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Thursday and Friday November 21–22, 2024 at Columbia University with plenary speakers:

Hubert Lacoin (IMPA). Strong disorder and very strong disorder are equivalent for directed polymers

The Directed Polymer in a Random Environment is a statistical mechanics model, which has been introduced (in dimension 1) as a toy model to study the interfaces of the planar Ising model with random coupling constants. The model was shortly afterwards generalized to higher dimensions. In this latter case, rather than an effective interface model, the directed polymer in a random environment can be thought of as modeling the behavior of a stretched polymer in a solution with impurities. The interest in the model model, triggered by its rich phenomenology  has since then generated a plentiful literature in theoretical physics and mathematics. An important topic for the directed polymer is the so-called localization transition. This transition can be defined in terms of the asymptotic behavior of the renormalized partition function of the model. If the finite volume partition function converges to an almost surely positive limit we say that weak disorder holds. On the other hand, if it converges to zero almost surely, we say that strong disorder holds. It has been proved that weak disorder implies that the distribution of the rescaled polymer converges to standard Brownian motion while some localization results have been proved under the strong disorder assumptions. Much stronger characterizations of disorder-induced localization have been obtained under the stronger assumption that the partition function converges to zero exponentially fast. This latter regime is known as the very strong disorder regime. It has long been conjectured that strong and very strong disorder are equivalent. In this talk we will provide an overview of the localization transition picture and sketch a proof of the equivalence of strong and very strong disorder. Joint with Stefan Junk.

Hao Shen (Madison). Quantum Yang-Mills and stochastic analysis

In this talk we will give an overview of the quantum Yang-Mills theory. Quantum Yang-Mills is a model that plays a central role in theoretical physics and poses many challenging questions to probability theory. We will first introduce the motivation and definition of the model, explain what are the fundamental questions that need to be addressed, and then discuss several approaches towards the understanding of different aspects of this model. In particular, depending on time, we will discuss the singular stochastic PDE approach to understand the small scale aspects of the Yang-Mills field, some deterministic analysis approach towards the construction of the state space and observables, the ergodicity approach to understand the large scale behavior of the lattice models, the integrable probability of the special two dimensional case, as well as the scaling limit or universality problems. Some of these methods and techniques are pervasively applicable to other problems in probability, so hopefully general audience will find these methods of interest.

Yilin Wang (IHES). The Brownian loop measure on Riemann surfaces and applications to length spectra

The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. More precisely, we show how the Brownian motion, the Laplace-Beltrami operator, and the lengths of closed geodesics on a hyperbolic surface are connected, and use the connection to obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. This is a joint work with Yuhao Xue (IHES).

Christian Webb (Helsinki). Bosonization in critical and near critical models of statistical field theory

Bosonization is a curious duality occurring in certain two-dimensional field theories. It allows relating bosonic (probabilistic) models with fermionic models. I will review some recent results about bosonization in the setting of the Gaussian free field, the sine-Gordon model, the Ising model, and free fermions. This is based on joint works with R. Bauerschmidt, B. Bayraktaroglu, K. Izyurov, S. Mason, S.C. Park, and T. Virtanen.

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IMPORTANT: REGISTRATION AND COLUMBIA CAMPUS ACCESS

Due to possible campus access restrictions it is necessary that all participants register in advance. YOU MUST REGISTER THROUGH THIS ONLINE FORM BY NOVEMBER 19TH AND BRING YOUR PHOTO ID ALONG WITH THE EMAIL PASS YOU WILL RECEIVE FROM COLUMBIA.

REGISTRATION FORM: https://forms.gle/D1c3LUnD75kSJKta9

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Schedule:

Thursday Nov 21
9-9:45 Refreshments (508 Math)
10-11 Yilin Wang (412 CEPSR)
11-11:15 Bathroom Break
11:15-12:15 Christian Webb (412 CEPSR)
12:15-1:45pm Lunch (Around Campus)
2:00-2:45 Short Talks (412 CEPSR)
2:45-3:00 Bathroom Break
3:00-4:00 Short Talks (412 CEPSR)
4:00-4:45 Tea (508 Math)
4:45-6:00 Short Talks (412 CEPSR)
6:15-7:15 Reception (508 Math)

Friday Nov 22
9-9:45 Refreshments (508 Math)
10-11 Hubert Lacoin (312 Math)
11-11:15 Bathroom Break
11:15-12:15 Hao Shen (312 Math)
12:15-1:45pm Lunch (Around Campus)
1:45-2:55 Short Talks (312 Math)
3:00-3:30 Tea (508 Math)
3:30-4:30 Short Talks (312 Math)

Location for the talks on Thursday will be Columbia University, Davis Auditorium (412 CEPSR), see http://www.cs.columbia.edu/theory/directions.html and all refreshments will be the Mathematics Building, Room 508 (the Cantor Lounge). Talks on Friday will be in the Mathematics Building, Room 312.

Afternoon short-talk schedule + Abstract Book

• Thursday 2:00-2:45: Joseph Chen, Miriam (Mira) Gordin, Seunghyun Lee, Anna Brandenberger
• Thursday 3:00-4:00: Pranav Chinmay, Mauricio Daros Andrade, Maximillian Newman, Malte Hassler, Jingheng Wang
• Thursday 4:45-6:00: Barkat Mian, Richard Groenewald, Brian Chao, Haotian Gu, Jack Piazza, Jinwoo Sung, Yuyang Feng
• Friday 1:45-2:55: Ron Nissim, Benjamin Budway, Andres Riveros, Georgy Gaitsgori, Manuel Arnese, Aaradhya Pandey
• Friday 3:30-4:30: Ran Tao, Jiemiao Chen, Yucheng Guo, Hongyi Chen, Kevin Hu

Women in Probability lunch
Women in Probability is glad to host another event at this year’s Northeast Probability Seminar. This year, we’ll host a lunch on Thursday, November 21. These events are a great opportunity for early career researchers to interact with peers as well as more established researchers. Anyone interested in joining can contact Tai Melcher (melcher@virginia.edu).

Applying for Funding

We gratefully acknowledge the generous funding from the National Science Foundation that makes this event possible. Those interested in receiving financial support for travel and lodging can apply while registering on the registration form linked earlier and must follow the instructions there. (No further funding remains for 2024.)

Applying for Short Talks

Graduate students, postdocs and early career participants are encouraged to apply to give a short talk. Those interested can apply while registering on the registration form linked earlier and must follow the instructions there.

The program was chosen by the seminar’s scientific committee:
Louis-Pierre Arguin, Yuri Bakhtin, Nayantara Bhatnagar, Paul Bourgade, Ivan Corwin, Julien Dubedat, Ramon van Handel, Elena Kosygina, Eyal Lubetzky, Carl Mueller, Daniel Ocone, Robin Pemantle, Victor de la Pena, Jay Rosen, Atilla Yilmaz

Local organizers:
Ivan Corwin, Matthew Junge, Roger Van Peski

Funding Acknowledgment:

Partially funded by NSF grant is DMS:2331449, the Columbia Mathematics Department, and Simons Foundation grant 929852.