precision algebra, software engineering aspects, principled evaluation of algorithms, and best modeling practices in mathematical optimization.

In addition to the lectures, there will be hands-on implementation sessions, working with state-of-the-art optimization software.

Exams are based on lecture content so active participation is highly recommended. Additional materials are given for optional further reading.