Seminars

Topological Data Analysis via persistent homology is a new emerging area of data analysis that uses methods from simplicial topology. The persistent homology of a data set can be calculated using a simple algorithm called reduction algorithm. In this talk, I will present a new construction of worst-case examples for this algorithm. Our constructions are similar to the worst-case examples introduced by Morozov, but replace the single-triangle arrangement with a strip formed by base and fin triangles. This structure allows us to give an explicit algorithm for their construction and to perform experiments comparing the runtime of different variants of the reduction algorithm. We further show that, after suitable edge and triangle subdivisions, these strip examples remain worst-case and can be realized as clique complexes of filtered graphs, and hence as Vietoris–Rips complexes of finite point clouds for a sequence of scale parameters.

We are concerned with the computation of the \(H\)-infinity norm for \(H\)-infinity functions of the form \(H(s) = C(s)D(s)^{-1}B(s)\), where the middle factor is the inverse of an analytic matrix- valued function, and \(C(s)\), \(B(s)\) are analytic functions mapping to short-and-fat and tall- and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is very large. We propose a subspace projection method to obtain approximations of the function H where the middle factor is of much smaller dimension. The \(H\)-infinity norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the largest singular value of the reduced function is maximized. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

We study Poncelet type problem about the number of convex \(n\)-gons inscribed into one convex \(n\)-gon and circumscribed around another convex \(n\)-gon. It is proved that their number is at most 4. The result is generalized for spherical and hyperbolic geometries. This contrasts with Poncelet type porisms where usually infinitude of such polygons is proved, provided that one such polygon already exists. An inequality involving ratio of lengths of line segments is used. Alternative way of using Maclaurin–Braikenridge’s conic generation method generalized by Brianchon is also discussed. Properties related to constructibility with straightedge and compass are also studied. A new proof, based on mathematical induction, of generalized Maclaurin–Braikenridge’s theorem is given. We also gave examples of regular polygons for which number 4 is realized.

The representation theory of finite groups was introduced at the end of the 1800s with the work of Frobenius and Schur. It was developed further through contributions by Brauer, Burnside, Green and others. Beginning around the mid-20th century, category theory started to develop and, together with the abstraction techniques it provides, rapidly spread into other fields. The aim of this talk is to describe the evolution of functorial representation theory and to discuss the theory of biset functors, one of the most effective techniques of this new style of representation theory, and to describe various applications of this technique.

\(p\)-permutation modules provide an algebraic link between permutation representations and modular representation theory. In this talk we will (i) recall the definition and main properties of \(p\)-permutation modules over a field \(k\), (ii) explain how the assignment \(G\mapsto\) (Grothendieck group of the category of \(p\)-permutation \(kG\)-modules) naturally has the structure of a (fibered) biset functor, and (iii) state and sketch proofs of the main structural results describing composition factors as (fibered) biset functor. This is a joint work with Robert Boltje and Cisil Karakguzel.

Prime numbers and their distribution is in the heart of analytic number theory. This talk will be about counting prime numbers with certain restrictions. We will cover some elementary techniques for detecting primes belonging to certain subsets of all residue classes modulo primes. And then we will discuss some classical analytic methods and their implications.

In this talk, we will briefly review Artin formalism and its \(p\)-adic variant. Artin formalism provides a factorization of \(L\)-functions whenever the associated Galois representation decomposes. We will explain why the \(p\)-adic Artin formalism becomes a nontrivial problem when there are no critical \(L\)-values. In particular, we will focus on the case where the Galois representation arises from the Rankin–Selberg product of a newform with itself. We will present the results in this direction, including our result in the case where the modular form in question is \(p\)-non-ordinary, and discuss future directions.

Associativity is a well-known property of some binary operations. Instead of the binary classification into associative and non-associative operations, more refined approaches for quantifying the degree of associativity have been proposed. One such method is the so-called associative spectrum, introduced by Csakany and Waldhauser in 2000. The associative spectrum of a groupoid \(G\) is an integer sequence whose \(n\)-th term equals the number of distinct term operations induced on \(G\) by the bracketings of \(n\) variables. In this talk, we provide a brief introduction to the associative spectrum, and give an overview on our work on the associative spectra of graph algebras and quasigroups. A quasigroup is a groupoid satisfying the condition that for all elements \(a\) and \(b\), there exist unique elements \(x\) and \(y\) such that \(xa=b\) and \(ay=b\). This talk is based on joint work with Tamas Waldhauser (University of Szeged).

Let \(s(n)\) denote the sum of proper divisors of an integer \(n\). The function \(s(n)\) has been studied for thousands of years, due to its connection with the perfect numbers. In 1992, Erdos, Granville, Pomerance, and Spiro (EGPS) conjectured that if \(\mathcal{A}\) is a set of integers with asymptotic density zero then \(s^{-1}(\mathcal{A})\) also has asymptotic density zero. This has been confirmed for certain specific sets \(\mathcal{A}\), but remains open in general. In this talk, we will give a survey of recent progress towards the EGPS conjecture. This talk is based on joint work with Kubra Benli, Giulia Cesana, Cecile Dartyge, Charlotte Dombrowsky, Paul Pollack, and Carl Pomerance.

A tile of a countable group is a finite subset whose non-overlapping left translations cover the entire group. We will review some recent results (including works related to Fuglede Conjecture) about the tilings of groups. It is unknown whether, for every countable group \(G\), any finite subset \(K\subset G\) is contained in a tile of \(G\). We prove this for hyperbolic groups (in the sense of Gromov).

There are various network descriptors for the understanding of complex networks. A major theme is to understand the relations between network descriptors and their connection to the networks function. The aim of this talk is to rigorously connect several network descriptors using mathematical tools. The central notion is entropy, which plays a fundamental role in quantifying disorder and complexity. Entropy is also closely related to the spectral radius of the graph adjacency matrix. Another central player is the so-called Randic index, introduced in the 1970s to study chemical compounds. We will see that the topological entropy is bounded from below by the Randic index. In social networks, vertices with high degree are often adjacent to other high-degree vertices, a tendency referred to as assortativity. We finish with breadth-first search ordering with decreasing degrees that characterizes the maximization of all these notions. This is a joint work with Fatihcan Atay, Bilkent University, Ankara.

I will present some commonly used proof methods and techniques in discrete mathematics and graph theory. We will see several nice theorems and their smart proofs based on few basic definitions and such techniques. This week we continue with the pigeon hole principle. At the end of the lecture, there will be a problem solving session from journals such as American Mathematical Monthly, Elemente der Mathematik, and Mathematics Magazine.

Cluster algebras offer a unifying framework that connects a diverse range of mathematical areas, including representation theory, string theory, Poisson geometry, integrable systems, knot theory, and combinatorics. In this talk, we will give a gentle introduction to cluster algebras, tracing their origins in representation theory and highlighting recent developments that connect cluster structures with crystal operators.

In this talk, I will give an overview of my research interests and current projects within the field of modern Mathematical Physics. A central focus is the mathematical derivation of the foundations of quantum mechanics. In particular, I highlight the concept of asymptotic emergence, a rigorous mathematical framework for understanding the classical and macroscopic limits of quantum systems. Key topics include spontaneous symmetry breaking, phase transitions, and entropy, explored through the lens of strict deformation quantization of Poisson manifolds, C*-algebras, and large deviation theory.

In this second talk of the series, we will continue with some further properties of the Mobius function such as the Mobius Inversion Formula. Then we will see some other arithmetic functions such as the divisor function and the Euler totient function. If time permits, we will start considering the summatory functions of these functions and cover a technique called partial summation.

In this series of lectures, we introduce finite group actions and Burnside rings. We begin with basic definitions, explore concrete examples, and cover operations such as the induction and restriction of group actions.

In this talk, we will explore recent advances in combinatorics and graph theory, with a focus on stability results, extremal problems, and the structural properties of discrete objects. We will introduce a stability version of Dirac’s classical theorem, providing a full characterization of near-Hamiltonian graphs, and discuss extensions of Posa’s theorem to hypergraphs. Additionally, I will present the resolution of a longstanding conjecture by Hakimi and Schmeichel on the maximum number of pentagons in planar graphs. Further, I will highlight connections between combinatorics and algebra, including results on intersecting families of polynomials over finite fields and higher-order extensions of Schur’s theorem.

Classically, jet bundles provide the framework for variational calculus as well as for both the Lagrangian and the Hamiltonian formalism in physics. In this talk, we will be concerned with the extension of jet theory to the noncommutative setting. Noncommutative differential geometry generalizes classical differential geometry by replacing the commutative algebra \(C^{\infty}(M)\) of smooth functions on a smooth manifold \(M\) with an arbitrary unital associative algebra \(A\) over a commutative ring \(\mathbb{k}\). We will see that this data is sufficient to construct, via homological algebra and category theory, a generalization of the classical notion of jet to this noncommutative setting. In particular, we will discuss differential operators and their corresponding principal symbols.

One of the central problems in number theory is the Birch and Swinnerton-Dyer conjecture, which asserts that the order of vanishing of the L-function of a rational elliptic curve \(E\) at the central value coincides with the rank of its Mordell-Weil group. A far-reaching generalization is the Bloch–Kato conjecture. In this talk, we recall the Bloch–Kato conjecture in the setting of GSp6-Shimura varieties and present the construction of an Euler system using a novel method that overcomes a major obstacle. As a consequence, we obtain the first non-trivial result towards the Bloch–Kato conjecture in this setting.

I will present some commonly used proof methods and techniques in discrete mathematics and graph theory. We will see several nice theorems and their smart proofs based on few basic definitions and such techniques. This week we start with the pigeon hole principle. At the end of the lecture, there will be a problem solving session from journals such as American Mathematical Monthly, Elemente der Mathematik, and Mathematics Magazine.

Characterization of the identity function using functional equations has been actively studied by many authors. In 1992, Claudia Spiro introduced the concept of additive uniqueness. A set \(E\subseteq\mathbb{N}\) is called an additive uniqueness set of a set of arithmetic functions \(\mathcal{F}\) if there is exactly one element \(f\in\mathcal{F}\) which satisfies \(f(m+n)=f(m)+f(n)\) for all \(m,n\in E\). In the present talk, we show that for \(k\ge 3\) the set of all nonzero hexagonal numbers is a new \(k\)-additive uniqueness set for the collection of multiplicative functions. This is a joint work with Poo-Sung Park (Kyungnam University, Republic of Korea) and Emil Inochkin (ADA University, Azerbaijan).

In this series of talks, we will define some commonly used arithmetic functions in number theory such as the Mobius function, Euler totient function and the divisor function, and we will cover their elementary properties. The talks will be self-contained and will not require a prerequisite.

The local-global principle in modular representation theory asserts that the invariants of blocks of finite groups are determined by the invariants of local subgroups. There are several outstanding conjectures revolving around this principle. Alperin’s block-wise weight conjecture (1987) predicts that the number of simple modules of a block is equal to the number of its weights. The finiteness conjectures of Donovan (1980) and Puig (1982) state that there are only finitely many blocks of finite groups with a given defect group, up to Morita and splendid Morita equivalence, respectively. In this talk, we prove a finiteness theorem in the spirit of Donovan’s and Puig’s conjectures, in terms of functorial equivalences. We also give a reformulation of Alperin’s conjecture in terms of diagonal \(p\)-permutation functors. Some parts of this work are joint with Boltje and Bouc.

In this talk, we will discuss an estimate for a discrete mean value of the Riemann zeta function and its derivatives multiplied by Dirichlet polynomials. Assuming the Riemann Hypothesis, we obtain a lower bound for the \(2k\)-th moment of all the derivatives of the Riemann zeta function evaluated at its nontrivial zeros. This is based on a joint work with Kubra Benli and Nathan Ng.

This is the first session in a seminar series dedicated to exploring finite groups and their representations. In this introductory session, we will cover foundational concepts and outline our plans for the upcoming semester.

Biset functors over a commutative and unitary ring \(k\) provide a powerful framework for studying finite groups and their actions. The biset category, whose objects are finite groups and morphism sets are given by Grothendieck groups \(B(G,H)\) of finite \((G,H)\)-bisets, serves as the foundation for this theory. Serge Bouc made significant contributions by introducing the slice Burnside ring and the section Burnside ring for a finite group \(G\), demonstrating that both naturally possess the structure of a Green biset functor. The classification of simple modules over the section Burnside ring is achieved through the fibered biset functor approach, as detailed in the article by Robert Boltje and Olcay Coskun. This is a joint work with Olcay Coskun.

This is the first of a series of seminars in which we develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the sub-quotients of a finite group \(G\).

This is the first of a series of seminars in which we develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a biset functor defined on the sub-quotients of a finite group \(G\). The obstruction groups for this theory are the reduced cohomology groups of a category \(D^*_G\) whose objects are the sections \((U,V)\) of \(G\), where \(V\) is a non-trivial normal subgroup of the subgroup \(U\) of \(G\), and whose morphisms are defined as a generalization of morphisms in the orbit category.