Monday, April 29, 2019 at 15:15, Research Seminar Mathematical Logic, Set theory and Model theory
Arthur Forey (ETH Zürich)
(Guest of Salma Kuhlmann)
Uniform bound for points of bounded degree in $\mathbb{F}_q[t]$
I will present a bound for the number of $\mathbb{F}_q[t]$ - points of bounded degree in a variety defined over $\mathbb{Z}[t]$, uniform in $q$. This generalizes work by Sedunova for fixed $q$. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.
Friday, May 10, 2019 at 13:30, Research Seminar Real Geometry and Algebra
Philipp J. di Dio (TU Berlin)
(Guest of Markus Schweighofer)
Carathéodory Numbers, Flat Extension and Evaluation Polynomials
Since the fundamental result of Richter in 1957 [Ric57] that every truncated moment functional is represented by a finite sum of Dirac measures this minimal number, the Carathéodory number, and the reconstruction of such atomic measures attracted much attention [dDS18b] [RS18] [dDS18a] [dDK19]. We present very recent results on lower bounds on the Carathéodory number [dDK19] and their implication on the flat extension [CF96] [CF98]. Flattening moment data provides possibilities to reconstruct atomic measures from moments and we show how evaluation polynomials are used. Depending on the shape of the moment cone we discuss the existence of such evaluation polynomials [dDS18a] and their approximation by sums of squares.
This is joint work with Mario Kummer and Konrad Schmüdgen.
References
[CF96] R. Curto and L. A. Fialkow, Solution of the truncated moment problem for flat data, Mem. Amer. Math. Soc. 119 (1996), no.568.
[CF98] $\underline{\quad \quad}$ , Flat extensions of positive moment matrices: recursively generated relations, Mem. Amer. Math. Soc. 136 (1998), no.648.
[dDK19] P. J. di Dio and M. Kummer, The multidimensional truncated Moment Problem: Carathéodory Numbers from Hilbert Functions and Shape Reconstruction from Derivatives of Moments, arxiv.org/abs/1903.00598.
[dDS18a] P. J. di Dio and K. Schmüdgen, The multidimensional truncated Moment Problem: The Moment Cone, arxiv.org/abs/1809.00584.
[dDS18b] $\underline{\quad \quad}$, The multidimensional truncated Moment Problem: Carathéodory Numbers, J. Math. Anal. Appl. 461 (2018), 1606--1638.
[Ric57] H. Richter, Parameterfreie Abschätzung und Realisierung von Erwartungswerten, Bl. Deutsch. Ges. Versicherungsmath. 3 (1957), 147--161.
[RS18] C. Riener and M. Schweighofer, Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions, J.Compl. 45 (2018), 22--54.
Monday, June 3, 2019 at 15:15, Research Seminar Mathematical Logic, Set theory and Model theory
Patrick Speissegger (McMaster University, Hamilton, Canada)
(Guest of Pantelis E. Eleftheriou)
Expansions of the real field by canonical products
We consider expansions of o-minimal structures on the real field by certain canonical Weierstrass products and/or associated functions, such as their logarithmic derivatives. We show that there are only three possible outcomes for the resulting structures: they are either o-minimal, or d-minimal but not o-minimal, or they define the set of integers.
This is joint work with Chris Miller.
Monday, June 17, 2019 at 15:15, Research Seminar Mathematical Logic, Set theory and Model theory
Makoto Fujiwara (Meiji University, Tokio and Zukunftskolleg, Konstanz)
Constructivism and weak logical principles in arithmetic
In early 20th century, L. E. J. Brouwer tried to reconstruct mathematics in favor of his intuitionism, a new philosophy (and first school) of constructive mathematics in which, intuitively speaking, everything has to be built from the ground up, including the meaning of the logical symbols. Brouwer's student A. Heyting tried to formalize Brouwer's 'proofs as constructions'-concept via his theory Heyting arithmetic (HA) and proposed an informal semantics, called the Brouwer-Heyting-Kolmogorov (BHK) interpretation. Note that HA is a variant of Peano arithmetic (PA) based on intuitionstic logic and one can obtain PA just by adding the law of excluded middle (A or not A) into the axioms of HA. Though all the sentences provable in HA are valid in the sense of the BHK interpretation, it seems not that all the sentences valid in the BHK interpretation are provable in HA. Thus some weak fragment of the law of excluded middle can be contained in arithmetic which exactly captures constructivism.
Motivated from this fact, I have studied the correlation between weak logical principles over Heyting arithmetic and its extension in all finite types. The hierarchy of the weak logical principles is closely related to the notion of constructivism. To show that one logical principle A does not imply another principle B, one typically uses appropriate forms of realizability (which is a kind of formal treatment of the BHK interpretation) to show that A has a certain semi-constructive interpretation which B does not.
In this talk, I would give an overview of the hierarchy of the weak logical principles from the perspective of constructivism. If time permits, I would also present some meta-mathematical results with respect to (classical and constructive) reverse mathematics where these logical principles play an important role.
Friday, June 28, 2019 at 13:30, Research Seminar Real Geometry and Algebra
Jesus A. De Loera (University of California, Davis)
(Guest of Claus Scheiderer)
Carathéodory theorem and its influence in Mathematics
Convex geometry has been an important tool in several areas of mathematics, e.g., convexity appears in the study of polynomials. My talk will show this continues to be the case today by focusing one influential theorem, Carathéodory's theorem from 1905. In its most basic form it describes the size of a minimal linear combination representing a vector in a cone as a sum of others, and it is among the most fundamental results in Convex Geometry and it has seen many variations and extensions. I will review some variations of Carathéodory's theorem that have interesting applications. In particular, I will talk about integer versions, given a system $Ax=b, x \geq 0$, what is the size of the sparsest solution integer? I will mention some open problems too. This talk is geared for non-experts, but all new results are joint work with Iskander Aliev, Gennadiy Averkov, Timm Oertel, and Chris O'Neill.
Monday, July 1, 2019 at 15:15, Research Seminar Mathematical Logic, Set theory and Model theory
Merlin Carl (Europa-Universität Flensburg)
Diproche - Automated Proof Checking for Didactical Applications
We present the Diproche (Didactical Proof Checking) system (a software inspired by the Naproche (Natural language Proof Checking) system (developed by Koepke, Cramer, Schröder and others) which uses techniques from computational linguistics and automated theorem proving to provide an automated proof checker for proofs written in a fragment of natural mathematical language specifically designed to express solutions to typical proving exercises for beginner students. The aim of Diproche is to serve as an automatic tutor for students learning how to prove. In addition to reporting gaps in proofs, Diproche provides a detailed feedback on linguistic mistakes, failures to meet the goal of the exercise or self-formulated subgoals, type errors and diagnosis of possible formal fallacies. Moreover, users can obtain hints when getting stuck while searching for a solution.
Friday, July 23, 2019 at 13:30, Research Seminar Real Geometry and Algebra
Mateusz Michalek (MPI MiS Leipzig, Aalto University Finland)
(Guest of Markus Schweighofer)
Algebraic Phylogenetics
Phylogenetics is a science that aims at reconstructing the history of evolution. Phylogenetic tree models are generalizations of well-known Markov chains. In my talk I will present so-called group-based models and their relations to algebra and combinatorics. To a model of evolution one associates an algebraic variety that is the Zariski closure of points corresponding to probability distributions allowed by the model. Many important varieties arise by this construction, e.g. secant varieties of Segre products of projective spaces. It turns out that group-based models provide toric varieties. In particular, they may be studied using tools from toric geometry relating to combinatorics of lattice polytopes.
Thursday, August 1, 2019 at 13:30, Research Seminar Real Geometry and Algebra
Santiago Laplagne (Universidad de Buenos Aires)
(Guest of Claus Scheiderer)
Algorithms for normalization of polynomial rings
In this talk we will study algorithms for constructing the normalization of polynomial rings. We will show how to reduce the task to smaller local problems and then recombine the results. The algorithms use Groebner bases and Puiseux series as tools, and we will make a short introduction to them.