We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley’s question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $\beta=1$ Grothendieck polynomials, and provide bounds for general $\beta$.
]]>About a year has passed since I became interested in integrating AI tools into my work tasks (predating the GPT-4
hype by several months). Over time, I have developed several workflows around math teaching and research that could be useful for my colleagues. With the help of these workflows, I experience a significant speedup of many mundane tasks. I describe my tools and workflows here.
Here is the list of tools I use more or less daily:
So, my combined monthly spending on these tools is around $20, which is the same as the cost of just the single ChatGPT Premium subscription.
To create a problem with solution for my undergrad probability course, I follow the steps:
mathpix
to capture the problem from a textbook or an old examVSCode
copilot
autocomplete the solutionAt the solution completion stage, the AI often outputs nonsense. The power of copilot
vs, ChatGPT
is that I can direct the AI solution step by step. Here is an example when I copy the problem from another exam. In this example, I have LaTeX
compilation running in the background in terminal, to update the PDF automatically.
Often, copilot
even tries to do computations for me, but fails miserably. Thus, I also need Mathematica
to check computations. For that, I copy the latex code of an expression to compute, and ask GPT-4
to convert it to Mathematica
code:
heygpt --model=gpt-4 "convert this expression to Mathematica, \
output just the mathematica expression \
\int_0^{\frac{1}{2}}\int_x^\infty 2\lambda e^{-\lambda y}\,dy\,dx " | tee >(pbcopy)
This request is automated through Alfred
. I then copy the expression to Mathematica
, and get the result. In the video below, even the step of the translation from LaTeX
to Mathematica
got wrong at the first try, since the order of integration got messed up. After noticing this and correcting, I got the right answer.
Here is a sample video of this workflow (sped up 2x):
Below is just a sample of tasks where I use AI tools in my research. I am always on the lookout to automate more.
You can ask GPT-4
or ChatGPT
to generate TikZ
code for pictures. Here is an example with an early ChatGPT-4
(April 2023). It generated the tikz code for a picture using this request. After minimal modifications, I put it into a paper - Figure 5 on page 31 in arXiv:2305.17747:
Unfortunately, this method can only generate rather simple pictures. For example, generate tikz code for an example of a six vertex configuration with domain wall boundary conditions
does not produce anything meaningful, but generate tikz code for a 4 times 4 dotted grid with axes (t,x) labeled and a red line from (0,0) to (1,1)
works reasonably fine. Moreover, it can modify your existing TikZ
pictures on the fly, like changing notation in a complicated table of vertex weights. For this I would use a free-form input from a file workflow, see Section 3.3 below.
Mathematica
can output any of its results in LaTeX
using the option TeXForm
. However, sometimes this output is not pretty and I can ask GPT-4
to make it better (for example, remove \left
and \right
braces unless this is strictly needed; organize factors in a large product such that they look like 1-q
instead of -q+1
, and so on). Here is an example of not fully polished Mathematica
output from a 2015 paper arXiv:1502.07374 (pages 26-27). I would prefer 1-q^J
instead of q^J-1
, but got lazy to fix this manually:
In another direction, translating from LaTeX
to Mathematica
was not generally avaiable to me until AI tools. Now, I can copy a piece of LaTeX
code from a paper I currently write, and check in Mathematica
that I did not make any typos. Better yet, I can snap a piece of code from a PDF of any paper I find, and use it for Mathematica
computations.
In this example, I look at arXiv:0905.0679, pick the formula for the weight w(x)
from Section 4, and check that the expression for w(x+1)/w(x)
given in Section 8.2 is indeed correct.
Here is the GPT-4 API
prompt for the second part of this task, where there is a typo in OCR w_{t, S}( & +1) / w_{t, S}(x)
, but GPT-4
does not care, and still produces a decent output:
heygpt --model=gpt-4 "convert this expression to Mathematica, output just the mathematica expression\
$\begin{aligned} w_{t, S}( & +1) / w_{t, S}(x)=\frac{q^{2 N+T-1}\
\left(1-\kappa^2 q^{2 x-t-S+3}\right)}{1-\kappa^2 q^{2 x-t-S+1}}\
\\ & \times \frac{\left(1-q^{x-t-N+1}\right)\left(1-q^{x-S-N+1}\right)\
\left(1-\kappa^2 q^{x-T+1}\right)\left(1-\kappa^2 q^{x-t-S+1}\right)}\
{\left(1-q^{x+1}\right)\left(1-q^{T-S-t+x+1}\right)\left(1-\kappa^2 q^{x+N-t+1}\right)\
\left(1-\kappa^2 q^{x+N-S+1}\right)}\end{aligned}$" | tee >(pbcopy)
There are of course manual caveats:
w_{t, S}( & +1) / w_{t, S}(x)=
from the Mathematica outputN
in Mathematica
is protected, so I need to replace it with NN
manually.Full video of the example is below. It is sped up 2x:
For bibliography, I maintain a giant bibtex file. For many years, I used google scholar
bibtex export feature, but it is imprecise:
google scholar
is doing it inconsistentlyRecently, I stopped using google scholar’s bibtex export, and instead wrote my own bibtex prompt. Basically, I give GPT-4
several examples, and add the info about the paper from two sources which I copypast from the web: arXiv and the journal website. Here is an example of an Alfred workflow which generates and executes a request to GPT-4
:
condition="I want you to make bibtex files in a format, from the data provided.\
Here are examples of my bibtex entries, use this format.\
IMPORTANT: 1. Abbreviate first names of authors, and journal names.\
2. Also, use double curly brackets around title. 3. Remove any month entries.
After END EXAMPLE I will give you data, output only bibtex entry for this data.
BEGIN EXAMPLE
@article{ayyer2022modified,
author = {Ayyer, A. and Mandelshtam, O. and Martin, J.B.},
journal = {arXiv preprint},
note = {arXiv:2209.09859 [math.CO]},
title = {{Modified Macdonald polynomials and the multispecies zero range process: II}},
year = {2022}}
@article{Baxter1972,
author = {Baxter, R. J.},
doi = {10.1016/0003-4916(72)90335-1},
journal = {Annals of Physics},
number = {1},
pages = {193--228},
title = {{Partition function of the Eight-Vertex lattice model}},
volume = {70},
year = {1972}}
@article{onsager1931reciprocal,
author = {Onsager, L.},
journal = {Phys. Rev.},
pages = {405},
publisher = {American Physical Society},
title = {{Reciprocal Relations in Irreversible Processes. I.}},
volume = {37},
year = {1931}}
<...A FEW MORE EXAMPLES...>
END EXAMPLE
remember: IMPORTANT: Abbreviate first names of authors, and journal names.\
Also, double curly brackets around title. And no month entries please.\
I just need the bibtex entry as output, and no comments.
"
e_condition=${(q)condition}
combined_query="${e_condition} {query}"
heygpt --model=gpt-4 "$combined_query" | tee >(pbcopy)
Here is an example of the input as {query}
to the above script. These are just copypastes of pieces from the journal webpage and arXiv:
Home Probability Theory and Related Fields Article
Published: 30 March 2013
Macdonald processes
Alexei Borodin & Ivan Corwin
Probability Theory and Related Fields volume 158, pages225–400 (2014)Cite this article
2990 Accesses
264 Citations
Metricsdetails
arXiv:1111.4408 [math.PR]
You see that this text has lots of garbage data. But ran through the above prompt, this becomes the following bibtex entry:
@article{BorodinCorwin2014,
author = {Borodin, A. and Corwin, I.},
journal = {Prob. Theory Relat. Fields},
note = {arXiv:1111.4408 [math.PR]},
pages = {225-400},
title = {{Macdonald processes}},
volume = {158},
year = {2014}
}
When I go to a conference, I like to have its calendar in my icloud. This way, I can always check my watch to see what is the next talk. Conferences rarely provide .ics
files or google calendars (which would be equally good), and one of the main reasons I see for this is that it’s a pain to nicely display a google calendar on the web. So, math conferences typically resort to one of two terrible things:
LaTeX
table, which is downloaded on click (and not displayed in browser)html
tableI do not know which one is worse for machine readability, but thanks to mathpix
OCR and GPT-4
, I can convert either of them into .ics
, which I can then add to my calendar.
Here is the prompt which more or less works for this conversion:
heygpt --model=gpt-4 "Make ical code for this event.\
This is in San Francisco, CA time zone, Pacific time, winter, year is 2023.\
Output the ical code only. I need the most complete information about\
the event or multiple events. Here is the data to process: {query}" | tee >(pbcopy)
Then I copy the text from the PDF or the webpage using OCR, day by day, and ask to convert it to .ics
. Here is an example video of how this works for one day of a random conference:
I usually do all my prose writing in VSCode
. This includes grant and proposal writing, where I use copilot
autocompletion to break the writing block. Then, I usually polish the final version of the text with Grammarly
on the web, as it suggests readability improvements.
Here is an example of me writing a blog post (this one) in VSCode
:
A large portion of writing is responding to emails. Would it be nice to have long autocomplete suggestions, much longer than what gmail
offers? You can also load several emails at once, and use the available context to generate a response.
I have created an applescript
shortcut (which I call from Alfred
) which can export selected message(s) in the Mail.app
to VSCode
. Of course, I used GPT-4
to come up with the applescript
code.
tell application "Mail"
set theMessages to selection
set theOutput to ""
set totalMessages to count of theMessages
repeat with i from 1 to totalMessages
set aMessage to item i of theMessages
set theOutput to theOutput & "Subject: " & subject of aMessage & return
set theOutput to theOutput & "From: " & (sender of aMessage) & return
set theOutput to theOutput & "To: " & ((address of to recipient of aMessage) as string) & return
set theOutput to theOutput & "Date: " & (date received of aMessage as string) & return
set theOutput to theOutput & "Content: " & content of aMessage & return
set theOutput to theOutput & "---------------------------------------------" & return
log "Processed message " & i & " of " & totalMessages
end repeat
return theOutput
end tell
Example usage, where I open Mail.app
, select a message, and run the hotkey in Alfred
. Then I write the response in VSCode
. I can then just copy the result, and put it in the email as an answer (not shown in the video).
I have an Alfred
workflow to ask GPT-4
arbitrary questions. This is often much faster than google
, when I just need a simple information like
what is the hotkey to record screen on Mac
what is the best video format for web
how to convert video from .mov to .mp4
I also have a workflow to read the question from file, and copy the answer to the clipboard when GPT-4
is done. Here is the heygpt
command for this
heygpt --model=gpt-4 `cat ~/GPT.txt` | tee >(pbcopy)
The | tee >(pbcopy)
part is responsible for copying the output to the clipboard, while at the same time the terminal shows the output. Here is a sample usage, where I write a request into file GPT.txt
, run the hotkey in Alfred
, then topy the result into the same text file, ask for funnier jokes, and run the request again. In the video, hg
is my alias for heygpt --model=gpt-4
.
I can summarize PDFs by cutting them into several chunks and sending the chunks separately to be summarized. The result is a subsequent summary of all chunks, which I can summarize later one more time to get a high-level overview.
This is achieved with this python
script (loosely based on this script, with my modifications which include heygpt
):
import sys
import subprocess
import fitz # This is PyMuPDF
from nltk.tokenize import sent_tokenize
from io import StringIO
def read_pdf(filename):
sys.stderr.write(f"Reading PDF file... {filename}\n")
context = ""
with fitz.open(filename) as pdf_file:
num_pages = len(pdf_file)
sys.stderr.write(f"Number of pages in PDF: {num_pages}\n")
for page_num in range(num_pages):
page = pdf_file[page_num]
page_text = page.get_text('text')
context += page_text
sys.stderr.write(f"Finished reading page {page_num + 1}/{num_pages}\n")
return context
def split_text(text, chunk_size=5000):
sys.stderr.write("Splitting text into chunks...\n")
chunks = []
current_chunk = StringIO()
current_size = 0
sentences = sent_tokenize(text)
for sentence in sentences:
sentence_size = len(sentence)
if sentence_size + current_size <= chunk_size:
current_chunk.write(sentence)
current_size += sentence_size
else:
chunks.append(current_chunk.getvalue())
current_chunk = StringIO(sentence)
current_size = sentence_size
sys.stderr.write(f"Created a new chunk. Total chunks so far: {len(chunks)}\n")
if current_chunk.getvalue():
chunks.append(current_chunk.getvalue())
sys.stderr.write(f"Total chunks created: {len(chunks)}\n")
return chunks
def heygpt_completion(prompt, model_name="--model=gpt-4-1106-preview"):
sys.stderr.write("Sending prompt to 'heygpt' subprocess...\n")
try:
result = subprocess.run(['heygpt', model_name], input=prompt, text=True, capture_output=True, check=True)
return result.stdout.strip()
except subprocess.CalledProcessError as e:
error_message = f"heygpt error: {e}\n"
sys.stderr.write(error_message)
return error_message
def summarize(document):
sys.stderr.write("Summarizing document...\n")
chunks = split_text(document)
summaries = []
for chunk in chunks:
prompt = "Please summarize the following text:\n" + chunk
sys.stderr.write("Sending chunk to be summarized...\n")
summary = heygpt_completion(prompt)
summaries.append(summary)
sys.stderr.write(f"Received summary for chunk: {len(summaries)}\n")
sys.stderr.write("Completed summarization of all chunks.\n")
return ' '.join(summaries)
if __name__ == "__main__":
if len(sys.argv) < 2:
sys.stderr.write("Usage: python script.py <pdf_filename>\n")
sys.exit(1)
pdf_filename = sys.argv[1]
try:
sys.stderr.write(f"Starting the summarization process for {pdf_filename}\n")
document = read_pdf(pdf_filename)
summarized_content = summarize(document)
print("Final summarized content:\n")
print(summarized_content)
except FileNotFoundError:
sys.stderr.write(f"File '{pdf_filename}' not found.\n")
except Exception as e:
sys.stderr.write(f"An error occurred: {e}\n")
If I need to look up MSC2020 (Math subject classification) codes by code, or have suggestions for such codes for an abstract of a paper, I use the following heygpt
prompt in Alfred
:
hg "1. If the input is a code or a few codes, look up in the MSC 2020 (math subject classification database)
and output the descriptions of the codes.
2. However, if the input is a mathematical text, look up the MSC 2020 (math subject classification database)
and output suggested primary and up to 5 secondary MSC codes, with summaries.
Here is the input: {query}"
The output on the abstract of this recent paper is quite appropriate:
Primary MSC code:
- **60K35** (Interacting random processes; statistical mechanics type models; percolation theory):
This code would be appropriate since the paper focuses on stationary measures for interrelated particles
in systems from a probabilistic and statistical mechanics perspective.
Secondary MSC codes:
- **82B23** (Exactly solvable models; Bethe ansatz):
Since the paper mentions the use of integrable stochastic vertex models and the Yang-Baxter equation,
this code seems relevant.
- **82C22** (Interacting particle systems in time-dependent statistical mechanics):
The work is centered around interacting particle systems, like the mASEP, TAZRP, and q-PushTASEP,
making this code appropriate.
- **81R50** (Quantum groups and related algebraic methods):
The paper uses algebraic structures such as the Yang-Baxter equation, which is closely related to quantum groups.
- **60J27** (Continuous-time Markov processes on discrete state spaces):
If the paper discusses Markov processes for modeling the time evolution of the systems, this would be a fitting code.
- **05E05** (Symmetric functions and generalizations):
If the paper details connections with nonsymmetric Macdonald polynomials,
which are a generalization of symmetric functions, this code could be relevant.
And here is the output if I just ask Alfred
to execute cmsc 60K35
:
1. The code 60K35 from the MSC 2020 database corresponds to:
**60K35**: Interacting random processes; statistical mechanics type models; percolation theory
This area of mathematics concerns with random processes that have interaction terms
or dependencies between them, which are often analyzed using methods from statistical mechanics.
Examples might include processes such as Ising models, exclusion processes,
and various models of percolation. These models and theories help to understand how
the structure and behavior at a local level can affect the system's global properties,
often studying phenomena like phase transitions and critical behavior.
Based on the joint work with Amol Aggarwal and Matthew Nicoletti.
]]>In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multispecies ASEP, or mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the q-Boson particle system; (3) the q-deformed Pushing Totally Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang-Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang-Baxter equation.
For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin. For the colored q-Boson process and the q-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer-Mandelshtam-Martin (1, 2, and Bukh-Cox. Our proofs of stationarity use the Yang-Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac-Evans-Mallick).
On the line and in a quadrant, we use the Yang-Baxter equation to establish a general colored Burke’s theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity.
]]>Solvable lattice models originated in mathematical physics (Ising model, square ice). They found numerous applications to combinatorics (enumerative and algebraic - keywords are nonsymmetric Macdonald polynomials and Hecke algebras), representation theory (keyword - quantum groups), and more recently, probability. We will start with classical results, such as the Izergin-Korepin determinant and the entropy of the square ice. We’ll select further topics based on the audience’s interests.
August 29, 2023
Leo Petrov - Introduction
September 5, 2023
Nick Sweeney - The Alternating Sign Matrix conjecture
after Chapter 1 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud
September 12, 2023
Petch Chueluecha - Dodgson condensation
after Section 3.5 of the book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David M. Bressoud
September 19, 2023
Mikhail Tikhonov - The Yang-Baxter equation and the Izergin-Korepin determinant
after Wheeler-Zinn-Justin 2015 (Section 4.2 with $u=1$) and Petrov 2020 (Yang-Baxter equation is Proposition 2.1; and the determinant is proven in Section 3.4 with $\gamma=1$, $s_0=0$)
September 26, 2023
Leo Petrov - From the Izergin-Korepin determinant to the number of alternating sign matrices
October 10, 2023
Jacob Campbell - R-matrices from quantum groups (after Kassel “Quantum Groups”, 1995, Section 8.1); see also this part on the definition of the quantum $sl_2$
October 17, 2023
Suren Kyurumyan - The stochastic six-vertex model (after Gorin-Nicoletti’s lecture notes, Section 4.1)
October 24, 2023
Break
October 31, 2023
Leo Petrov - From the stochastic six-vertex model to asymptotics (after Gorin-Nicoletti’s lecture notes, Section 5; and also papers on Schur measures)
17-21 • Pasadena, CA • AIM workshop “All roads to the KPZ universality class.”
16-20 • CIRM, Marseille, France • Conference in honor of Richard Kenyon
7-11 July 2025 • Angers, France • InterPlay between Integrable Probability and Interacting Particle Systems
]]>By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.
]]>