September 24, 2023

There are no events currently scheduled for September 24th.

September 25, 2023

• 10:00 - 11:00 am EDT

  Journal Club

  11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:30 - 5:00 pm EDT

  TLN Working Group

  Group Work - 10th Floor Classroom

September 26, 2023

• 9:00 - 10:30 am EDT

  Tutorial

  Tutorial - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

September 27, 2023

• 9:00 - 10:00 am EDT

  Professional Development: Ethics I

  Professional Development - 11th Floor Lecture Hall
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

September 29, 2023

• 8:30 - 8:50 am EDT

  Check In

  11th Floor Collaborative Space
• 8:50 - 9:00 am EDT

  Welcome

  11th Floor Lecture Hall

  • Brendan Hassett, ICERM/Brown University
• 9:00 - 9:45 am EDT

  Formulas for Macdonald polynomials via the multispecies exclusion and zero range processes

  11th Floor Lecture Hall

  • Olya Mandelshtam, University of Waterloo

  Abstract

  We describe some recently discovered connections between one-dimensional interacting particle models and Macdonald polynomials and show the combinatorial objects that make this connection explicit. The first such model is the multispecies asymmetric simple exclusion process (ASEP) on a ring, linked to the symmetric Macdonald polynomials P_\lambda through its partition function, with multiline queues as the corresponding combinatorial object. The second particle model is the multispecies totally asymmetric zero range process (TAZRP) on a ring, which was recently found to have an analogous connection to the modified Macdonald polynomial H_\lambda. The combinatorial objects interpolating between probabilities of the TAZRP and the modified Macdonald polynomials turn out to be tableaux with a queue inversion statistic. We explain the plethystic relationship between multiline queues and queue inversion tableaux, and along the way, derive a new formula for P_\lambda using the queue inversion statistic. This plethystic correspondence is closely related to fusion in the setting of integrable systems.
• 10:00 - 10:30 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 10:30 - 11:15 am EDT

  Dimers in 3D

  11th Floor Lecture Hall

  • Catherine Wolfram, MIT

  Abstract

  A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. Given a compact region R in R^d, consider regions R_n in (1/n) Z^d which approximate R (and some boundary condition). What do random dimer tilings of R_n look like for large n? The 2D version of this question was answered by Cohn, Kenyon, and Propp in 2000. I will talk about how to answer this question in 3D. In both cases, it turns out that a random tiling of R_n is exponentially more likely to lie close to a fixed ""limit shape"" (and more generally, there is a large deviation principle, meaning the probability of lying close to any possible limiting configuration is given by a function called the ""rate function""). While the results are analogous, I will explain that the methods of proof are very different, because many of the key tools for studying dimers are special to two dimensions. This talk is based on https://arxiv.org/abs/2304.08468, which is joint work with Nishant Chandgotia and Scott Sheffield.
• 11:30 am - 12:15 pm EDT

  Dimer models and local systems

  11th Floor Lecture Hall

  • Haolin Shi, Yale University

  Abstract

  The dimer model studies a natural probability measure on the space of perfect matching (""dimer covers"") of an edge-weighted graph; the probability of a dimer cover is proportional to the product of its edge weights. When the graph is bipartite, a useful change in viewpoint is to view the edge weights as defining a ""C∗ local system"", that is a line bundle with connection. This leads us to consider natural generalizations using higher-rank bundles, in particular SLn -local systems. We will talk about a collection of results with emphasis on calculating connection probabilities in double and triple dimer models.
• 12:25 - 12:30 pm EDT

  Group Photo (Immediately After Talk)

  11th Floor Lecture Hall
• 12:30 - 2:30 pm EDT

  Lunch/Free Time
• 2:30 - 3:15 pm EDT

  Diagram algebras

  11th Floor Lecture Hall

  • Rosa Orellana, Dartmouth College

  Abstract

  One of the best-known planar diagram algebras is the Temperley-Lieb algebra. This algebra is defined combinatorially using non-intersecting matchings and can be realized as a centralizer of Lie algebras (or quantum groups) acting on tensor space. They have a wide range of applications, most notably to knot theory and statistical mechanics. The partition algebra is a generalization of the Temperley-Lieb algebra which was proposed by Martin to study higher dimensional statistical models. In this talk I will discuss planar subalgebras of the partition algebras related to the Temperley-Lieb algebra. This is joint work with N. Wallace and M. Zabrocki.
• 3:30 - 4:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 4:00 - 4:45 pm EDT

  Restricted permutations

  11th Floor Lecture Hall

  • Richard Kenyon, Yale University

  Abstract

  We discuss large permutations with restricted permutation matrices, that is, whose permutation matrix has no 1s in some region. We give enumerative results and limit shapes.
• 5:00 - 6:30 pm EDT

  Reception

  11th Floor Collaborative Space

• 11:00 - 11:30 am EDT

  Recurrent network models for predictive processing

  Post Doc/Graduate Student Seminar - 10th Floor Classroom

  • Bin Wang, University of California, San Diego

  Abstract

  Predictive responses to sensory stimuli are prevalent across cortical networks and are thought to be important for multi-sensory and sensorimotor learning. It has been hypothesized that predictive processing relies on computations done by two separate functional classes of cortical neurons: one specialized for “faithful” representation of external stimuli, and another for conveying prediction-error signals. It remains unclear how such predictive representations are formed in natural conditions, where stimuli are high-dimensional. In this presentation, I will present some efforts on characterizing how high-dimensional predictive processing can be performed through recurrent networks. I will start with the neuroscience motivations, define the mathematical models and mention some related mathematical questions that we haven't yet solved along the way.
• 11:30 am - 12:00 pm EDT

  Fishing for beta: uncovering mechanisms underlying cortical oscillations in large-scale biophysical models

  Post Doc/Graduate Student Seminar - 10th Floor Classroom

  • Nicholas Tolley, Brown University

  Abstract

  Beta frequency (13-30 Hz) oscillations are robustly observed across the neocortex, and are strongly predictive of behavior and disease states. While several theories exist regarding their functional significance, the cell and circuit level activity patterns underlying the generation of beta activity remains uncertain. We approach this problem using the Human Neocortical Neurosolver (HNN; hnn.brown.edu), a detailed biophysical model of a cortical column which simulates the microscale activity patterns underlying macroscale field potentials like beta oscillations. Detailed biophysical models potentially offer concrete and biologically interpretable predictions, but their use is challenged by computationally expensive simulations, an overwhelmingly large parameter space, and highly complex relationships between parameters and model outputs. We demonstrate how these challenges can be overcome by combining HNN with simulation based inference (SBI), a deep learning based Bayesian inference framework, and use it to characterize the space of parameters capable of producing beta oscillations. Specifically, we use the HNN-SBI framework to characterize the constraints on network connectivity for producing spontaneous beta. In future work, we plan to compare these predictions to higher level neural models to identify which simplifying assumptions are consistent with detailed models of neural oscillations.
• 1:30 - 3:00 pm EDT

  Topology+Neuro Working Group

  Group Work - 10th Floor Classroom
• 3:00 - 3:30 pm EDT

  Coffee Break

  11th Floor Collaborative Space

September 30, 2023

• 8:30 - 9:15 am EDT

  A Sanov-type theorem for Unimodular Marked Random Graphs, and its applications

  11th Floor Lecture Hall

  • Kavita Ramanan, Brown University

  Abstract

  We establish a large deviation principle in a strong topology for the component empirical measure of several sequences of marked random graph models, including Erdos-Renyi random graphs, random regular graphs, and more general configuration models. We show that the corresponding rate function is given by a relatively tractable formula involving the relative entropy functional. We also describe several applications of this result, such as Gibbs conditioning principles. This talk is based on joint work with I-Hsun Chen and Sarath Yasodharan.
• 9:30 - 10:15 am EDT

  Permutation limits (Permutons)

  11th Floor Lecture Hall

  • Sumit Mukherjee, Columbia University

  Abstract

  Permutation limit theory arises by viewing a permutation as a probability measure on the unit square, and is motivated by dense graph limit theory. Using the theory of permutation limits (permutons), we can compute limiting properties of various permutation statistics for random permutations, such as number of fixed points, number of small cycles, pattern counts, and degree distribution of permutation graphs. We can also derive LDPs for random permutations. Our results apply to many non uniform distributions on permutations, including the the celebrated Mallows model, and mu-random permutations. This is based on joint work with Bhaswar Bhattacharya, Jacopo Borga, Sayan Das and Peter Winkler.
• 10:30 - 11:00 am EDT

  Coffee Break

  11th Floor Collaborative Space
• 11:00 - 11:45 am EDT

  Higher-Order Graphon and Permuton Theories: Fluctuations, Inference, and Degeneracies

  11th Floor Lecture Hall

  • Bhaswar Bhattacharya, University of Pennsylvania

  Abstract

  Motifs (patterns of subgraphs), such as edges and triangles, encode important structural information about the geometry of a network. Consequently, counting motifs in a large network is an important statistical and computational problem. In this talk we will consider the problem of estimating motif densities and fluctuations of subgraph counts in an inhomogeneous random graph sampled from a graphon. We will show that the limiting distributions of subgraph counts can be Gaussian or non-Gaussian, depending on a notion of regularity of subgraphs with respect to the graphon. Using these results and a novel multiplier bootstrap for graphons, we will construct confidence intervals for the motif densities. We will also present parallel results for patterns in random permutations through the lens of permuton theory. Finally, we will discuss various structure theorems and open questions about degeneracies of the limiting distribution.
• 12:00 - 1:30 pm EDT

  Lunch/Free Time
• 1:30 - 2:15 pm EDT

  Interactive workshop on planning a new active learning probability course for research-level students in math-adjacent fields.

  11th Floor Lecture Hall

  • Robin Pemantle, University of Pennsylvania
• 2:30 - 3:00 pm EDT

  Coffee Break

  11th Floor Collaborative Space
• 3:00 - 4:00 pm EDT

  Poster Session

  11th Floor Collaborative Space

  • Slides
• 4:00 - 5:00 pm EDT

  Active Learning Process

  11th Floor Lecture Hall

  • Robin Pemantle, University of Pennsylvania

  Abstract

  The workshop will continue in an optional evening session, where we will make a plan for a course that fits the most populations of those attending the evening session. We will look over some existing materials, plan a shared materials archive, and discuss some of the more crucial mechanics of active classrooms.