WS 2022/2023

Montag, 14.11.2022 um 15:15 Uhr, Logic Colloquium

Neil Barton(Universitetey of Oslo)

(Gast von Salma Kuhlmann and Carolin Antos-Kuby)

Engineering Set-Theoretic Concepts

On one view of set theory, the paradoxes precipitated a radical clarifica-
tion of our concept of set, culminating with the isolation of the iterative conception of set. In this talk, I’ll present some work from a book I’ve been working on (also entitled ’Engineering Set-Theoretic Concepts’). I’ll argue that in fact the iterative conception admits of further splitting into multiple conceptions of set, and that we are ourselves at a conceptual crossroads motivated by a kind of paradox. In one direction, we are pushed to the standard picture of ZFC. In the other, we are pushed towards a conception of set on which every set is countable. I’ll also (for the logic folks) present some of the mathematics behind this latter less familiar picture, and situate ZFC-based set theory within it.

Freitag, 18.11.2022 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Daniel Panazzolo (Université de Haute Alsace)

(Gast von Salma Kuhlmann)

Group of real powers and translations

Motivated by Hilbert's 16th Problem, we consider the group of (germs of  maps) generated by real translations x->x+a  and real powers x -> x^r, that is, maps of the form

         x -> ( (x^r1 + a1)^r2 + a2) + ..)^rn + an

Our main results are (1) an estimate for the number of real fixed points (in terms of n) and (2) a generalization of a result of Cohen proving that G has the structure of a free product.

Montag, 21.11.2022 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Vincent Bagayoko (University of Konstanz)

(Gast von Mateusz Michalek and Salma Kuhlmann)

Hyperseries and surreal numbers

Hyperseries are extensions of power series (and of transseries) with a formal exponential, logarithm, and so-called transfinite iterators thereof. Those transfinite iterators are characterized by conjugacy equations (they can for instance conjugate the real exponential to a translation), and can be used to solve functional equations in a formal setting.
On the other hand, surreal numbers are abstract magnitudes, defined in a game theoretic and set theoretic vein by John H. Conway, which extend the real numbers with transfinite and infinitesimal elements. It turns out that there is a strong connection between hyperseries and surreal numbers, which I will illustrate.

Montag, 28.11.2022 um 15:15 Uhr, Oberseminar Complexity Theory, Model Theory, Set Theory.

Carlo Collari (University of Pisa)

(Gast von Mateusz Michalek and Salma Kuhlmann)

Multipath homology and the path poset

The aim of this talk is to present some homology theories for graphs (and, in particular, multipath homology), describe their mutual relationship, and see how these theories are connected with other homology theories (i.e. homology theories for knots and algebras).

We start by describing some classical theories for graphs. Then, we will focus on the construction, due to Helme-Guizon and Rong, of chromatic homology. This theory categorifies the chromatic polynomial of graphs, and its construction is based on Khovanov homology. In particular, we will present a result due to J. Przytycki, which relates Khovanov homology, chromatic homology, and Hochschild homology.

Afterwards, we will describe Turner and Wagner's approach to the "extension" of chromatic homology to oriented graphs while preserving an analogue of Przytycki's result.

Finally, we will define multipath homology and compare it with Turner-Wagner homology.

In the last part of the talk, time permitting, we will give a topological description of the path poset, which is a key ingredient in the definition of Turner-Wagner and multipath homologies, and some techniques to compute it.

This talk is based on joint work with L. Caputi (University of Aberdeen), S. Di Trani (Sapienza Università di Roma), and J. Smith (Nottingham Trent  University)

Montag, 19.12.2022 um 15:15 Uhr, Logic Colloquium

Laura Wirth (Universität Konstanz)

(Gast von Salma Kuhlmann and Carolin Antos-Kuby)

The Interplay of Languages, Automata and Monadic Second-Order Logic

A basic tool from Theoretical Computer Science for the specification of formal languages are finite automata. Research on the logical aspects of the theory of finite automata began in the early 1960s with the work of Buchi, Elgot and Trakhtenbrot on monadic second-order logic in the context of words. The basic idea of their approach is to use formulas of monadic second-order logic over a suitable signature to describe properties of words. Thus, monadic second-order logic provides another tool for the specification of languages. Buchi, Elgot and Trakhtenbrot independently derived that the two concepts – finite automata and monadic second-order logic – are even expressively equivalent. Hence, their equivalence result, referred to as Buchi-Elgot-Trakhtenbrot Theorem, establishes an early connection between Automata Theory and Mathematical Logic. In this talk, we provide an introduction to the above-mentioned concepts. Moreover, we present an extension of the Buchi-Elgot Trakhtenbrot Theorem to formulas, involving free variables, whereas the original statement addresses only sentences. If time permits, we will further outline quantitative extensions of the above-mentioned concepts and results.

Montag, 16.01.2023 um 15:15 Uhr, Logic Colloquium

Konstantin Genin(Universität Tübingen)

(Gast von Salma Kuhlmann and Carolin Antos-Kuby)

On Falsifiable Statistical Hypotheses

Popper argued that a statistical falsification required a prior methodological decision to regard sufficiently improbable events as ruled out. That suggestion has generated a number of fruitful approaches, but also a number of apparent paradoxes and ultimately, no clear consensus. It is still commonly claimed that, since random samples are logically consistent with all the statistical hypotheses on the table, falsification simply does not apply in realistic statistical settings. I claim that the situation is considerably improved if we ask a conceptually prior question: when should a statistical hypothesis be regarded as falsifiable. To that end I propose several different notions of statistical falsifiability and prove that, whichever definition we prefer, the same hypotheses turn out to be falsifiable. That shows that statistical falsifiability enjoys a kind of conceptual robustness. These notions of statistical falsifiability are arrived at by proposing statistical analogues to intuitive properties enjoyed by exemplary falsifiable hypotheses familiar from classical philosophy of science. That demonstrates that, to a large extent, this philosophical tradition was on the right conceptual track. Finally, I demonstrate that, under weak assumptions, the statistically falsifiable hypotheses correspond precisely to the closed sets in a standard topology on probability measures. That means that standard techniques from statistics and measure theory can be used to determine exactly which hypotheses are statistically falsifiable. In other words: the proposed notion of statistical falsifiability both answers to our conceptual demands and submits to standard mathematical techniques.

Freitag, 20.01.2023 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Jannik Wesner (Technische Universität Dortmund)

(Gast von Claus Scheiderer)

Hexagons, Heptagons and their Adjoint Curves

For a (not necessarily convex) polygon with n vertices in the projective plane, we consider the intersections points of the lines containing the edges, which are not the vertices. If no three of these lines have a common point, then there is a unique curve of degree n-3 going through these residual points, called the adjoint curve. If the polygon is real and convex, then the adjoint curve is a hyperbolic curve. For a given non-singular hyperbolic cubic we give a method constructing all associated hexagons. Furthermore we characterize when this construction gives convex hexagons. If time permits, we prove that a very general quartic is adjoint to exactly 864 heptagons, using intersection theory.
 

Freitag, 03.02.2023 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Hanieh Keneshlou (Universität Konstanz)

(Gast von Mateusz Michalek)

On the construction of regular maps to Grassmannians

A continuous map f:C^n--->C^N is called k-regular, if the image of any k distinct points in C^N are linearly independent. The study of existence of regular map was initiated by Borusk 1957, and later attracted attention due to its connection with the existence of interpolation spaces in approximation theory, and certain inverse vector bundles in algebraic topology. In this talk, based on a joint work with Joachim Jelisiejew, we consider the general problem of the existence of regular maps to Grassmannian. We will discuss the tools and methods of algebra and algebraic geometry to provide an upper bound on N, for which a regular map exists.

Freitag, 10.02.2023 um 13:30 – 15:00 Uhr, Oberseminar Relle Geometrie und Algebra

Ngoc Hoang Anh Mai (Universität Konstanz)

(Gast von Markus Schweighofer)

Several issues on applications of real algebraic geometry to optimization and machine learning

In the first part of the talk, I will briefly present my previous works with co-authors on applications of real algebraic geometry to optimization and machine learning. In the second part,  I will focus on the relationship between singularity theory, sums of squares, and optimization.