I will outline the Hodge Theorem for the first cohomology group of a Riemann surface and explain how the theory of Higgs bundles can be viewed as providing a non-abelian version of this.

Popescu’s Theoprem is a renown and famously hard theorem from commutative algebra. It says, colloqually statet, that if a morphism of a noetherian scheme (not necessarily of finite type) has smooth fibers, the morphism itself is smooth (i.e., it is a submersion). This is an instance of the princple “information about the fibers of a morphism imply global properties of the morphism itself”. Other examples are Zariski’s Main Theorem ar the “Flatness Miracle” for flatness of morphisms. Popescu’s theorem has a myriad of applications (e.g, for Artin approximation or the Bass-Quillen conjecture,) and it found its way into Séminaire Bourbaki and a whole chapter of the stacks project. In the seminar, we will try to understand at least the formulation of the theorem and the overall strategy of its proof.

Rational parametrizations play a crucial role in algebraic geometry. In differential algebra, a similar concept is that of so-called realizations. In this talk we see when and how realizations and its inverse operation, implicitization, can be successfully performed, which is strongly linked to finding rational parametrizations. We will also consider the problems of finding polynomial realizations and that of realizations involving parameters in an optimal way.
This talk is based on joint works with Rafael Sendra, Alexey Ovchinnikov and Dmitrii Pavlov.

Hardy fields are an algebraic setting for a tame part of asymptotic analysis. In this talk, after explaining some basic definitions around Hardy fields, I will focus on analytic Hardy fields, which are those that mainly arise in practice. I will explain a recent, in some sense ultimate, extension theorem for analytic Hardy fields, and I will give a few applications and state some open questions. (Joint work with L. van den Dries and J. van der Hoeven.)

Optimal transport is the general problem of moving one distribution of mass to another one as efficiently as possible, typically keeping track of the ambient geometry. In this seminar I will present recent results on the optimal transport problem between algebraic hypersurfaces of the same degree in complex projective space. I will explain how this problem, which is defined through a constrained dynamical formulation, is equivalent to a Riemannian geodesic problem away from the discriminant. I will discuss the main properties of the distance obtained in this way on the space of hypersurfaces, which has the meaning of an inner Wasserstein distance. For instance, this distance is of Weil-Petersson type. Time permitting, I will connect to the condition number of polynomial system solving and the problem of regularity of roots of polynomials. This is joint work with P. Antonini and F. Cavalletti

CR maps (for most of our talk that will just mean holomorphic maps taking a real submanifold in a complex space into a real-algebraic subset) enjoy certain unique determination properties. We will survey some of the known results starting with the original observation by Poincaré, and move on to some newer results showcasing the importance of complex analytic structure inside of the targets.

Popescu’s nested approximation theorem states the following: Let \(F\) be a polynomial system of equations in two sets of variables \(x,y\) with a formal power series solution \(\widehat{y}(x)\) (i.e., \(F(x,\widehat{y}(x)=0\)) that is nested (i.e., each \(\widehat{y_i}(x)\) only depends on the first \(x_1,\ldots,x_{n_i}\)). Then for any natural number \(c\) there exists an algebraic powerseries \(y(x)\) such that

1. \(F(x,y(x))=0\),
2. \(y(x)\) and \(\widehat{y}(x)\) agree up to degree \(c\) and
3. \(y(x)\) is nested.

In general, the proof of this theorem requires Popescu’s theorem on smoothing morphisms. This talk will first give a more elementary way to prove the theorem if there are only two nests and the first nest is only dependent on \(x_1\).

The second part of the talk will be an outline of the general proof.

It is known since the time of Abel that a univariate holomorphic function which is algebraic, i.e., zero of a polynomial equation, is also differentially finite, i.e., solution of a linear differential equation with polynomial coefficients. For instance, \(y(x) = \sqrt{1+x} – \sqrt[3]{1+x}\) solves a polynomial equation \(P(x,y)=0\) of degree \(6\) in \(y\) and satisfies the differential equation \(6(1+x)^2 y^{\prime \prime} + (1+x)y’ + y = 0\).

It remained a prominent mystery (and still is, though many cases could be clarified), which differential equations arise in this way, say, stem from algebra. The notorious Grothendieck-Katz \(p\)-curvature conjecture predicts a criterion for this via reduction modulo prime numbers. In all generality, the conjecture is unsolved.

In the lecture, which addresses a non-expert audience, we give two proofs of Abel’s result, and further insight about this intriguing question.

Let \(R\) be the polynomial ring \(\mathbb{C}[x_1,\ldots,x_n]\) in \(n\) variables with complex coefficients. Let \(f\) be a non constant polynomial in \(R\). We denote by \(\mathrm{Syz}_R(f_1,…,f_n,f)\) the \(R\)-module of syzygies among the partial derivatives \(f_i\) of \(f\) w.r.t. \(x_i\).

To this syzygy module we associate two (different but related) modules over the Weyl algebra \(D:=D(R)\) of linear differential operators on the polynomial ring \(R\). We will show how, in some interesting cases, how these \(D\)-modules can help to compute the (singular) cohomology of the complement in \(\mathbb{C}^n\) of the hypersurface defined by \(f=0\).

How do the roots of a monic polynomial whose coefficients depend smoothly on parameters depend on those parameters? What about the spectral decomposition of a linear operator? I will give a survey revolving around these questions. The subject, which started with Rellich’s work in the 1930s, enjoyed sustained interest through time that intensified in the last two decades, bringing some definitive optimal results.

Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that exponential periods and the Euler constant are exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.

The Jacobi triple product identity is one of the most mysterious equation between infinite products and infinite sums. There are some interpretations of this equation and many mathematicians research its generalization according to each interpretation. In this talk, we consider one of them from the viewpoint of singular weight Jacobi forms and give a characterization of root systems.

The extended Catalan arrangement of type \(B_2\) is defined as the coning of lines \(x=k\), \(y=k\), \(x\pm y=k\)  (\(k=-m,\dots,m\)).

Its logarithmic vector field is known to be free, but explicit shape of the basis was not known until recently.

 I will give a short introduction on the (mulit-)arrangement theory and explain how to construct an explicit basis for the extended Catalan arrangement of type \(B_2\).

In the 1960’s, Deligne, Mumford and Knudsen wrote seminal papers about the compactification of moduli spaces of “stable \(n\)-pointed curves”. The simplest example occurs when one takes \(n\) distinct points on the projective line \(\mathbb{P}^1\) and considers their isomorphism classes under Möbius transformations. Compactifying the space of isomorphism classes of such \(n\)-tuples boils down to define suitable limits as some of the points come together and coalesce.

The mentioned papers use quite a bit of machinery from algebraic geometry. We will present a down-to-earth approach using phylogenetic trees. They play the role of a manual (Bedienungsanleitung) for finding proofs: combinatorial manipulations with the trees result in outlines of proofs which then just have to be formulated rigorously in algebraic terms. It is almost miraculous how nicely this works.

The talk is understandable for Master’s students.

In the 1960’s, Deligne, Mumford and Knudsen wrote seminal papers about the compactification of moduli spaces of “stable \(n\)-pointed curves”. The simplest example occurs when one takes \(n\) distinct points on the projective line \(\mathbb{P}^1\) and considers their isomorphism classes under Möbius transformations. Compactifying the space of isomorphism classes of such \(n\)-tuples boils down to define suitable limits as some of the points come together and coalesce.

The mentioned papers use quite a bit of machinery from algebraic geometry. We will present a down-to-earth approach using phylogenetic trees. They play the role of a manual (Bedienungsanleitung) for finding proofs: combinatorial manipulations with the trees result in outlines of proofs which then just have to be formulated rigorously in algebraic terms. It is almost miraculous how nicely this works.

The talk is understandable for Master’s students. This second part of the lectures about moduli spaces does not presuppose having attended the first half.

In the 1960’s, Deligne, Mumford and Knudsen wrote seminal papers about the compactification of moduli spaces of “stable \(n\)-pointed curves”. The simplest example occurs when one takes \(n\) distinct points on the projective line \(\mathbb{P}^1\) and considers their isomorphism classes under Möbius transformations. Compactifying the space of isomorphism classes of such \(n\)-tuples boils down to define suitable limits as some of the points come together and coalesce.

The mentioned papers use quite a bit of machinery from algebraic geometry. We will present a down-to-earth approach using phylogenetic trees. They play the role of a manual (Bedienungsanleitung) for finding proofs: combinatorial manipulations with the trees result in outlines of proofs which then just have to be formulated rigorously in algebraic terms. It is almost miraculous how nicely this works.

The talk is understandable for Master’s students.

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

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.

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.