WS 2021/22

Montag, 8. November 2021 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Irem Portakal (MPI MIS Leipzig)

(Gast von Mateusz Michalek)

Polynomial systems arising from equilibria in Game Theory

In this talk, we discuss the polynomial systems satised by the Nash, correlated and dependency equilibria of an n-person game in normal form. This is an ongoing joint work with Bernd Sturmfels.

Montag, 15. November 2021 um 15:15 Uhr, Logic Colloquium

Philip Welch (University of Bristol)

(Gast von Salma Kuhlmann)

Quasi-induction

Induction, whether appearing in the role of a proof by induction ("if 1 has property P, and (n has property P implies that n+1 has property P, therefore all n have property P") or as  inductive definitions (eg that of the set of well formed formulae in a formal language) is a principle tool in mathematics. Whilst many inductive definitions involve only a passage through finite stages, and are often presented as definitions by recursion, many also involve transfinitely many stages before completion.  (We give an example involving "infinite chess".) The theory of such inductive definitions over general structures was magisterially laid down by Moschovakis in the 1970's (in "Elementary Induction on Abstract Structures").

We consider here a broader class of "quasi-inductive" processes that alter the rules of induction at limit stages of their production to a "liminf" rule rather than simple "union". We give some examples that have arisen in the philosophical theory of truth, from computer science, and from set theory. Such q.i.-processes extend inductive ones and also result in a rich theory to which pleasingly many of the results from the Moschovakian theory can be extended.

Freitag, 19. November 2021 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Noémie Combe (MPI MIS Leipzig)

(Gast von Mateusz Michalek and Gabriela Michalek)

Exponential varieties, statistical manifolds and Frobenius structures

We consider a class of manifolds corresponding to statistical models, related to exponential families. Exponential manifolds have been considered from the point of view of information geometry and independently real algebraic geometry. However, investigations on this object have evolved independently on both sides, creating a certain gap. The aim is to try to unify both approaches, by first establishing a dictionary between them. Secondly we connect them by introducing our web algebraization theorem, for the case of Frobenius statistical manifolds. This theorem relies on the theory of three-webs, introduced and developed by Blaschke and Cartan. We prove that the 3-webs of Frobenius exponential manifolds are algebraizable, i.e. that the r-dimensional foliations of the web belong to a hypercubic, answering an open question of Amari.

Freitag, 26. November 2021 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Steven Sam (University of California, San Diego)

(Gast von Mateusz Michalek)

Bi-graded Koszul modules, K3 carpets, and Green's conjecture

I will explain a recent approach to alternate proofs of Voisin's solution to Green's conjecture on syzygies of canonical embeddings of generic curves using the theory of Koszul modules and their generalizations. This is joint work with Claudiu Raicu.

Freitag, 10. Dezember 2021 um 13:30 – 15:00 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Mareike Dressler (MPI MIS Leipzig)

(Gast von Salma Kuhlmann)

Algebraic Perspectives on Signomial Optimization

Signomials are obtained by generalizing polynomials to allow for arbitrary real exponents. This generalization offers great expressive power, but has historically sacrificed the organizing principle of “degree” that is central to polynomial optimization theory. In this talk, I introduce the concept of signomial rings that allows to reclaim that principle and explain how this leads to complete convex relaxation hierarchies of upper and lower bounds for signomial optimization via sums of arithmetic-geometric exponentials (SAGE) nonnegativity certificates. In the first part of the talk, I discuss the Positivstellensatz underlying the lower bounds. It relies on the concept of conditional SAGE and removes regularity conditions required by earlier works, such as convexity of the feasible set or Archimedeanity of its representing signomial inequalities. Numerical examples are provided to illustrate the performance of the hierarchy on problems in chemical engineering and reaction networks.
In the second part, I provide a language for and basic results in signomial moment theory that are analogous to those in the rich moment-SOS literature for polynomial optimization. That theory is used to turn (hierarchical) inner-approximations of signomial nonnegativity cones into (hierarchical) outer-approximations of the same, which eventually yields the upper bounds for signomial optimization.
This talk is based on joint work with Riley Murray.

Montag, 13. Dezember 2021 um 15:15 – 16:45 Uhr, Logic Colloquium

Hadil Karawani (Leibniz-ZAS Berlin)

(Gast von Salma Kuhlmann und Carolin Antos-Kuby)

strength, duality and social meaning - a perspective from experimental semant

It is a "usual assumption" of literature on modality that (epistemic) must is the semantic dual of might (c.f. von Fintel & Gillies 2010, Willer 2013, Lassiter 2016). However, this view is not universal. Building on previous work (Karawani 2014, Crespo, Karawani and Veltman 2018, Karawani and Waldon 2017), I propose a dynamic analysis of epistemic modality according to which might p is weaker than may p; and that in theory, at least, may p (and not might p) is the semantic dual of must p; while might p is the dual of the plain (non modal) proposition p. My discussion will appeal to database analysis of English and an experimental paradigm, and discuss the role of social meaning as an expressivist strategy. This will help determine where and why the literature has been lead astray. I then go on to discuss the semantics of must. Within this theory, epistemic must expresses universal quantification over epistemically likely worlds (a subset of epistemic possibility); evidential must however expresses something weaker, namely quantification over a larger set. This discrepancy between epistemic and evidential must has puzzled researchers trying to figure out a unified semantic account of must. I argue that must in its evidential reading behaves like a weak necessity modal due to quantifying over the evidence set (a superset of epistemic possibility). By doing so, I provide a witness that falsifies the typological assumption according to which "the world's languages do not allow for weak-necessity epistemic modals (but only allow for weak-necessity deontic modals like ought and should)" (Fintel and Iatridou 2008, Iatridou and Zeijlstra 2013, Mirrazi & Zeijlstra 2021) . I conclude with a discussion of how the experimental paradigm may be extended and generalized to evaluate other claims that have been made in the literature regarding semantic duality (e.g. w.r.t. would, cf. Ward 2008).

Montag, 10. Januar 2022 um 15:15 – 16:45 Uhr, OS Complexity Theory, Model Theory, Set Theory

Lasse Vogel (HHU Düsseldorf)

(Gast von Salma Kuhlmann)

On the classification of definable sets over pseudofinite structures

Since finite structures cannot have elementary extensions they are impractical
for standard model theory. To bypass this issue one can instead study infinite structures which behave suciently similar to finite ones, such structures are called pseudo-finite. A simple way to construct examples is as the ultraproduct of finite structures. We will limit our study to such ultraproducts for the sake of simplicity. One thing someone might now be interested in is the shape that denable sets may have in those structures (up to denable bijection). In this talk we will recall the notion of nonstandard cardinality, which is an invariant for denable bijections, and investigate to which degree we can classify denable sets with its help.

Montag, 17. Januar 2022 um 15:15 – 16:45 Uhr, Logic Colloquium

Luca San Mauro and Giorgio Venturi (Sapienza University of Rome, University of Campinas)

(Gast von Salma Kuhlmann and Carolin Antos-Kuby)

Illocutionary acts in mathematics

In this talk, we present our broad research project that aims to revisit the language of mathematics through the lens of contemporary linguistics and philosophy of language. We argue, against a certain received view, that pragmatic phenomena occur in mathematics in a peculiar fashion. In particular,
1. we adopt speech act theory to examine the illocutionary forces that shape
theorems and proofs;
2. we offer a novel understanding of mathematical proofs as complex speech acts
which combine a commissive and a directive nature;
3. we finally characterize mathematical assertions as those assertions in which
the success of the illocution binds the success of the perlocution.

Freitag, 21. Januar 2022 um 13:30 – 15:00 Uhr, OS Reelle Geometrie und Algebra

Jan Draisma (University of Bern)

(Gast von Mateusz Michalek)

No short polynomials vanish on bounded-rank matrices

Given a variety X in K^n, one may ask what is the smallest number t of terms of a nonzero polynomial in K[x_1,...,x_n] that vanishes on X. In general, this is quite a subtle question, but there are three situations where we have established the answer precisely: sufficiently general linear spaces in K^n, the variety of rank-r matrices in K^{m x n}, and the variety of skew-symmetric rank-r matrices in K^{m x m}. In these cases, we also have a characterisation of the shortest nonzero polynomials. I'll discuss these results and their proofs.

Freitag, 28. Januar 2022 um 13:30 – 15:00 Uhr, OS Reelle Geometrie und Algebra

Paul Breiding (MPI MIS Leipzig)

(Gast von Markus Schweighofer)

Facet Volumes of Polytopes

We consider what we call facet volume vectors of polytopes. Every full-dimensional polytope in R^d with n facets defines n positive real numbers: the n (d-1)-dimensional volumes of its facets. For instance, every triangle defines three lenghts; every tetrahedron defines four areas. We study the space of all such vectors. We show that for fixed integers d\geq 2 and n\geq d+1 the configuration space of all facet volume vectors of all d-polytopes in R^d with n facets is a full dimensional cone in R^n, and we describe this cone in terms of inequalities. For tetrahedra this is a cone over a regular octahedron. (Joint work with Pavle Blagojevic and Alexander Heaton)

Freitag, 4. Februar 2022 um 13:30 – 15:00 Uhr, OS Reelle Geometrie und Algebra

Antonio Lerario (SISSA Trieste)

(Gast von Markus Schweighofer)

The zonoid algebra

In this seminar I will discuss the so called "zonoid algebra", a construction introduced in a recent work (joint with Breiding, Bürgisser and Mathis) which allows to put a ring structure on the set of zonoids (i.e. Hausdorff limits of Minkowski sums of segments). This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. Moreover this algebra plays the role of a probabilistic intersection ring for compact homogeneous spaces. (joint work with P. Breiding, P. Bürgisser and L. Mathis)

Montag, 07. Februar 2022 um 15:15 – 16:45 Uhr, OS Complexity Theory, Model Theory, Set Theory

Sebastian Krapp (University of Konstanz)

(Gast von Salma Kuhlmann)

Topological properties of Ordered Abelian Groups and Definable Henselian Valuations

In my talk, I will firstly present ordered abelian groups as topological subspaces of their divisible hull and discuss topological properties such as density and closure. Secondly, I will outline the basic concept of henselian valuations and motivate the study of definable henselian valuation rings in fields and ordered fields. Moreover, I will present topological conditions on the value group that ensure the definability of the corresponding valuation ring independent of the specific field.
All valuation theoretic notions will briefly be introduced during the talk.

Freitag, 11. Februar 2022 um 13:30 – 15:00 Uhr, OS Reelle Geometrie und Algebra

Tobias Sutter (University of Konstanz)

(Gast von Salma Kuhlmann)

From infinite to finite programs: explicit error bounds and applications

While the Generalized Moment Problem (GMP) is known to have great modelling power and applications in various fields, computing its numerical solutions is difficult. In this talk, we present a numerical approximation scheme to certain GMPs. More precisely, we consider Linear Programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first-order methods, leading to a priori as well as a posteriori performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost optimal control problems in the context of Markov Decision Processes on Borel spaces.