7 DEC 2023
11:00 - 12:30

The Ehrenfeucht-Fraïssé games and their variants are a natural game-theoretic tool for investigating the similarity and dissimilarity between two mathematical structures. Although the study of these games is quite different in finite and infinite model theory, one might ask: if not all natural numbers are equal, why should all infinities be equal? In fact, it is within the framework of countably infinite model theory, not infinite model theory, that these games are often successful!
Even at the level of the first uncountable cardinal, the picture is very mixed, but this is not surprising: The Sikorski extension theorem vs. non-trivial forcing axioms, countability and uncountability in Shelah's classification theory, descriptive set theory vs. generalised descriptive set theory.

In this series of lectures (for some n> 1) I will talk about an area of research that started in the early 1990s and is an intersection of several independent topics.
The lectures will start with a brief introduction to infinite games, then continue with an introduction to the classical Ehrenfeucht-Fraïssé games, and then look at these games for uncountable structures. We will see how they lead to large Aronszajn trees and the notion of weak embeddability between them. The notion of weak embeddability between ordered structures goes back to the early post-Cantor era (e.g. in a 1915 paper by Hartogs), possibly in a different terminology. The notion has attracted attention in various mathematical contexts after being explored and popularised by Todorcevic, Mekler and Väänänen. Part of the talk will be devoted to this topic. I will discuss the concept of weak embeddability between ordered structures, emphasising its historical significance and mathematical importance. I will mention recent progress on the famous problem of a universal tree in the class of (wide) Aronszajn trees. If time permits, I will also present a proof sketch showing that the universality number of (wide) Aronszajn trees is never the first uncountable cardinal number.


Venue: Niavaran, Lecture Hall 2

6 DEC 2023
14:00 - 15:00

In this talk, we will begin by reviewing the key findings and providing an update on the famous Hedetniemi conjecture. We shall demonstrate how this conjecture links Combinatorics with Topology and subsequently, we will provide our latest discoveries on the Topological version of this conjecture.

This week's talk will be on our Zoom room:

https://us06web.zoom.us/j/83407340293?%20pwd=cXarB4geUkQrP9MMwe8Z71K94wYPpT.1
Meeting ID : 834 0734 0293
Passcode : 362880
Venue: Niavaran, Lecture Hall 1

30 NOV 2023
11:00 - 12:30

The Ehrenfeucht-Fraïssé games and their variants are a natural game-theoretic tool for investigating the similarity and dissimilarity between two mathematical structures. Although the study of these games is quite different in finite and infinite model theory, one might ask: if not all natural numbers are equal, why should all infinities be equal? In fact, it is within the framework of countably infinite model theory, not infinite model theory, that these games are often successful! Even at the level of the first uncountable cardinal, the picture is very mixed, but this is not surprising: The Sikorski extension theorem vs. non-trivial forcing axioms, countability and uncountability in Shelah's classification theory, descriptive set theory vs. generalised descriptive set theory. In this series of lectures (for some n> 1) I will talk about an area of research that started in the early 1990s and is an intersection of several independent topics. The lectures will start with a brief introduction to infinite games, then continue with an introduction to the classical Ehrenfeucht-Fraïssé games, and then look at these games for uncountable structures. We will see how they lead to large Aronszajn trees and the notion of weak embeddability between them. The notion of weak embeddability between ordered structures goes back to the early post-Cantor era (e.g. in a 1915 paper by Hartogs), possibly in a different terminology. The notion has attracted attention in various mathematical contexts after being explored and popularised by Todorcevic, Mekler and Väänänen. Part of the talk will be devoted to this topic. I will discuss the concept of weak embeddability between ordered structures, emphasising its historical significance and mathematical importance. I will mention recent progress on the famous problem of a universal tree in the class of (wide) Aronszajn trees. If time permits, I will also present a proof sketch showing that the universality number of (wide) Aronszajn trees is never the first uncountable cardinal number.
Venue: Niavaran, Lecture Hall 2

29 NOV 2023
17:30 - 19:00

The Nonvanishing Conjecture and the Abundance Conjecture are longstanding open problems in the Minimal Model Program. I am going to present some unexpected generalizations which appeared in the literature in the last few years and to discuss a few variants of them.

https://zoom.us/join
Meeting ID: 9086116889
Passcode: 362880
Venue: , (Online)

29 NOV 2023
15:30 - 17:00

The (one-phase) Bernoulli problem deeply influenced the modern free boundary regularity theory and is still an object of intensive research. Free boundary problems are often PDE-described problems with a priori unknown (free) interfaces or boundaries. Such types of problems appear in Physics, Geometry, Probability, Biology, or Finance, and the study of solutions and free boundaries uses methods from PDE, Calculus of Variations, and Geometric Measure Theory. We first review the very classical theory, which is about how to understand the regularity of free boundaries. Then, we present some recent results for the Bernoulli problem with the p-Laplace operator.
Venue: Niavaran, Lecture Hall 1

29 NOV 2023
14:00 - 15:00

As the role of robots in human life is broadening, roboticists face new planning problems that are beyond the classical motion planning problem, which aims to plan a sequence of actions that moves a robot from a source location to a goal location. This talk presents several new planning problems with complex goals. All these problems are solved using automata-theoretic approaches. In these approaches, an optimal plan is synthesized on a product automaton that is constructed from a discrete structure that models all possible behaviors of the system and an automaton that specifies all the desired behaviors for planning. The first problem aims to choose a minimum number of sensors that are placed in the environment such that using the chosen sensors, the system can determine whether an agents execution in the environment fits a pre-disclosed itinerary or not. The second problem focuses on planning to optimally observe an unpredictable environment to capture a sequence of events that meets specification provided by the user. This problem is useful when a robot is tasked with automatically recording a structured video or a documentary. The next problem considers optimal temporal logic planning given both hard and soft specifications of the mission where the soft specifications might be conflicting. The challenge lies in choosing an optimal selection of the soft constraints to satisfy. The ideas of the last two problems are combined to define and solve a multi-objective temporal logic planning problem. Finally, the talk discusses several future directions of these problems.

https://us06web.zoom.us/j/83407340293?pwd=cXarB4geUkQrP9MMwe8Z71K94wYPpT.1
Meeting ID : 834 0734 0293
Passcode : 362880

Venue: Niavaran, Lecture Hall 1

29 NOV 2023
12:45 - 13:45

We investigate some properties such as nuclearity, local lifting property and weak expectation property in the operator space.
Venue: Niavaran, Lecture Hall 1