This talk provides an elementary survey of D. Popescu’s General Néron Desingularization Theorem, stating that regular morphisms of noetherian rings, that is flat morphisms with geometrically regular fibers, are filtered colimits of smooth morphisms. The focus of the talk is on elucidating the theorem’s key concepts, particularly the notion of smooth algebras. Applications, notably in the context of Artin approximation, will be briefly outlined.

Wave maps are the prime example of geometric wave equations with important applications in geometry and theoretical physics. One intriguing feature is that wave maps tend to form singularities in finite time via self-similar solutions. Understanding the stability of these solutions is key and leads to challenging spectral problems for Fuchsian equations. I will explain this connection and outline a novel method to treat these spectral problems rigorously. This is based on joint work with O. Costin (Ohio State) and I. Glogic (U Vienna).

A function is said to be D-finite (“differentially finite”) if it satisfies a linear differential equation with polynomial coefficients. D-finite functions are ubiquitous in number theory and combinatorics. In a seminal article from 1980, Richard Stanley asked whether it is possible to decide if a given D-finite function is algebraic or transcendental. Several very useful sufficient criteria for transcendence exist, e.g., using asymptotics, but none of them is also a sufficient condition. Characterizing the algebraicity of a D-finite function is highly nontrivial even if the sequence of its Taylor coefficients satisfies a recurrence of order one: this question was completely settled only in 2023 by Florian Fürnsinn and Sergey Yurkevich, based on important articles by Gilles Christol (1986) and by Frits Beukers and Gert Heckman (1989). In this talk, I will present answers to Stanley’s question and illustrate them through several examples coming from number theory and combinatorics.

We will give a general overview of some old and new ideas in geometric quantization, which aims at “quantizing” symplectic manifolds. As we will see, in the case of Kahler manifolds, recent developments show that flows of Hamiltonian vector fields analytically continued to imaginary time play a key role. We will mention, without technical details, some concrete examples like the complex plane, reductive Lie groups and toric manifolds.

Hardy fields form a natural domain for a “tame” part of asymptotic analysis. In this talk I will sketch how a recent theorem which permits the transfer of statements concerning algebraic differential equations between Hardy fields and related structures yields applications to some classical linear differential equations. (Joint work with L. van den Dries and J. van der Hoeven.)

Our motivation is the question whether there exists a constant $c_0$ such that the differential operator

\( (-1)^n \frac{d^{2n}}{dx^{2n}} + \frac{c}{x^{2n}} \)

gives rise to a unique self-adjoint operator on the half-line for \(c\geq c_n\). In the case \(n=1\) this is the radial
part of the Schrödinger equation and the answer \(c_1=3/4\) is well-known. 
 
The answer boils down to the question of how many solutions of the underlying differential equation are square integrable.
Since the equation is of Fuchs type this reduces to the question of how many zeros of the indicial polynomial have real part \(> -1/2\).
Our answer is based on some old results about zeros of polynomials including the Grace-Heawood theorem and Orlando’s formula. 

Based on joint work with Fritz Gesztesy and Markus Hunziker

Let \(X,Y\) be affine Schemes and \(\varphi:Y\rightarrow X\) be a morphism (called change of basis). Let \(Z\subseteq X\) be a closed subscheme, \(\pi:\widetilde{X}\rightarrow X\) be the blow up of \(X\) along \(Z\) and \(\widetilde{X}\times_{X}Y\) be the fibre product of \(\widetilde{X}\) and \(Y\) over \(X\). Furthermore, let \(\tau:\widetilde{X}\times_{X}Y\rightarrow Y\) be the projection onto \(Y\). Then the blowup of \(Y\) along \(S=\varphi^{-1}(Z)\) is the restriction \(\widetilde{\tau}:\widetilde{Y}\rightarrow Y\), where \(\widetilde{Y}\) is the Zariski-closure of the preimage \(\tau^{-1}(Y\setminus S)\).
In order to prove this theorem, I will revise the universal property of blowups and both the construction of blow ups via ring extensions and the construction of blow ups via graphs in projective space.

Depending on the interests of the audience I will tell a story beginning with elementary irrationality proofs for zeta values and passing through the topics of the title. 

Comment: This lecture will be an informal and gentle introduction to some aspects of the items in the title. It is particularly accessible and recommended for students.

A constant term sequence is a sequence of rational numbers whose \(n\)-th term is the constant term of \(P(x)^nQ(x)\), where \(P(x)\) and \(Q(x)\) are multivariate Laurent polynomials. While the generating functions
of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify those diagonals which are constant terms. In this talk I will explain such a classification in the case of sequences satisfying linear recurrences with constant coefficients. I will also showcase the still unresolved case of hypergeometric sequences.

The talk is based on joint work with Alin Bostan and Armin Straub.

A vector field \(D\) on \(\mathbb{R}^n\) is called regular at \(a\) if \(D\) does not vanish at \(a\). A simple use of the inverse function theorem shows that any such vector field can be brought by a local diffeomorphis at a into the normal form of a partial derivative, say \(\partial_{x_1}\).

The story becomes more intricate when one wants to rectificate several vector fields simultaneously into \(\partial_{x_1}, …, \partial_{x_k}\). As the latter commute with each other, the same condition is necessary for the original vector fields (since pull-backs of vector fields preserve the Lie-bracket). This is Frobenius’ (first) integrability condition. His theorem then asserts that this necessary condition is also sufficient. In this way one obtains an efficient characterization of simultaneous rectifibility. (The second integrability condition concerns modules of vector fields).

The nice thing about Frobenius’ theorem is that it’s proof is pure commutative algebra. We will try to work this out with all details in the lecture.

In this talk I will discuss Luna’s slice theorem, a result in algebraic geometry that characterizes particular quotients of varieties by reductive groups.
After giving some background info about group actions I will present the precise statement and give a few examples. Then we will look at the outline of the proof as well as some applications.

In this talk I will define a differential extension of the power series over a field \( k\) of characteristic \(p\) making use of logarithms. I will explain, that every ordinary linear differential equation with polynomial or power series coefficients over \(k\) admits a basis of solutions in this extension, in particular the exponential differential equation \(y’=y\) has a solution \(\exp_p\). Such solutions have remarking properties, which we will explore.
This talk is based on joint work with H. Hauser and H. Kawanoue.