23 APR 2025
16:00 - 17:00

This lecture is intended as an introduction to some of the metaphysical and epistemological issues that are usually raised in connection with mathematical thought. These issues include, among other things, the nature of abstract objects like numbers and sets, some of the well-known arguments for the existence of such objects as well as the epistemological problems that the existence of those objects incur. There will also be some brief discussions of certain anti-realist approaches to mathematical ontology.

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Venue: Niavaran, Lecture Hall 1

23 APR 2025
14:00 - 15:00

‎Let A=[a_₀, a_₁, ..., a_{n-1}] ∈ ℂ^ℓ be a finite complex sequence of length ℓ periodically continued with period ℓ. ‎The periodic autocorrelation function PAF(A,s) of A at lag s is‎ PAF(A, s) =∑ {j=0}^{ℓ-1} a_joverline{a_{j+s}}, s=0,1, ..., ℓ-1. ‎Two sequences‎ ‎A and B of the same length ℓ. ‎form a Legendre pair (A,B) if‎ ‎PAF(A,s)+PAF(B,s)=-2, s=1,2, ?, ⌊n/2⌋. ‎Binary Legendre pairs of length ℓ are pertinent to the construction of Hadamard matrices of order ℓ +2 which can exist only if ℓ is odd‎. ‎It is conjectured that there is a binary Legendre pair of every odd length ℓ which is stronger than the Hadamard-matrix conjecture‎. ‎Several infinite classes of Legendre pairs are known‎. ‎The most prominent constructions are defined via characters of finite fields‎. ‎In particular‎, ‎there are Legendre pairs of length ℓ for every prime ℓ =p and every Mersenne number ℓ =2^s-1‎, ‎respectively‎. ‎The smallest undecided case is ℓ=11. ‎In the first part of the talk we summarize the known results on binary Legendre pairs‎. ‎In the second part we discuss quaternary Legendre pairs of length ℓ‎. ‎In contrast to binary Legendre pairs they can exist for even ℓ as well‎. ‎Very recently‎, ‎the first infinite construction was found by Jedwab and Pender completing our earlier semi-construction‎. ‎We give also constructions of quaternary Legendre pairs of length ℓ for all remaining even ℓ < 42‎. ‎The smallest open case is ℓ =42 and the case ℓ =46 is particularly interesting since the existence of a quaternary Legendre pair of this length would imply the existence of a quaternary Hadamard matrix of order 94. ‎Finally‎, ‎we mention generalizations to k-ary sequences‎.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

17 APR 2025
14:00 - 16:00

We address Krasner's problem (1956) for Polish ultrametric spaces and provide a (hopefully) satisfactory solution. In particular, we discuss a correspondence between the isometry groups of Polish ultrametric spaces belonging to some natural subclasses and various collections of generalized wreath products proposed in the literature. This is joint work with R. Camerlo and A. Marcone.

Zoom room information:
https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618
Venue: Niavaran, Lecture Hall 1

16 APR 2025
14:00 - 15:00

Digital signature schemes play a vital role in ensuring secure communication, authentication, and data integrity across numerous applications, including secure email, financial transactions, and blockchain systems. However, classical schemes like RSA and ECDSA are vulnerable to quantum attacks, prompting a global shift toward post-quantum cryptographic alternatives. As part of this transition, NIST has already standardized three post-quantum cryptographic schemes: (i) ML-KEM (FIPS 203) for key encapsulation, based on CRYSTALS-Kyber; (ii) ML-DSA (FIPS 204) for digital signatures, derived from CRYSTALS-Dilithium; and (iii) SLH-DSA (FIPS 205), a stateless hash-based signature scheme based on $ m SPHINCS^+$. Hash-based digital signature schemes are particularly important because their security is based on the properties of cryptographic hash functions, rather than number-theoretic problems, offering a more robust foundation for post-quantum security.
An important type of digital signature schemes is Group Signature Schemes which enable members of a group to sign messages anonymously on behalf of the group while a designated authority is able to reveal the signer?s identity when necessary hence it ensures accountability. Such functionality is critical in privacy-preserving applications like direct anonymous attestations and reputation systems. Fully dynamic GSSs are especially valuable as they allow users to join or be revoked without requiring system-wide updates?an essential property for real-world scenarios.
In this talk, after introducing digital signatures and their everyday applications, I will review several hash-based group signature scheme proposals, including G-Merkle, DGM, DGMT, and SPHINX-in-the-Head, highlighting their limitations in terms of scalability and efficiency. I will then present DGSP, our newly proposed scalable and efficient fully dynamic group signature scheme, and compare it with existing post-quantum alternatives to demonstrate its advantages.
Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

16 APR 2025
17:30 - 19:00


Venue: Online

9 APR 2025
14:00 - 15:00

A classic result of Vizing tells us the number of colours required for a proper edge-colouring of a graph. More recently, a related problem has been proposed, where the goal is to simultaneously colour the edges of multiple graphs. In this talk, I will give an introduction to the general topic and then present some new bounds for simultaneous edge-colourings. Based on joint work with Simona Boyadzhiyska, Allan Lo and Michael Molloy.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

6 MAR 2025
14:00 - 16:00

Mathematicians have used machines and datasets to aid their research for millennia. In our time, computational tools such as computer algebra systems, SAT/SMT solvers, Reinforcement Learning, and LLMs play an increasingly central role in mathematical discovery. Complementing these, formal verification and proof assistants ensure the correctness of mathematical arguments, providing mathematicians with instant and extremely high confidence in formalized results and enabling large-scale collaboration. This formal proof revolution is transforming how we produce, organize, and interact with mathematical knowledge.
In this talk, I will introduce Lean, a popular proof assistant based on dependent type theory. I will give an overview of Lean's community-based mathematical library Mathlib. I will survey the state of the art in using Lean in mathematical research. Finally, I will highlight Lean's powerful metaprogramming framework, and how we are employing it in a new collaborative project to bring Homotopy Type Theory (HoTT) and synthetic homotopical reasoning into Lean4.

Zoom room information:
https://us06web.zoom.us/j/81916335336?pwd=5zQT4lutMiBoY5Xp1pkFGtbqiaGozg.1
Meeting ID: 819 1633 5336
Passcode: 559618
Venue: Niavaran, Lecture Hall 1

5 MAR 2025
17:30 - 19:00

Seen through the lens of the Minimal Model Program, the classification of algebraic varieties can be summarised into two main steps: firstly, the algorithm of the MMP allows to decompose a variety with mild singularities, birationally, into a tower of fibrations whose general fibres have ample, anti-ample, or numerically trivial canonical divisor. In view of this decomposition, the natural second step to take is to study these 3 classes of algebraic varieties in detail, for example, studying their moduli theory and/or any other property that could shed light on ways to understand all possible elements that belong to such classes. It turns out a very important property to understand in this process is boundedness: a collection of varieties is bounded when the elements of the given collection can be parametrised using a finite type geometric space. The property of boundedness plays a crucial role in the construction of proper moduli spaces of finite type. Moreover, if a given collection of algebraic varieties is bounded (in char 0), then the topological types of their underlying analytic spaces belong to only finitely many homeomorphism classes: hence, all of their topological invariants come in just finitely many possible different versions. While over the past 15 years, several breakthroughs have completely settled the question of boundedness (and the subsequent construction of moduli spaces) in the case of log canonical models (varieties/pairs with ample canonical divisor) and Fano varieties (those with anti-ample canonical divisor), the situation is still quite unclear in the case of trivial numerical divisor. In this seminar, I will try to explain what is known, or not, and what those challenges are that make the situation quite more complicated than the other 2 cases. I will moreover explain how we can overcome most of the issues if we assume that a K-trivial variety is endowed with a fibration structure of relative dimenesion one. The seminar includes results from various works I developed over the past 10 years with G. Di Cerbo, C. Birkar, S. Filipazzi, and C. Hacon. Moreover, I will talk about current work in progress with P. Engel, S. Filipazzi, F. Greer, M. Mauri were we show various new boundedness results for K-trvial varieties fibered in K3 surfaces or abelian varieties.

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Passcode: 362880
Website: http://math.ipm.ac.ir/agnt/
Venue: , (Zoom Lecture)

5 MAR 2025
14:00 - 15:00

Locally recoverable codes (LRCs) play a vital role in distributed storage systems where the failure or unavailability of storage devices is a common occurrence. The purpose of LRCs is to facilitate the repair processes required to recover lost or damaged data in such systems. In this talk, we discuss new constructions of (r, δ)-LRCs. Initially, we provide generator matrices for (r, 2)-LRCs, which is Singleton-Type Bound (STB)-optimal. Also, we present a method for recovering an erased symbol in a codeword of our (r, 2)-LRC. We argue an example illustrating that the Gray image of particular Z_{p^{s+1 }} -LRCs of length n results in LRCs of length np^s over F_p, which may not necessarily be linear. This introduces a novel class of LRCs, prompting further exploration of the connections between existing nonlinear LRCs in finite fields and linear ring-based LRCs. Next, we present the parity-check matrices for another family of (r, δ)-LRCs over Z_{p^s} and construct two instances of STB-optimal (r, δ)-LRCs.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880
Venue: Niavaran, Lecture Hall 1