27 SEP 2023
16:00 - 17:00

I will start by displaying a model of a free boundary problem which by now belongs to the folklore. Then, I will go over some well-known facts in regard to the Poisson problem with zero Dirichlet boundary conditions. Finally, I will introduce the optimization problems which will lead us to the aforementioned free boundary problems. Some open problems and intriguing questions will be shared as well.

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

27 SEP 2023
14:00 - 15:00

A Hadamard matrix $H$ is said to be skew Hadamard if $H+H^t=2I$, where I is the identity matrix. Skew Hadamard matrices are useful in constructing several combinatorial objects, such as association schemes, self-dual codes, and strongly regular graphs. The notions of type and profile have been proved to be very useful in classification and equivalence check of Hadamard matrices. Here, we define skew type for skew Hadamard matrices. We make use of it to classify some types of skew Hadamard matrices of order $36$. This is a joint work with Hadi Kharaghani and Ali Mohammadian.

https://us06web.zoom.us/j/83407340293?pwd=cXarB4geUkQrP9MMwe8Z71K94wYPpT.1

Meeting ID: 834 0734 0293
Passcode: 362880
Venue: Niavaran, Lecture Hall 1

21 SEP 2023
10:00 - 12:00

Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. The trace of $M$, denoted by $ r_R(M)$, is defined as the sum of ideals $phi(M)$, where the sum is taken over all $R$-module homomorphisms $phi:Mlongrightarrow R$. The trace of a module have been considered in various contexts, in particular to better understand the center of the ring of endomorphisms of a module. When the canonical module $omega_R$ of $R$ exists, the significance of the trace of $omega_R$ arises from the fact that it describes the non-Gorenstein locus of $R$. The study of commutative rings lying between Gorenstein and Cohen-Macaulay rings has been very intense in the last years. There are several different ways to approach this study. One of them was recently initiated by Herzog, Hibi, and Stamate in [2] introducing the notion of nearly Gorenstein ring. A Cohen-Macaulay local ring $R$ admitting a canonical module $omega_R$ is said to be nearly Gorenstein if $ r(omega_R)$ contains the maximal ideal (there is a similar notion for positively graded algebras over a field). In these talks, we study the trace ideal of semidualizing modules, give some characterizations of them and compare them with annihilators of specific Ext modules. Then, two new classes of rings, namely quasi nearly Gorenstein and weakly nearly Gorenstein, are introduced as generalizations of nearly Gorenstein rings; [1].
references:
1. M. Bagherpoor and A. Taherizadeh. Trace ideals of semidualizing modules and two generalizations of nearly Gorenstein rings, Communications in Algebra {51}(2) (2023), 446-463.
2. J. Herzog, T. Hibi, and D. I. Stamate. The trace of the canonical module. Isr. J. Math. 233 (2019), 133-165.
3. H. Lindo, Trace ideals and centers of endomorphism rings of modules over commutative rings, Journal of Algebra {482} (2017), 102-130.
online in Zoom: https://zoom.us/join
Meeting ID: 9086116889
Passcode: 362880
Venue: Niavaran, Lecture Hall 2

16 AUG 2023
15:30 - 17:00

Strongly-correlated quantum systems are often extremely fragile and notoriously hard to control, which poses challenges for possible technological applications. That is why a certain subclass of quantum states, the so-called topological phases of matter, recently attracted much attention. These are characterised by a certain degree of stability and robustness under perturbations, rooted in their special mathematical properties. Apriori, it is not always clear whether a given quantum state of matter is topological or not. We propose a mathematical criterion, which we call ''the geometric test", to tell whether a state of matter is in a topological phase. We then apply our test to strongly-interacting states of matter in Quantum Hall effect, observed in certain 2d materials (Gallium-arsenide, graphene, ...) at low temperatures and in strong magnetic fields. I will explain the idea of the test (which works pretty well) and the results, based on recent and upcoming works with Dimitri Zvonkine (Paris) and with Igor Burban (Paderborn).

https://meet.google.com/you-qymk-ybu
Venue: , (Online)

19 JUL 2023
15:30 - 17:00

A nodal domain refers to a connected region where the eigenfunction is nonzero. The simplest topological invariant of nodal domains is the nodal count. The celebrated Pleijel nodal domain theorem gives an upper bound on the asymptotic behaviour of the nodal count of the Laplace eigenfunctions with Dirichlet boundary condition. There have been advancements in this direction including its extension to the Neumann problem and more generally to the Robin problem with a non-negative parameter as well as improvements of the Pleijel upper bound for the Dirichlet problem. In this talk, we discuss an implication of these results. In particular, we focus on the extension of the Pleijel-type nodal domain theorem to the Robin problem without restriction on the sign of the Robin parameter. This is joint work with David Sher.

Google Meet joining info:
//meet.google.com/fwh-kebs-yci

Here you can find more information about our speaker:
https://research-information.bris.ac.uk/en/persons/asma-hassannezhad/publications/
Venue: , (Online)

5 JUL 2023
16:00 - 17:00

This is a general overview of some of the most prominent Programs in Mathematics. We give a very short (and mainly historical) account of programs proposed by Riemann (1854) Klein (1872) Poincarè (1892) Hilbert (1900/1921) Weil (1949) Langlands (1967) Grothendieck (1984) and Mori (1988). We also review some of the most famous classification programs, such as ?classification of von Neumann algebras? (1936) ?classification of finite simple groups? (1972) and ?classification of separable simple nuclear unital C*-algebras? (1993). We give a short account of two programs proposed for the current century by Smale (1999) and Simon (2000). This is not a technical review and we hope to give a glimpse, or if successful, an essence of all these.

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