The moment our Universe burst into being, when all of space-time, energy, and the laws that govern them were compressed into the tiniest conceivable domain, is hidden behind a veil of extremes known as the Planck scale. This seminar probes that first 10−43 s and 10−35 m, where classical notions of gravity, causality, and even dimensionality dissolve into quantum foam. We synthesize insights from loop quantum gravity, string theory, and causal-set approaches to show how discrete space-time may avert the Big-Bang singularity, replacing it with a bounce or emergent phase. Recent advances in primordial gravitational-wave detection, cosmic microwave-background polarization, and high-energy particle phenomenology are examined for the faint signatures they might carry from this pre-inflationary epoch. Finally, we discuss the philosophical ramifications of a Universe whose origin is not a singular event but a quantum phase transition, and highlight open questions, ranging from vacuum selection to the nature of time, that define the cutting edge of quantum-cosmological research. Participants will leave with an integrated roadmap of theoretical frameworks and observational strategies poised to illuminate the Planck-era frontier.

Stem cell differentiation is governed by complex epigenetic dynamics involving reactive and diffusive biochemical components. This talk presents a novel approach integrating reaction-diffusion theory with machine learning to model these dynamics in hematopoiesis. By coupling reactive-diffusive components, such as transcription factors and mRNA, with purely reactive elements, including cis-regulatory elements, enhancers, and histone modifications, we construct stage-specific reaction-diffusion models for four critical stages of haematopoiesis: embryonic stem cells (ESC), hemangioblasts (HB), hemogenic endothelium (HE), and hematopoietic progenitors (HP). Coefficients for the reaction kinetics of these models are quantified through machine learning applications on experimental datasets, capturing the intricate feedback and regulatory mechanisms underpinning stage-specific epigenetic transitions. Numerical simulations of these models demonstrate high consistency with biologically observed phenomena, providing new insights into the spatial and temporal coordination of epigenetic components. This integrative framework bridges theoretical and experimental biology, offering a new approach for understanding the spatial heterogeneity and stage-specific regulation in stem cell differentiation. The approach not only advances reaction-diffusion modeling in epigenetics but also highlights the potential of machine learning to decipher complex biological systems. These findings pave the way for future applications in regenerative medicine and stem cell engineering, with implications for both fundamental research and therapeutic development. 

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Fractional calculus is the study of integral and derivative operators to non-integer orders. As soon as this idea is conceived, it gives rise to various questions. What does it mean? How is it defined? Why does it matter? Where did it come from? What can we do with it? None of these questions has a single simple answer. This talk will cover the motivation and history behind fractional calculus, the foundational definitions and references of the field, potential applications with advantages and drawbacks, and then attempt to put some structure on the many different types of fractional integrals and derivatives that have been defined so far, categorising them into broad classes and focusing on the relationships between them.

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In this talk we talk about some dendrites that can be obtained as a generalized inverse limit spaces with a single upper semi-continuous bonding function.

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 Islamic Geometric Patterns (IGP) represent one of the most profound intersections of mathematics and art in human culture. These intricate designs, which flourished throughout the Islamic world, transform mathematical principles into stunning visual compositions. At their foundation lies the mathematical DNA of wallpaper groups – the seventeen distinct ways patterns can repeat infinitely across a plane. 

This talk explores how symmetry groups generate these mesmerizing patterns, examining historical examples from diverse Islamic traditions. We'll trace their evolution from traditional compass-and-straightedge construction to the polygon method, demonstrating how each pattern embodies specific wallpaper groups. Through the lens of group theory, we'll unravel how the mathematical classification of symmetry groups provides deeper insight into both their structure and complexity. Our discussion will illuminate how these patterns make abstract mathematical concepts tangible, serving as sophisticated bridges between mathematical precision and cultural heritage.

                                                                      هاوجێبوونی ناکۆتا: بیرکاری پشت هونەری نەخشەی ئیسلامی

هونەری نەخشەی ئیسلامی بە یەکێک لە هەرە خاڵە هاوبەشەکانی نێوان بیرکاری و هونەر لە کەلتووری مرۆڤایەتیدا دادەنرێت. ئەم دیزاینە قەشەنگانە کە لە سەرتاسەری جیهانی ئیسلامیدا (بە کوردستانیشەوە) بە ڕوونی دەبینرێت سوودی لە پرەنسیپەکانی بیرکاری بینیوە بۆ دروستکردنی دیزاینی ئاڵۆز و دەوڵەمەند لە ستایل و شێوازدا. لەم وتوێژەدا هەڵدەستین بە پیشاندانی بەشێکی زۆر لەم نەخشانە و دواتر بیرکاری هاوکاربووە لە دروستکردنیان دواتر هەڵدەستین بە پۆلێنکردنیان بەسەر ١٧ بەشدا لەسەر بنەمای سیمەتری (هاوجێبوون).

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A linear operator on a Hilbert space may be approximated with finite matrices by choosing an orthonormal basis of the Hilbert space. In this work, we establish an approximation of the numerical range of unbounded operator matrices by projection method. Application to Shrodinger operator is given.

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In this talk, I will begin by reviewing Fraïssé's construction method, a model-theoretic technique for constructing infinite homogeneous structures, and its foundational role in the field. I will then discuss key correspondences between the combinatorial properties of these structures and the group-theoretic properties of their automorphism groups. The focus will then shift to the Urysohn universal metric space (and Urysohn sphere), introduced by Pavel Urysohn in the 1920s. This remarkable space is a homogeneous metric space into which any separable metric space can be isometrically embedded. Finally, I will present selected results and observations concerning its isometry group, emphasizing their properties and broader implications.

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In this talk, we explore the saturation number of monomial ideals in the polynomial ring \( S = K[x_1, \ldots, x_n] \) over a field \( K \). We focus on various classes of ideals, including \(\mathbf{c}\)-bounded strongly stable, Veronese-type, irreducible monomial ideals, and monomial ideals in two variables. Additionally, we examine the saturation numbers of ordinary and symbolic powers of specific families of monomial ideals, relating them to the saturation numbers of their irreducible components.

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Mathematical modeling is a powerful tool for understanding complex phenomena across different fields. This talk explores two areas: bubble dynamics and battery modeling. In fluid dynamics, we study cavitation bubble collapse near boundaries using boundary integral and finite volume methods, with applications in medicine and cleaning. In electrochemistry, we examine battery behavior using physics-based models like the Doyle-Fuller-Newman framework, essential for improving energy storage. By presenting these different fields, we show how modeling helps solve diverse problems. We'll discuss our findings and highlight areas needing further research due to time, knowledge, or resource limitations. This talk aims to encourage collaboration and inspire new research in both bubble dynamics and battery modeling.

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In this talk we will discuss some recent interesting connections between properties of sets of nonnegative integers with different kinds of geometries. Starting with a primitive integer vector, we examine certain associated functions and relate the setting to statements in convex and algebraic geometries.

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We consider the convexity properties of distortion functionals, particularly the linear distortion, defined for homeomorphisms of domains in Euclidean n-spaces, n ≥ 3. The inner and outer distortion functionals are lower semi-continuous in all dimensions and so for the curve modulus or analytic definitions of quasiconformality it follows that if {fn}∞ n=1 is a sequence of K-quasiconformal mappings (here K depends on the particular distortion functional but is the same for every element of the sequence) which converges locally uniformly to a mapping f, then this limit function is also K-quasiconformal. Despite a widespread belief that this was also true for the geometric definition of quasiconformality (defined through the linear distortion H(fn)), T. Iwaniec gave a specific and surprising example to show that the linear distortion functional is not always lower semicontinuous on uniformly converging sequences of quasiconformal mappings. Here we show that this failure of lower semicontinuity is common, perhaps generic in the sense that under mild restrictions on a quasiconformal f, there is a sequence {fn}∞ n=1 with fn → f locally uniformly and with lim supn→∞ H(fn) < H(f). Our main result shows this is true for affine mappings. Addressing conjectures of F.W. Gehring and Iwaniec, we show the jump up in the limit can be arbitrarily large and give conjecturally sharp bounds: for each α < √ 2 there is fn → f locally uniformly with f affine and α lim sup n→∞ H(fn) < H(f) We conjecture √ 2 to be best possible.

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