**UP Math Seminar **

The University of Prishtina Math Department Seminar (UP Math Seminar) is organized approximately once a month. This academic year it usually takes place on a Thursday, from 1:00--2:00 PM (CET). Talks are in English and the format is that of a webinar. Video recordings are usually posted online shortly after a talk is given. For individual videos, chek the links below in this page.

Everyone is welcome, including first-year undergraduate students as talks are intended for a general math audience. To receive updates as well as the zoom link for each talk, you need to register (by sending me a request at my @uni-pr.edu address).

**Upcoming talks:**

**Erida Gjini (Instituto Superior Técnico, Lisbon)**

11 May 2023, 1:00--2:00 PM (CET)

Title: Eco-evolutionary dynamics in co-colonization systems with multiple strains

Abstract: The high number and diversity of microbial strains circulating in host populations has inspired extensive research on the mechanisms that maintain biodiversity. While much of this work focuses on strain-specific parameters and cross-immunity interactions, the role of co-infection (co-colonization) and co-infection interactions between strains has received less attention. Diversity in such interaction coefficients enables strains to create dynamically their niches for growth and persistence, and 'engineer' their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability-complexity regimes in such multi-player endemic systems remain poorly understood. In this talk, I will present results from recent work, where we analyze such an interacting system and the eco-evolutionary dynamics that emerge. Adopting slow-fast dynamic decomposition of the original coinfection model, we obtain a replicator equation, which sheds light on the key coexistence principles and evolutionary trajectories of a system.

**Hamdi Dërvodeli (The University of Warwick)**

25 May 2023, 1:00--2:00 PM (CET)

Title:

Abstract:

**Talks already given during this academic year:**

**Shend Zhjeqi (University of Michigan)**

27 April 2023, 1:00--2:00 PM (CET)

Title: Bend and Break

Abstract: Bend and Break is an elegant and powerful method in Birational geometry for showing that the geometry of varieties is rich. The method was developed by Shigefumi Mori in the 70's for the purpose of proving Hartshorne's conjecture: "A projective complex manifold whose tangent bundle is ample is in fact isomorphic to projective space." We will see how bend and break works and we will aim to show an easier application of Bend and Break: Fano Varieties, an important class of varieties that show up in Algebraic Geometry, are uniruled.

**Tony Shaska (University of Oakland) **

6 April 2023, 1:00--2:00 PM (CET)

Title: Machine learning in the Moduli space of curves

Abstract: We propose new methods to apply machine learning to various databases which have emerged in the study of the moduli spaces of algebraic curves. We find that with such methods one can learn many significant quantities to astounding accuracy in a matter of minutes and can also predict unknown results making this approach a valuable tool in pure mathematics.

**Arbër Selimi (Instituto Superior Técnico, Lisbon) **

30 March 2023, 1:00--2:00 PM (CET)

Title: Higher Structures on Higher Bundles and Local Connections

Abstract: The subject of principal bundles with connections (gauge theory) is of special interest in mathematics and physics. In this talk, I will provide a brief overview of gauge theory and the importance of bundles with connections. Different approaches on the theory of higher bundles, which are the categorified versions of principal bundles, will be introduced. The goal of this talk is to compare those definitions. Also a comparison of local connections and smooth functors from path groupoids of smooth manifolds to higher groups will be shown.

**Islam Foniqi (University of East Anglia) **

23 February 2023, 1:00--2:00 PM (CET)

Title: Decision and algorithmic problems in algebraic structures

Abstract: One of the most fundamental algorithmic problems in algebraic structures, defined by presentations on generators and relations, is the word problem: it asks whether there is an algorithm that takes two expressions on generators and decides if they represent the same element. In the 1930s Magnus proved that one-relator groups have decidable word problem. Our focus will be on one-relator monoids, where the analogous problem is still open, and we will provide some connections to other decision problems about monoids, inverse monoids, and groups. Furthermore, we will discuss some recent results on the undecidability of the submonoid membership problem on positive one-relator monoids.

**Gramoz Goranci (ETH)**

12 January 2023, 1:00--2:00 PM (CET)

Title: Vertex sparsification in dynamic algorithms and beyond

Abstract: In the era of big data, there has been an ever-growing interest in designing fast algorithms. In classic algorithm design, the input data is revealed upfront, and the goal is to design algorithms that run in near-linear time in the size of input. However, in many real-world applications involving graphs, the input is subject to frequent changes. This motivates the study of dynamic graph algorithms, which are data structures that maintain relevant graph information under vertex/edge updates. In this talk, we’ll discuss a general algorithmic tool for designing dynamic graph algorithms known as vertex sparsification. This is a compression paradigm that reduces large graphs into smaller ones while provably preserving properties or features of interest. In particular, we show that black-box efficient constructions of vertex sparsifiers and their data-structure variants lead to dynamic maintenance of effective resistances on graphs. We will also briefly discuss the implications of this technique in dynamically maintaining Laplacian systems and maximum flows.

**Besfort Shalaj (University of Bristol) **

22 December 2022, 1:00--2:00 PM (CET)

Title: The Probabilistic Zeta Function of a Finite Lattice

Abstract: The main object relevant to this talk is Brown's probabilistic zeta function of a finite lattice, as a generalization of that of a finite group. We propose a natural alternative or extension that may be better suited for non-atomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, obtained through a more general Möbius inversion procedure. Unlike for groups, the Dirichlet series for lattices need not be ordinary. In this context, we are interested in lattices with probabilistic zeta function given by ordinary Dirichlet series. We focus on the partition lattice and the d-divisible partition lattice. As an application of the prime number theorem, we show that the probabilistic zeta function of the latter typically fails to be an ordinary Dirichlet series.

**Vjosa Blakaj (Technical University of Munich)**

24 November 2022, 1:00--2:00 PM (CET)

Title: Quantum Shannon theory: entropy-constrained sets

Abstract:

In this talk, I will give a brief introduction to quantum Shannon theory, which, in short, is concerned with the study of the ultimate capability of noisy physical systems subject to the laws of quantum mechanics to preserve information and correlations. I will then address the question of the (im)possibility of "single-letter" formulas/criteria, in particular for asymptotically defined quantities, and focus on the transcendental properties of entropy-constrained sets.**David Kalaj (University of Montenegro) **

20 October 2022, 1:00--2:00 PM (CET)

Title: Curvature of minimal graphs

Abstract:

In this paper we solve the longstanding Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. This conjecture states the following. For any minimal graph lying above the entire unit disk, the Gaussian curvature at the point above the origin satisfies the sharp inequality $|\mathcal{K}|< \pi^2/2$. The conjecture is first reduced to the estimation of the Gaussian curvature of certain Scherk type minimal surfaces over some bicentric quadrilaterals inscribed in the unit disk containing the origin. Then we make a sharp estimate of the Gaussian curvature of those minimal surfaces over those bicentric quadrilaterals at the point above the zero. Our proof uses complex-analytic methods since minimal surfaces that we consider allow conformal harmonic parameterization.**Talks given during the academic year 2021-2022:**

**Travis Schedler (Imperial College London)**

23 May 2022, 12:30--13:30

Title: Symplectic representation theory

Abstract: Recently, a unifying theme has emerged in representation theory---the mathematical study of linear symmetries---wherein these symmetries can be seen as coming from a rich geometric object, a symplectic resolution of singularities. These objects have their origin in physics---Hamiltonian classical mechanics---and representation theory emerges from the passage to quantum mechanics. I will explain these ideas in the elementary context of representations of two-by-two matrices. I will then outline a program to classify symplectic resolutions, which is a vast extension of the celebrated Cartan--Killing--Dynkin classification of simple complex Lie algebras via Dynkin diagrams. No familiarity with these ideas (or with physics) will be assumed.

**Qamil Haxhibeqiri (Academy of Sciences and Arts of Kosovo)**

11 April 2022, 12:30--13:30

Title: To Shape theory via procategories

Abstract: Shape theory is a new branch of topology. Like homotopy theory, shape theory is devoted to study of global properties of topological spaces. However, the tools of homotopy theory are of a such nature that they yield interesting results only for spaces behave well locally (e.g.Absolute Neighborhood retracts (ANR spaces) or CW-complexes). On the other hand the tools of shape theory are so disegned that they also yield interesting results in the case of bad local behavior. Moreover shape theory agrees with homotopy theory on class of ANR's and CW-complexes i.e. on spaces with good local properties.

There are two approaches to define shape theory: one is Borsuk's which use the notion of fundamental sequences and the other is Mardesic-Segal's approach which use the notion of inverse systems of ANR's. K. Borsuk introduced the shape theory for compact metric spaces. After that S. Mardesic and J.Segal developed shape theory for compact Hausdorff spaces and S. Mardesic generalized shape theory for arbitrary topological spaces. The aim of this talk is to define shape category by Mardesic's approach. The shape category is defined by an arbitrary pair of categories and then are given special cases of shape category for compact Hausdorff spaces and some other spaces.

**Alban Rrustemi (London) **

14 March 2022, 12:30--13:30

Title: Emergence of large language models

Abstract: Language models (LMs) are becoming pervasive across a wide range of applications including search, machine translation, and speech recognition. Over the past two years, modern LMs have produced some of the most impressive demonstrations of artificial intelligence (AI) -- e.g. they can write newspaper articles, summarise books, take part in coding competitions, etc. In this talk, I'll provide a brief overview of LMs and aspects of machine learning that power these models. I'll focus on the scaling properties that unlock some of these advancements. I'll also comment on the overall role that LMs play in the field of AI.

**Bahtijar Vogel (Malmö University) **

7 February 2022, 12:30--13:30

Title: Sustainable Design Principles for Internet of Things (IoT): Towards open and secure IoT systems

Abstract: The Internet of Things (IoT) market is predicted to grow from an installed base of 30.7 billion devices in 2020, to 75.4 billion in 2025. There are different types of platforms available that often are referred to as IoT platforms, such as device-to-device, cloud-based and device-to-cloud platforms (which are often also referred to as enterprise platforms that face a vendor lockdown). The diversity of IoT platforms and their complex offerings creates confusion among developers and researchers, as well as end users. As such, today the IoT is mainly associated with vertically integrated systems that often are closed and fragmented in their applicability. With such closed nature and fragmentation in the market, developers usually struggle to reach critical mass, and even end users need to navigate through different brands and understand which devices are compatible in relation to which IoT platforms. Commercial or proprietary IoT platforms carry a pricing model and often promote vendor lock-in. Thus, often IoT platform providers lack support of new protocols, tools and data formats in time due to a constantly changing IoT landscape. Openness in IoT systems offer a multitude of benefits, even though security is never assured. While security is often recognized as a top priority for organizations and a push for competitive advantage, repeatedly, IoT products have become a target of diverse security attacks. Thus, orchestrating smart services and devices in a more open, standardized and secure way in IoT environments is yet a desire as much as it is a challenge. In order to address some of the above mentioned challenges, in this presentation, Dr. Vogel will be talking about his research related to the need for open and secure design principles for Internet of Things (IoT).

**Uran Meha (University of Lyon) **

20 December 2021, 12:30--13:30

Title: Plactic monoids via rewriting theory

Abstract: Two powerful approaches to the study of algebraic objects are homological algebra and representation theory. Recently the field of rewriting theory has supplied an algorithmic viewpoint for the homological study of monoids given via adequate presentations by generators and relations. In this talk we show how rewriting theory applies to certain monoids arising from the representation theory of Lie algebras (plactic monoids), and we obtain explicit 3-dimensional homological information for two classes of plactic monoids.

**Vlerë Mehmeti (Paris-Saclay University) **

22 November 2021, 12:30--13:30

Title: Local--global principle and non--Archimedean geometry

Abstract: We know since the 19th century, by the works of Galois, that we can't always explicitly determine the solutions of polynomial equations, we can at most approximate them. Ever since, the following question has been given a central place in Number Theory: when do solutions exist? I will be speaking of one (of the numerous) approaches to study this problem, called the local--global principle. Among other things, I will introduce for this purpose a geometry (called non-Archimedean), which satisfies surprising properties that are often incompatible with our Euclidean intuition.

[Back]