The seminar take place every Friday at 12:00 in room M2 (third floor of School of Sciences) and online via the following link.
Zoom Link: https://authgr.zoom.us/j/92857333993?pwd=RVZNZ1ROemIzSlp1U1RSU3VDaFdHZz09
Meeting ID: 928 5733 3993
Passcode: 002865
March 15th,2024
Speaker: Theofanis Chatzidiamantis, Universität Bonn
Title: Higher Categories and Homotopy (Type) Theory
Abstract: The main aim of this talk is to give some topological motivation for the notions of ∞-Groupoids and ∞-Categories, and outline the definitions of two different, but in some sense equivalent, models: Quasicategories (Boardman-Vogt, 1973) and Complete Segal Spaces (Rezk, 2001). We will then discuss the basics of homotopy type theory, an alternative system of foundations of mathematics where types, its "objects", can be interpreted as ∞-Groupoids. Finally, we will briefly describe how the Complete Segal Space model of ∞-Categories can be formulated using an extended version of homotopy type theory, Riehl and Shulman's simplicial type theory.
March 15th,2024
Speaker: Theofanis Chatzidiamantis, Universität Bonn
Title: Higher Categories and Homotopy (Type) Theory
Abstract: The main aim of this talk is to give some topological motivation for the notions of ∞-Groupoids and ∞-Categories, and outline the definitions of two different, but in some sense equivalent, models: Quasicategories (Boardman-Vogt, 1973) and Complete Segal Spaces (Rezk, 2001). We will then discuss the basics of homotopy type theory, an alternative system of foundations of mathematics where types, its "objects", can be interpreted as ∞-Groupoids. Finally, we will briefly describe how the Complete Segal Space model of ∞-Categories can be formulated using an extended version of homotopy type theory, Riehl and Shulman's simplicial type theory.
March 8th,2024
Speaker: Panteleon Ioannidis Iason Ioannis
Title: Prover-Skeptic dialogues
Abstract: Prover-skeptic dialogues are an alternative perspective on the idea of formal proof. By nature, they are connected to the Curry-Howard correspondence, which provides a bridge between Computation Theory and Mathematical Logic. Historically, this bridge followed the Brouwer-Heyting-Kolmogorov interpretation of proof as construction, namely the so-called Intuitionistic Logic, which rejects the principle of the excluded middle.
An introduction to this logic will be made through the BHK interpretation. Then we will explore the Curry-Howard syntactic correspondence through the untyped λ-calculus computation model and arrive at an application of it, which is the prover-skeptic dialogues as an alternative method of formal proof.
February 23th, 2024
Speaker: Athanasios Beslikas, Jagiellonian University, Krakow.
Title: Composition Operators on the Dirichlet Spaces of the Unit Disk.
Abstract: In this talk, we will present a new sufficient condition for the boundedness of the composition operator C_φ f = f ◦ φ on the Dirichlet spaces with weight. Our approach differs from the classical works of Shapiro, Pau, Perez, where the main tool is the generalized Nevanlinna counting function. We will apply results on the boundedness of composition operators on weighted Bergman spaces A^p_a (D^2 ) of the bidisk to prove the main result, as well as establish a connection with the reproducing kernel of De Branges-Rovnyak.
January 12th, 2024
Speaker: Theodosis Papazoglou, Imperial College of London
Title: Decoding COVID-19: Insights from Frequentist/Bayesian Generalised Linear Models
Abstract: Linear models are a powerful tool in analyzing and predicting the behavior of complex systems and are heavily used in financial and biological data. They describe a continuous response as a function of one or more predictor variables. However, what happens when the response is not continuous, e.g. binary, count etc.? How do we generalize the idea of regression to deal with the relationship between categorical/count responses and the set of regression variables?
In this presentation, we will introduce the concept of Generalized Linear Models (GLMs), specifically focusing on the cases of Logistic and Poisson Regression to handle binary and count data, respectively. Furthermore, we will not restrict ourselves to the well-known frequentist case of GLMs but will also expand on the Bayesian framework of GLMs.
In Bayesian GLMs we approach the GLMs within the context of Bayesian inference. Instead of treating the regression coefficients as unknown fixed constants, we treat them as random variables assuming a prior distribution and make inference based on their posterior distribution using Bayes’ Theorem, rather than receiving single values.
We will provide a detailed analysis of real COVID data to provide insight on the relationship between cases of death/hospitalization and the age covariate. Our focus will be the analysis of the data from both Frequentist and Bayesian perspectives using the R programming software and the probabilistic language Stan within R.
December 7th,2023
Speaker: Anastasios Slaftsos, Università degli Studi di Padova
Title: A visual introduction to tilting theory
Abstract: Representation theory is an area of Mathematics that studies the properties of actions of abstract objects (such as rings or groups) on other objects (such as abelian groups or vector spaces). One of the recurrent problems in this area is to classify all these actions that are called representations. In this talk, we focus mainly on the representation theory of a quiver, i.e. an oriented graph, and we give an heuristic view (example driven) of some of the ideas and tools of representation theory and in particular, tilting theory.
November 10th, 2023
Speaker: Georgios Nikolaidis, Aristotle University of Thessaloniki
Title: Logarithmic Conjugation Theorem
Abstract: The present speech will be developed within the framework of harmonic functions defined in the complex plane. Specifically, we will examine the relationship between harmonic functions and analytic functions defined in simply connected domains. Then, we will wonder how this result can be generalized to multiply connected domains, which is the essence of the Theorem of Logarithmic Convexity. After proving the theorem in a simple special form, we will mention how the general case is handled.
The tools to be used have been introduced in undergraduate courses on analysis, which is why the speech is recommended for graduate students as well as advanced undergraduate students.
October 20th, 2023
Speaker: Kyriaki Iliadi-Charmana, Aristotle University of Thessaloniki
Title: Multivariate Normal Distribution
Abstract: The normal distribution is considered the most important in statistics and probability theory. It allows for a satisfactory approximation of experimental data and other distributions and serves as a foundation for statistical inference. The multivariate normal distribution is a generalization of the univariate one.
To study the multivariate normal distribution, it is important to establish the framework for the analysis of multivariate distributions, which means presenting random vectors and their properties. Based on this, the multivariate normal distribution and some of its extensions are defined and studied. Finally, estimation within the normal distribution is of interest, using unbiased estimators of minimum variance and maximum likelihood estimators.
Engaging with the multivariate normal distribution enables us to focus not only on the study of a characteristic expressed by a random variable but also on multiple factors constituting a random vector and observing their joint behavior.
August 31st, 2023
Speaker: Clément Leroy,Institut Polytechnique de Paris
Title: Exploring Elliptic Curves for Cryptography
Abstract: The talk "Exploring Elliptic Curves for Cryptography" delves into the multifaceted field of elliptic curves, explaining their fundamental principles and cryptographic applications. Elliptic curve cryptography (ECC) has emerged as a cornerstone of modern cryptographic systems due to its remarkable ability to provide strong security with relatively small key sizes, making it exceptionally efficient for resource-constrained environments. The foundation of elliptic curve theory is established, unraveling the algebraic properties of elliptic curves and their associated mathematical operations. This paves the way for comprehending the notions of points, addition laws, and finite fields, creating a solid basis for exploring their cryptographic potential. The manifold applications of elliptic curves in cryptography are exposed, showcasing how ECC is a vital component of public key cryptography, enabling secure key exchange, digital signatures, and encryption. The security of these applications hinges on the intricate nature of the elliptic curve discrete logarithm problem, endowing ECC with resistance against classical attacks. To augment the security of elliptic curve-based cryptosystems, techniques like pairings are investigated. Pairings extend the utility of elliptic curves to new cryptographic constructs, facilitating protocols like identity-based encryption. The intricacies of pairing-based cryptography are explored, highlighting its potential to enhance the privacy and efficiency of cryptographic operations. In conclusion, "Exploring Elliptic Curves for Cryptography" equips attendees with a comprehensive understanding of the theoretical foundations and practical applications within the field of elliptic curve cryptography. By delving into the nuances of the bases of elliptic curve theory and exploring more complex concepts like pairings, participants gain insights to harness the robust security offered by elliptic curves.
May 4th, 2023
Speaker: Skylar Werner, University of Göttingen
Title: An introduction to Categorical Logic
Abstract: Category theory is the study of the algebraic properties of morphisms in an extreme generalized way. Because of this generalization, many areas of mathematics that seemed to be distinct as mathematical subjects are now connected. This talk will derive many Categorical thoughts and show how Propositional logic is embedded into the Categorical framework while hinting at how this new philosophy relates logic to geometry. At the end, we will define what a presheaf is and its connection to logic.
April 6th, 2023
Speaker: Karegar Baneh Kohal Quentin, Universidad Nacional Autónoma de México (UNAM)
Title: Introduction to index theory via Lie groupoids
Abstract: The problem of computing the Fredholm index of an elliptic operator on a compact manifold in terms of topological data has been solved by Atiyah and Singer in the 60's. Since then lots of work has been devoted to solving analog problems in more general situations, and to developing new formalisms to handle such generality. In this talk, we will sketch how Lie groupoids, C*-algebras and K-theory allow us to recover the original index formula of Atiyah and Singer.
Slides: [PDF]
March 23rd, 2023
Speaker: Sotirios Asimakopoulos, University of Patras
Title: Furstenberg's proof of van der Waerden Theorem
Abstract: The aim of this talk is to present Furstenberg's proof of the van der Waerden theorem through topological dynamics. First we give some basic definitions and examples for topological dynamical systems. Next we formulate the van der Waerden theorem, which states that: given any coloring of the natural numbers, we can find a monochromatic numerical progression, however large (of length). To prove this theorem, we should define the shift map representation and give (without a proof) some theorems that hold for the reset points and multiple reset points of a dynamical system. Finally, through these theorems and the shift map, we manage to reach our initial goal, namely the proof of the van der Waerden theorem.
March 16th, 2023
Speaker: Skylar Werner, University of Göttingen
Title: Divisor theory in Algebraic Geometry
Abstract: When working in Projective space, it is know that given a nonsingular projective curve C in 2-dimensional projective space. Then for any line L, we have that the size intersection of C and L contains exactly the degree of C when counting with proper multiplicity. Thus we can define a divisor on C as the sum of all the intersection points times their multiplicities. We will abstract this nature and define two different divisors called Weil and Cartier Divisors, how they are related, the invariant power they bring, and how they are used to study geometry.
February 9th, 2023
Speaker: Evangelos Kotzafilios, Aristotle University of Thessaloniki
Title: Exotic behaviour at the complex plane.
Abstract: In this talk we will review the motivation for studying the dynamics of complex functions, define Fatou and Julia sets, and discuss some of the pathologies of the latter. We will see how many objects introduced into the topology purely for "counterexample" reasons arise naturally in the dynamics.
January 19th, 2023
Speaker: Dimitrios Konstantinidis, Aristotle University of Thessaloniki
Title: Oidification of Lie's Third Theorem
Abstract: The aim of the talk is to present a generalization of Lie's Third Theorem. First, we introduce the notions of vector bundles, Lie algebras and Lie groups. Then, we give the definitions of Lie algebroids and groupoids, as a natural generalization of the above. Finally, the appropriate generalization of the theorem is presented, since its classical version directly connects Lie algebras and Lie groups.
January 12th, 2023
Speaker: Dimitrios Nikolakopoulos, University of Patras
Title: The proof of Wedderburn's little theorem
Abstract: The aim of this talk is to prove Wedderburn's little theorem which claims that, every finite division ring is a field. The proof we are about to give is due to Ernst Witt. In order to get to the proof, we first need to recall the notions of cyclotomic polynomials and some basic algebra definitions.
Slides: [PDF]
December 15th, 2022
Speaker: Anastasios Iltsoglou, National Technical University of Athens
Title: Hausdorff measure and applications
Abstract: We start with the definition of the Hausdorff measure. Then, we study some of its properties and see how it relates to the Lebesgue measure. Next, we see some applications of the Hausdorff measure. Finally, we define the Hausdorff dimension and see cases of sets whose dimension is not a natural number.
Slides: [PDF]
December 8th, 2022
Speaker: Georgios Simantiras, Aristotle University of Thessaloniki
Title: Linear approximations of vector fields and Euler-like vector fields
Abstract: In this talk we start with the study of the pullback, as well as the normal bundle. Next, we are introduced to double vector bundle and in particular to the double tangent bundle. With the help of Linear Algebra we define the normal functor, and through it we study the linear approximation of a vector field, corresponding to the Taylor theorem for vector fields. Finally, we refer to Euler-like vector fields and their correspondence with tubular neighbourhoods, giving a simple example, which we generalize.
Slides: [PDF]
December 1st, 2022
Speaker: Evangelos Papapetros, University of Patras
Title: Similarity and w*-Similarity Problem
Abstract: In this talk there is going to be an introduction to concepts of Operator Theory and in particular to the very basic concepts of C*-algebra as well as von Neumann algebra. We also present two open problems from the theory of C*-algebras, the Similarity Problem as well as the w*-Similarity Problem, on which we suggest affirmative answers.
Slides: [PDF]
November 24th, 2022
Speaker: Anastasios Fotiadis, Aristotle University of Thessaloniki
Title: Symplectic Toric Manifolds and the Delzant's Theorem
Abstract: In this talk we give an introduction to the basic concepts of Symplectic Geometry. Specifically, starting with a quick review of symplectic vector spaces, we define symplectic manifolds and state basic examples of them, most significantly the cotangent bundle of a smooth manifold. We then study Hamiltonian actions on them in order to define a symplectic toric manifold. Finally, as a key result we mention the Delzant's Theorem which gives us a characterization of symplectic toric manifolds in terms of Delzant polytopes.
Slides: [PDF]
November 10th, 2022
Speaker: Vasileios Oikonomou, University of Patras
Title: Fundamental group and Seifert-Van Kampen Theorem
Abstract: In this presentation we introduce the notion of path homotopy and define the fundamental group of a topological space. The main goal is to construct an algebraic structure on a topological space. We proceed by studying the fundamental group of the circle $S^1$ and prove that it is isomorphic to the additive group of integers using covering spaces. Along the way we study conditions of the invariance of the fundamental group in order to extend our calculations and we study its relation to topological constructions such as the topological product. So it is natural to see some applications of the results we have, that is, we list some fundamental group calculations. Finally, we give a brief introduction to free groups and prove the Seifert-Van Kampen Theorem which is a powerful tool in computing fundamental groups.
Slides: [PDF]
November 3rd, 2022
Speaker: Emmanouela Makrimanolaki, University of Crete
Title: The notion of the absolute value of real numbers: Didactic issues and the opinions and attitudes of Greek teachers
Abstract: The purpose of the research that has been carried out was to formulate the opinions of mathematicians on the teaching of absolute values in secondary education and also the difficulties of the students.
October 20th, 2022
Speaker: Florias Papadopoulos, Aristotle University of Thessaloniki
Title: Cryptography in the Post-Quantum era
Abstract: In this talk we begin by mentioning basic concepts from the science of Cryptography and Quantum Computing in order to analyze the so-called "Quantum Threat" to Cryptography. We continue by presenting the main way in which this threat can be addressed, Post-Quantum Cryptography, which we analyze by describing some families of cryptographic schemes and systems that belong to it. Finally, we close the talk with a reference to the current and future views and actions of organizations actively engaged in the topic, in order to give a global perspective of the topic (eg Cryptography of Codes, Cryptography of Networks).
Slides: [PDF]
October 13th, 2022
Speaker: Athanasios Beslikas
Title: Univalent functions in spaces of analytic functions
Abstract: In this talk, we study some characterizations of univalent homomorphic functions and conformal mappings belonging to Hardy and Bergman spaces. We refer to earlier results using integral mean functions, and then give recent characterizations of conformal mappings on Hardy, Bergman spaces using conformally invariant quantities of the complex plane. Prerequisite knowledge for understanding the speech are some basic definitions of analytic function spaces, Riemann's conformal mapping theorem, and elements of hyperbolic distance, harmonic measure, and Green's function.
Slides: [PDF]
May 19th, 2022
Speaker: Georgios Nikolaidis, Aristotle University of Thessaloniki
Title: An introduction to Banach Algebras
Abstract: An important branch of Mathematics is Functional Analysis in which we study Banach spaces and properties of Operators. A Banach algebra is an extension of these concepts to a more abstract level that gives us a better understanding of the behavior of these sets. In addition to the definition and some basic properties of a Banach algebra we study the concept of spectrum and see some important properties of it, using simple results of Complex Analysis, such as the Liouville Theorem and the Laurent Series theorem.
Slides: [PDF]
May 5th, 2022
Speaker: Anastasios Slaftsos, Aristotle University of Thessaloniki
Title: An introduction to Algebraic Geometry
Abstract: Systems of polynomial equations appear almost everywhere in Mathematics. The purpose of Algebraic Geometry is to study (with algebraic methods) the geometric structure of the solution space of such systems. In our presentation, we define affine varieties as such solution spaces endowed with a suitable topology called Zariski's Topology and study their connection with the k-algebras of their coordinate rings. Finally, we generalize the above by making an introduction to the spectrum of a ring (origin of affine schemes).
Slides: [PDF]
April 14th, 2022:
Speaker: Evangelos Kotzafilios, Aristotle University of Thessaloniki
Title: The Ahlfors-Schwarz Lemma
Abstract: The purpose of the talk is to present a form of the Ahlfors-Schwarz Lemma, as well as some of its consequences. The result provides "elementary" proofs of important theorems of classical Complex Analysis, such as Picard's little theorem and the great theorems of Picard and Montel. Prerequisites for understanding most of the talk are: Schwarz Lemma and Schwarz - Pick Lemma, Egregium Theorem, Gaussian Curvature.
Slides: [PDF]