SS 2021

Freitag, 16. April 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Philipp di Dio (TU Berlin)
(Gast von Markus Schweighofer)

Introducing PDEs in the Moment Problem with the heat equation as an example

Partial differential equations (PDEs) and the theory of moments in real algebraic geometry (RAG) are two highly developed fields in mathematics. In this talk we want to show how to combine both fields and hopefully establishing a fruitful interaction enabling the usage of PDEs, their methods, and results in RAG and the other way around. We demonstrate this attempt with the heat equation.

Freitag, 23. April 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Kristin Shaw (University of Oslo)
(Gästin von Mateusz Michalek)

Real phase structures on matroid fans

In this talk, I will propose a definition of real phase structures on polyhedral complexes. I’ll explain that in the case of matroid fans, specifying a real phase structure is cryptomorphic to providing an orientation of the underlying matroid. Then I’ll define the real part of a polyhedral complex with a real phase structure. This determines a closed chain in the real part of a toric variety. In the case when the polyhedral complex is a non-singular tropical variety, the real part is a PL-manifold. Moreover, for a non-singular tropical variety with a real phase structures we can apply the same spectral sequence for tropical hypersurfaces, obtained by Renaudineau and myself, to bound the Betti numbers of the real part by the dimensions of the tropical homology groups. This is joint work in progress with Johannes Rau and Arthur Renaudineau.

Montag, 26. April 2021 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Michele Serra (Universität Konstanz)
(Gast von Salma Kuhlmann)

The automorphism group of a valued field of generalised formal power series

Hahn fields are fields of generalised power series over a given coefficient field and with exponents in an ordered abelian group. They are equipped with a canonical valuation. We will study the group of valuation preserving automorphisms of such fields.  To do this, we study the lifting property i.e., the possibility of lifting automorphisms of the coefficient field and exponent group to automorphisms of the Hahn field. For a Hahn field with the lifting property, a result of Hofberger will be generalised, providing a 4-factor semidirect product decomposition of the automorphism group. We will also introduce the stronger notion of canonical lifting property, and describe a large class of Hahn fields satisfying this property. If time permits, I will focus on the group of strongly additive automorphisms (automorphisms that commute with infinite sums) which will be described in terms of the groups of automorphisms of the base field, the exponent group, and the units in the valuation ideal.This generalises results of Schilling and Deschamps on automorphisms of Laurent, respectively Puiseux series fields.

References:
B. Deschamps - Des automorphismes continus d’un corps de séries de Puiseux
H. Hofberger - Automorphismen formal reeller Körper (PhD thesis)
O. F. G. Schilling - Automorphisms of fields of formal power series

Freitag, 30. April um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Hunter Spink (Stanford University)
(Gast von Mateusz Michalek)

Tautological Classes of Matroids

In this talk, I will introduce a new framework based on toric vector bundles which unifies a decade’s worth of disparate connections between matroid theory and algebraic geometry.

(Based on a joint work with Andy Berget, Chris Eur, Dennis Tseng).

Freitag, 07. Mai 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Gennadiy Averkov (Brandenburgische Technische Universität)

(Gast von Salma Kuhlmann)

Expressive power of the semidefinite programming

This talk is about the expressive power of semidefinite programming when one allows to lift and to use finitely many LMIs of a given size k. Hamza Fawzi showed that the expressive power grows strictly by passing from size 2 to size 3. I present a result implying that the expressive power grows strictly by passing from size k to size k+1 for every k. To put it informally: each given size is a new semidefinite world. Size 1 is the world of polyhedra, size 2 is the word of second-order cone representable sets, size 3 is an even larger world etc.

As a consequence of the main result, it can be derived that the well known semidefinite formulations of the standard cones from polynomial optimization (such as SOS cones) are optimal in terms of the size of the LMIs: even though these formulations have such a huge size, they use just the right amount of the expressive power in general.

The proof of the main result relies on Ramsey's theorem for hypergraphs.

Montag, 17. Mai 2021 um 15:15 Uhr, Logic Colloquium.

Beau Mount (Universität Konstanz)
(Gast von Salma Kuhlmann)

On the Second-Order Squeezing Argument

In a famous essay, Georg Kreisel (1967) made two claims about therelationship between model-theoretic validity (truth in all set-sized structures) andintuitive validity (roughly, truth in all structures whatsoever): (1) in the first-ordercase, they can be shown to coincide by a ’squeezing argument’; (2) in the second-order case, they cannot. In a recent essay, Juliette Kennedy and Jouko V ̈a ̈an ̈anen(2017) have questioned (2). They suggest that there is a squeezing argument forsecond-order logic, but it uses truth in all Henkin models rather than truth in allstandard models in the key step. On their view, the familiar complaint that Henkinmodels do not capture the second-order quantifier is misguided here: the distinctionbetween a standard and a Henkin model can only be made ’from outside’, and itsuse is out of place when one is carrying on fundamental reasoning in one’s homelanguage. I argue against their position: I show that, given an extremely convincingprinciple about intuitive validity due to George Boolos (1985), the falsity of thethesis about Henkin models needed for their argument is simply a theorem in aframework that all parties to the debate should accept. No view ’from outside’ isrequired: no greater degree of semantic reflection is needed to carry out the proofthan to set up Kreisel’s problem in the first place.

Freitag, den 28. Mai 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Kathlén Kohn (KTH Stockholm)

(Gästin von Mateusz Michalek & Gabriela Michalek, KWIM Vorlesungsreihe)

The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices

We study the maximum likelihood (ML) degree of multivariate Gaussian models that are described by linear conditions on the concentration matrix. We obtain new formulae for the ML degree, one via Schubert calculus, and another using Segre classes from intersection theory. This allows us to characterize the extreme cases on the ML degree spectrum: models with ML degree zero and models with maximal ML degree. It turns out that models with non-maximal ML degree are (up to Zariski closure) exactly those linear spaces for which strong duality in semidefinite programming fails. The subvariety of the Grassmannian formed by these linear spaces is a union of certain coisotropic hypersurfaces of determinantal varieties. We illustrate our results and the underlying geometry in the case of trivariate models: here we give a full, finite list of geometric types of linear subspaces in the space of symmetric 3x3 matrices incl. their ML degrees.

This talk is based on 3 joint works with 1) C. Améndola, L. Gustafsson, O. Marigliano, A. Seigal; 2) Y. Jiang, R. Winter; 3) S. Dye, F. Rydell, R. Sinn.

Montag, 7. Juni 2021, um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Visu Makam (School of Mathematics, Princeton)
(Gast von Mateusz Michalek)

Singular tuples of matrices is not a null cone

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING(n,m), consisting of all m-tuples of n x n matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SING(n,m) will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SING(n,m) is not the null cone of any reductive group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SING(n,m). This is joint work with Avi Wigderson.

Freitag, den 11. Juni um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Maria Infusino (Universität Cagliari)

(Gästin von Salma Kuhlmann & Gabriela Michalek, KWIM Vorlesungsreihe)

The truncated moment problem on unital commutative real algebras

In this talk we investigate under which conditions a linear functional L defined on a linear subspace B of a unital commutative real algebra A admits an integral representation with respect to a nonnegative Radon measure supported on a prescribed closed subset K of the space of homomorphisms of A endowed with the weak topology. This is a generalization of the classical truncated moment problem and it has the advantage of encompassing also infinite dimensional instances, e.g. when A is not finitely generated and B is finite dimensional or when A is finitely generated algebra and B infinite dimensional subspace of A.

We first consider the case when A is equipped with a submultiplicative seminorm and provide a criterion for the existence of an integral representation for L. Then we build on this result to prove a Riesz-Haviland type theorem, which also holds when A is not necessarily equipped with a topology. This theorem allows us to extend some well-known results for the truncated moment problem for polynomials in finitely many variables to situations when the monomial diagram associated to B contains infinitely many monomials in one of the variables, e.g. for rectangular or sparse truncated moment problems.

We also apply our result to the moment problem for point processes, which is particularly interesting for its applications in statical mechanics though very little is known in literature about its solvability. Last but not least, we closely analyze the relation between the full and the truncated moment problem in our general setting, obtaining a result in the spirit of Stochel’s theorem, which easily applies to full moment problems for localized algebras.

This is a joint work with Raúl Curto, Mehdi Ghasemi, and Salma Kuhlmann.

Montag, 14. Juni 2021 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Mickael Matusinski (Université de Bordeaux)
(Gast von Salma Kuhlmann)

About rationality of formal multivariate power series

Based on a joint work with Michel Hickel (Bordeaux) on algebraicity of formal power series.

A formal power series y is rational if it is the expansion of a quotient of two polynomials P/Q. I will discuss some characterisations of the rationality of formal multivariate power series, by answering the two following problems:

- given a fraction of two polynomials P/Q, expand it as a series y by deriving its coefficients from those of P,Q;

- given a formal multivariate power series y which is rational, reconstruct from its first coefficients polynomials P,Q such that y=P/Q.

This extends well known results from the 1-variable case, such as Kronecker's characterization of rational series based on Hankel matrices.

Freitag, den 18. Juni 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Victor Vinnikov (Ben-Gurion University of Negev)
(Gast von Salma Kuhlmann)

Real hyperbolic polynomials, complex stable polynomials and their determinantal representations

I will discuss the relationship between real homogeneous hyperbolic polynomials, complex polynomials that are stable (i.e., have no zeroes) with respect to a tube domain in the complex Euclidean space, and complex polynomials that are stable with respect to a compact domain (especially a bounded symmetric domain which is a compact realization of a tube domain over a homogeneous cone). I am particularly interested in how this relationship bears on determinantal representations certifying hyperbolicity and stability, and on linear matrix inequality representation of hyperbolicity cones.

This is a joint work with A. Grinshpan, D. Kaliuzhnyi-Verbovetskyi, and H. Woerdeman.

Montag, 21. Juni 2021 um 15:15 Uhr, Logic Colloquium.

Sam Roberts (Universität Konstanz)
(Gast von Salma Kuhlmann)

Sets as structures

Structuralism in the philosophy of mathematics is the view that mathematics concerns structures. Ordinary mathematical objects, like natural numbers, real numbers, sets, etc, are then taken to be places in those structures. Structuralism comes in two distinct flavours. According to eliminative structuralism, a structure is given by a collection of ordinary objects (like tables, chairs, stars, etc). According to non-eliminative structuralism, structures are sui generis mathematical objects. Both versions face a number of problems, however. The aim of this talk is to articulate a new take on structuralism according to which mathematical objects like sets are not places in structures but rather structures themselves. I will argue that this solves many of the problems facing traditional forms of structuralism, and show how, with the right assumptions, it can be used to make sense of the modern set theory.

Freitag, 02. Juli 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Harm Derksen (Northeastern University)

(Gast von Markus Schweighofer)

Maximum Likelihood Estimates for Matrix and Tensor Normal Models

For matrix normal models and tensor normal models we will discuss how many samples are needed such that: (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. Our techniques are based on invariant theory, the representation theory of quivers and the castling transform for tensors. This is joint work with Visu Makam and Michael Walter.

Montag, 5. Juli 2021, um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Anna Seigal (University of Oxford)

(Gästin von Mateusz Michalek & Gabriela Michalek, KWIM Vorlesungsreihe)

Invariant Theory for Maximum Likelihood Estimation

Maximum likelihood estimation is an optimization problem over a statistical model, to obtain the parameters that best fit observed data. I will focus on two settings: log-linear models and Gaussian group models. I will describe connections between maximum likelihood estimation and notions of stability from invariant theory. This talk is based on joint work with Carlos Améndola, Kathlén Kohn and Philipp Reichenbach.

Montag, 12.Juli 2021 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Laura Wirth (Universität Konstanz)

(Gästin von Salma Kuhlmann & Gabriela Michalek, KWIM Vorlesungsreihe)

Weighted Automata and Formal Power Series

Weighted automata generalize classical automata, which are a concept from theoretical computer science. On the other hand, formal power series are mathematical objects, which particularly occur in algebra. In this talk, we will point out a fascinating connection between the two. More precisely, we derive Schützenberger’s generalization of Kleene’s classical result on the coincidence of rational and recognizable languages in the realm of formal power series over semirings. First, we introduce basic concepts from formal language theory and algebra. Next, we generalize classical automata by equipping their transitions with weights to obtain so-called weighted automata. Consequently, weighted automata recognize formal power series, whereas classical automata recognize formal languages. However, we explain how formal power series generalize languages. Then, we define operations on the set of formal power series that correspond to the language-theoretic operations union, concatenation, and Kleene-iteration. With these operations, we introduce rational formal power series, which generalize the rational languages. After that, we prove the Kleene–Schützenberger Theorem. If time permits, we give a short introduction to the work of Droste and Gastin, who similarly generalized the Büchi–Elgot–Trakhtenbrot Theorem on the coincidence of recognizable and MSO-definable languages in the realm of formal power series. The graphical representation of these connections can be found in the attachement.

Freitag, 16. Juli 2021 um 13:30 Uhr, Oberseminar Relle Geometrie und Algebra

Tim Kuppel (Universität Konstanz)

(Gast von Mateusz Michalek)

Asymptotics of Plethysm

Plethysm is the operation of composing Schur functors, like symmetric and exterior powers, which give irreducible representations of the general linear group. Decomposing these compositions into irreducible components is a nontrivial task, and in particular combinatorial formulas for multiplicities are only known in a few simple cases. On the other hand, decomposing tensor products of symmetric powers is well understood due to Pieri‘s famous (combinatorial) formula. We will relate the asymptotic behaviour of these decompositions when the outer parameter is fixed, after discussing preliminaries from the representation theory of symmetric and general linear groups.

Montag, 19. Juli 2021, um 15:15 Uhr, Logic Colloquium.

Joel Hamkins (University of Oxford)
(Gast von Salma Kuhlmann)

Naturality in mathematics and the hierarchy of consistency strength

An enduring mystery in the foundations of mathematics is the observed phenomenon that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. For any two of the familiar large cardinal hypotheses, one of them generally proves the consistency of the other. Why should this be? Why should it be linear? Some philosophers see the phenomenon as signi cant for the philosophy of mathematics? It points us toward an ultimate mathematical truth. Meanwhile, the linearity phenomenon is not
strictly true as mathematical fact, for we can prove that the hierarchy of consistency strength is actually ill-founded, densely ordered, and nonlinear. The counterexample statements and theories, however, are often dismissed as unnatural. Linearity is thus a phenomenon only for the so-called "naturally occurring" theories. But what counts
as natural? Is there a mathematically meaningful account of naturality? In this talk, I shall criticize this notion of naturality, and attempt to undermine the linearity phenomenon by presenting a variety of natural hypotheses that reveal ill-foundedness, density, and incomparability in the hierarchy of consistency strength.