Lunch seminars for PhD students The problem that I would like to ask about is metamathematical, but I hope the question is appropriate.
I would like to know if there exist mathematical departments that run a regular seminar for all their PhD students both pure and applied. So that you have talks by PhD students ranging from geometry and number theory to statistics and string theory. If such successful seminars exist, I would love to hear the details how this works in practice; what do students speak about, their research or something else; are such seminars run by students themselves or by staff.
 A: The Graduate Student Seminar at IUPUI focuses on one specific topic per semester. These are chosen in consultation with students in order to make them appealing, and to complement the typical core course subjects.
The seminar is semi-formal, with students giving all the lectures... under supervision. This is because part of the goal is to guide students in self-study, and to have them practice giving presentations in a controlled setting (and to lose the fear of asking questions).
A: The University of Pennsylvania had a successful grad-student-organized seminar series called the "Pizza Seminar" when I was there (2005-2010).  It appears to still be active.  Some people talked about their research, others  about just math they found interesting.    Grad-student-only audience, with the occasional faculty speaker.  At the time it probably had more pure talks than applied just because of the department population; it looks like this is still true.
A: At UCLA, there is the GSO (Graduate Student Outreach) seminar, which features talks from graduate students (both pure and applied). The talks are (as far as I am aware) almost never research oriented. This is because speakers are encouraged to make their talks accessible to graduate students from all research groups, which makes the talks less advanced and specific in practice. Speakers often present on topics which are wholly unrelated to their own work or area. One can get a feel for the kind of talks that are given by clicking the link above and perusing some of abstracts. The seminar runs weekly, provided there is someone available to speak, which is typically the case.
I don't think there are any rules prohibiting faculty from attending, but regardless, it is almost exclusively attended by graduate students, with postdocs and undergraduates occasionally appearing among the audience members. It was common for pizzas (or some other kind of snack) to be ordered using departmental support. Afterwards, it wasn't uncommon for some subset of the students attending to go out together. I really enjoyed these seminars when I was a grad student, and my impression is that the GSO seminar is broadly well-liked as an event.
A: This isn't a university, but a tech company I used to work for (~100 engineers) used to do this.
Often the talks would be about the project we happened to be doing. It was a good way for everyone to keep in touch with the various things going on within the organisation.
Often too, the talks might be a more in-depth technical dive into a subject. This would be a fairly quick sketch of the state of the art in some area, some of the history of how we got here and why it's important, and the immediate challenges to push forwards with the field.
And occasionally it was completely unrelated. Sometimes we would have someone doing a quick talk through a hobby such as photography which had no relevance to work but still might be generally of interest.
The key thing to recognize here is that the technical details may not be important. Most people are really bad at communicating, especially in front of a large audience who may not all have the same technical chops or background.  Academics get a job because of their technical ability, but that job also usually involves a teaching role to undergraduates, and most lecturers (I speak here from personal experience on my uni course!) are tragically awful at this part of their job. Some also may prepare a script to read from, but fall apart when asked more free-form questions.
Presenting material to an audience is a skill you can learn though - witness the various organisations such as Toastmasters. If your department's leadership can see the value in this, then it's a great way to upskill your people. Like Toastmasters, it needs some way for the audience to feed back how the presenter can improve though. The learning process has to be two-way: the audience learn about a subject; and the presenter learns how to do better presentations.
A: At Warwick Mathematics Institute in the UK they do indeed have such a seminar, called the Postgraduate Seminar.  It's organised by PhD students for PhD students only and the talks are given by PhD students.
In answer to your question, students can either present updates on research they are working on or they give a survey of some existing work (see, for example, my talk on noncommutative geometry and conformal field theories).
https://warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/postgraduate/
A: If such successful seminars exist, I would love to hear the details how this works in practice; what do students speak about, their research or something else; are such seminars run by students themselves or by staff.
The Math Department of the University of Colorado at Boulder has a seminar like this, run by graduate students. To answer the question, ``What do students speak about'', let me copy a few of the recent titles (= Fall 2021).
Wed, Sep. 1 5pm (MATH 350)  Levi Lorzenzo (CU Boulder)
Connes' Embedding Problem, Kirchberg's conjecture, and MIP*= RE
Thu, Sep. 16
5pm (MATH 350)  Peter Rock (CU Boulder)
Gauss's Greatest Hit (or Another look at Gauss's Theorema Egregium)
Wed, Sep. 22
5pm (MATH 350)  Andrew Campbell (CU Boulder)
Party Hardy Spaces: An Introduction to Complex and Real Analysis
Wed, Oct. 6
5pm (MATH 350)  Chase Meadors (CU Boulder)
The transfinite subway to Hilbert's Hotel
Wed, Oct. 13
5pm (MATH 350)  Emily Montelius (CU Boulder)
Fault Free Tileability of Rectangles, Cylinders, Tori, and Möbius Strips with Dominoes
A: I don't know for how long it has ran, but Peter Neumann started the Kinderseminar at Oxford, specifically for graduates to talk about anything they wanted. The recent past events are here. I used it to talk about algebra mostly, but also combinatorics, cryptography, electrochemistry, social choice theory, and a few other even more random things. Allegedly someone once gave a seminar on the laws of cricket.
It was running since at least the 1980s, probably earlier. I think I remember him once saying that he brought it back from Reinhold Baer's Kinderseminar that ran in Oberwolfach.
After Peter retired, I ran it for a while, until I left Oxford, and it was in turn passed on. It appears to still be running, albeit with a presumably Covid-related suspension. The main original feature was that it did not take place in the departmental building, but in a College office. This tradition at least appears to have now been lost.
A: My undergraduate university (Saint Louis University) hosted talks every semester usually on Fridays where a speaker from another university or sometimes one from the department (whether that was a professor in the department or a grad student talking about their research or an interesting problem they came across) would give a lecture for an hour. Starting at 3:00 was tea and cookies and then the talk would begin 30 to 45 minutes later. Afterwards the speaker and some professors from the department would all go out for dinner.
I was the department bus boy so I'd help set up and facilitate these talks and even sometimes join them for dinner too. They were a lot of fun! It was a great way to learn about lots of different fields and meet people from different backgrounds. This process really helped inform my own graduate school application process. It was very valuable to me to hear all these very intelligent and kind people talk about their research, how they got into it, grad school life, etc. I highly recommend any student participate in such an endeavor if their department has something like this. If not, perhaps you could set one up!
A: When I was there, UT Austin had Sophex, a seminar organized and attended by first- and second-year grad students. I think it was advertised during orientation to let incoming first-years know about it. We'd meet at the end of the day on Friday and walk to the pub near the math building afterward. One of us would speak and another would bring snacks (as with Many Cheerful Facts). The atmosphere was friendly, and I always enjoyed going!
People would speak about things in their research area, things they found fun, or a mix of the two (as with U. Penn's Pizza Seminar). I think they generally succeeded at making their talks approachable for all the attendees, who came from a variety of fields. The past talk titles listed on the seminar page give a sense of what people spoke about.
Speakers could, and sometimes did, invite people other than first- and second-year students to attend their talks. I think they mostly did this when they were talking about things in their research areas, and wanted to share with coworkers in later years. I don't recall anyone inviting faculty when I was there, although I think it would've been fine. I think invited attendees generally participated considerately and tried not to dominate the conversation.
The organizer(s) would typically pass down their duties after one semester, e-mailing out a call for volunteers to replace them. As a result, we had a different organizer or pair of co-organizers each semester (although a co-organizer once became the next sole organizer). Both first- and second-years have organized; I've seen outgoing organizers ask specifically for first-years or rising second-years to replace them.
The seminar page seems to have last been updated in fall 2018, so I don't know whether Sophex is still running. If you've participated, feel free to comment with additions or corrections!
A: Princeton has a Graduate Student Seminar which, at least when I was there, was a lunch seminar offering pizza. It is part of the pure math department but I think people from the program for applied and computational mathematics sometimes attended or spoke.
I attended a seminar that was designed for a mix of mathematicians and physicists (not grad students, mainly postdocs), which seems similar to your idea - based on this I have some thoughts.
(1) Whether such a seminar would work at your university depends a lot on the culture and structure particular departments you want to include. The more interaction there already is between pure and applied math people, the easier it will be. At a minimum you need the pure mathematicians to be interested in learning some applied math and vice versa. Mentioning statistics and string theory suggests you may be interested in more departments than just those two, although maybe applied math at your university includes those.
(2) I think, if you want a seminar uniting two groups, it is vitally important that you make clear to the speaker and the audience that talks should be pitched at whatever group the speaker is not in. So pure mathematicians giving talks should focus on what applied mathematicians can understand and not try to necessarily say anything novel to other pure mathematicians. To help with this, you'll want to try to get people from each side asking questions about the other. The organizers of the seminar can start this off by asking questions of their own (but probably not asking questions on talks whose material they know well).
A: Ohio State runs a seminar called "What is...?" that I think meets your criteria.
From the webpage

The seminar's main goal is to expose culturally ambitious participants
to some mathematical notions not taught in standard courses.  These
topics form an important part of mathematical folklore, and may prove
useful for doing research and enhancing teaching.
Lectures will be given mostly by graduate and undergraduate student
participants.

A: The MIT math department has something pretty much exactly like this, called PuMaGraSS (=Pure Math Grad Student Seminar): https://math.mit.edu/pumagrass/. I co-organized PuMaGraSS one year when I was a PhD student. It has been going on for quite a long time, not sure the complete history. There is also a separate, similar lunch seminar for applied math students, called SPAMS, if I recall correctly.
EDIT: Here is the SPAMS website: https://math.mit.edu/spams/. I guess since MIT breaks the pure and applied into two seminars it does not exactly match your request, but is pretty close.
A: When I was at UC Berkeley, there was a seminar of this sort called "Many Cheerful Facts" (see here if you want to know why). It was organized by graduate student. Every week, one student would be recruited to speak and another would be recruited to bring baked goods, and then everyone ate the baked goods plus whatever lunch they brought and enjoyed the talk. The talks were usually expository about some major result or concept. It was a lot of fun!
A: This has existed at U of C for ages, over a decade.   See
https://math.uchicago.edu/~pizzaseminar/
I can't tell you more, because faculty are not allowed in.
A: 7 years ago I started the StReeTs (Student Research Talks) series at George Mason University.
It is for graduate students and run by graduate students (although I personally ran it for the first few years to get things going).  All subject areas are appropriate.
The seminar runs about twice a month.  There generally is pizza and drinks (the department pays).  Faculty generally do not attend (this is by design so students are not intimidated and are comfortable asking questions).
Talks are usually about 50 minutes with some time for questions at the end (there have been exceptions though; both longer and shorter talks).
Here is the website: https://streets-gmu.wikidot.com/
COVID has shaken the regularity, but it is still running.
Speakers mostly talk about their thesis work, but sometimes they talk about things they are learning about they find interesting that is not covered in foundational course work, and sometimes they bring in graduate students from local universities (our department reimburses local travel in those cases).
Our department is very friendly overall, and I think that students pick up on that, and so our seminars are pretty friendly too.  This one is no exception.
One last thing, we also have the Mason Experimental Geometry Lab (MEGL) which is a community of vertically integrated research teams conducting research explorations and community engagement activities.  We have, in the past, had collaborations between StReeTs and MEGL.
