9 JAN 2025
14:00 - 16:00

We aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the effect of large cardinal assumptions on this comparison. Using the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains an Aronszajn subtree, if there is an inaccessible cardinal. This is stronger than a therm due to Komjath where he proves the same consistency from two inaccessible cardinals. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree T such that if U ⊂ T is a Kurepa tree with the inherited order from T, then U has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: If MAω2 holds and ω2 is not a Mahlo cardinal in L then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's ρ function which might be useful in other contexts. This is a joint work with Stevo Todorcevic.

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

8 JAN 2025
15:30 - 17:00

This will be an expository talk on how algebraic geometry and more specifically Hodge theory have helped to prove some longstanding conjectures in combinatorics and the theory of matroids. These are the results of June Huh, Karim Adiprasito, Eric Katz, Petter Branden, and many more. These results earned June Huh his 2022 Fields Medal.

Online via Zoom:
Meeting ID: 824 7830 2122
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

1 JAN 2025
15:30 - 17:00

In this talk, we are going to uncover a paper written by E. Artin one hundred years ago, in 1924, entitled (in German) "Ein mechanisches System mit quasi-ergodischen Bahnen". "A Mechanical System with Quasi-Ergodic Trajectories " The connection between continued fractions and geodesics in the hyperbolic plane goes back to Humbert (1916) and Smith (1877) and it is used in an ingenious way by E. Artin in the above paper provided the first example of a dense geodesic trajectory on a special Riemann surface, the so-called "modular surface". In the first part of the talk, we will try to cover some preliminaries and generalities about discrete subgroups of the group of isometries of the hyperbolic plane and their relations to the various tessellations of the hyperbolic plane / Poincare disc. In the second part, we will try to talk about a special tessellation called Farey tessellation of the hyperbolic plane and its relation to continued fractions. One of the applications of this connection is the existence of a dense geodesic on the modular surface. If time permits, we will also try to discuss the relationship between the Diophantine approximation of an irrational number and the behavior of the corresponding geodesic on the modular surface.

Online via Zoom:
Meeting ID: 897 3971 8132
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

26 DEC 2024
14:00 - 16:00

It is well-known that the family of automorphisms of any structure constitutes a topological group. In the late 1980s and 1990s, significant progress was made in the understanding of the automorphism group of models of Peano arithmetic (PA). In particular, Richard Kaye provided a landmark characterization of the closed normal subgroups of the automorphism group of a countable recursively saturated model M of PA. His work established an infinite Galois-like correspondence between these closed normal subgroups and invariant initial segments of M. He also conjectured the existence of another correspondence between the non-closed normal subgroups of the automorphism group of M and its invariant initial segments; a conjecture which has remained unproved. In this talk, I will first revisit Kaye�s characterization, and then explore some progress on his conjecture, confirming it in some cases.

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

25 DEC 2024
16:00 - 17:00

This talk is based on a project started since 2016 on representations theory of affine Lie superalgebras. The study of representations over affine Lie (super)algebras has a long history; the first attempt in this regard dates back to 1974 when a certain class of modules over affine Lie algebras were studied and classified by the eminent mathematician, Victor Kac. In this talk, we will discuss different classes of modules over an affine Lie (super)algebra and state the difficulties appearing to obtain a classification and that how we can overcome these difficulties.

Subscribing the Mathematics Colloquium mailing list:
https://groups.google.com/g/ipm-math-colloquium
Venue: Niavaran, Lecture Hall 1

25 DEC 2024
14:00 - 15:00

Recently, motivated by applications in spatial biology, we explored the interactions between color classes in a colored point set from a topological perspective. We introduced MST-ratio as a measure for quantifying the mingling of points with different colors. Investigating this measure raises intriguing questions in discrete geometry, which is the primary focus of this talk.
In this talk, I will introduce the concept of the MST-ratio, present the best-known bounds and key complexity results for computing its maximum, and share new findings on its behavior in both random and arbitrary point sets. Finally, I will highlight several open questions in discrete and stochastic geometry that arise from this work.

Zoom room information:
https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1
Meeting ID: 852 3726 0136
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

12 DEC 2024
14:00 - 16:00

A map $g:{0,1}^n o {0,1}^m$ ($m > n$) is a hard proof complexity generator for a proof system P if and only if for every string $b in {0,1}^msetminus Range(g)$, the formula $ au_b(g)$ naturally expressing $b otin Range(g)$ requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is the Nisan-Wigderson generator. Razborov [Annals of Mathematics 2015] conjectured that if $A$ is a suitable matrix and $f$ is an NP $cap$ Co-NP function hard-on-average for P/poly, then NW$_{{f,A}}$ is a hard proof complexity generator for Extended Frege. In this talk, we prove a form of Razborov's conjecture for AC$^circ$-Frege. We show that for any symmetric NP $cap$ Co-NP function $f$ that is exponentially hard for depth two AC$^circ$ circuits, NW$_{{f,A}}$ is a hard proof complexity generator for AC$^circ$-Frege in a natural setting.

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

11 DEC 2024
14:00 - 15:00

In this talk, we present binomial complexities in sequences of symbols, a branch of combinatorics on words. We explore variations of factor complexity, specifically focusing on k-binomial complexity. Binomial complexities, initially introduced by Rigo and Salimov in 2015, provide insight into the structural properties of sequences by considering equivalence relations based on subword occurrences. The talk covers key findings on bounded binomial complexities observed in Sturmian words, Thue-Morse words, and fixed points of Parikh-collinear morphisms. We also address open questions concerning the relationship between factor complexity and binomial complexity, examining the conditions that contribute to their stability or divergence. The presented research revolves around two main questions: Can factor complexity coincide with binomial complexity under certain conditions, and do there exist words for which these complexities become equivalent at larger k-values? We investigate bounded and unbounded complexities and attempt to describe words that exhibit specific behaviors based on their complexity, using examples like Thue-Morse and Fibonacci words. Ultimately, we aim to uncover how different types of complexity can reveal hidden patterns and structures within infinite words, offering fresh perspectives on the vast field of symbolic sequences.

Zoom room information:
https://us06web.zoom.us/j/85237260136?pwd=MFSZoKdmRXAjfaSaBzbf19lTaaKglf.1
Meeting ID: 852 3726 0136
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

11 DEC 2024
17:30 - 19:00

A Tevelev degree is a type of Gromov-Witten invariant where the domain curve is fixed in the moduli. After reviewing the basic definitions and previously known results, I will report on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto, where we improve the Lian-Pandharipande bound on asymptotic enumerativity of Tevelev degrees of hypersurfaces and provide counterexamples to asymptotic enumerativity for certain other Fano varieties.

https://zoom.us/join
Meeting ID: 9086116889
Passcode: 362880
Email: agnt@ipm.ir
Venue: , (Zoom Lecture)