Seminars and Colloquia by Series

Random matrix theory and supersymmetry techniques

Series
Job Candidate Talk
Time
Monday, January 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tatyana ShcherbynaPrinceton University
Starting from the works of Erdos, Yau, Schlein with coauthors, significant progress in understanding universal behavior of many random graph and random matrix models were achieved. However for random matrices with a spatial structure, our understanding is still very limited.  In this talk I am going to overview applications of another approach to the study of the local eigenvalue statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). The SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY  to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. 
 

Geometry and analysis of degenerating Calabi-Yau manifolds

Series
Job Candidate Talk
Time
Thursday, December 5, 2019 - 12:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruobing ZhangSUNY Stony Brook

This talk concerns a naturally occurring family of Calabi-Yau manifolds that degenerates in the sense of metric geometry, algebraic geometry and nonlinear PDE. A primary tool in analyzing their behavior is the recently developed regularity theory. We will give a precise description of arising singularities and explain possible generalizations. 

Inferring computation from structure in neuronal networks

Series
Job Candidate Talk
Time
Thursday, December 5, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hannah ChoiUniversity of Washington

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. Specifically, neuronal networks at multiple scales utilize their structural complexities to achieve different computational goals. In this talk, I will discuss functional implications that can be inferred from the architecture of brain networks.

The first part of the talk will focus on a generalized problem of linking structure and dynamics of the whole-brain network. By simulating large-scale brain dynamics using a data-driven network of phase oscillators, we show that complexities added to the spatially embedded brain connectome by idiosyncratic long-range connections, enable rapid transitions between local and global synchronizations. In addition to the spatial dependence, I will also discuss hierarchical structure of the brain network. Based on the data-driven layer-specific connectivity patterns, we developed an unsupervised method to find the hierarchical organization of the mouse cortical and thalamic network. The uncovered hierarchy provides insights into the direction of information flow in the mouse brain, which has been less well-defined compared to the primate brain.

Finally, I will discuss computational implications of the hierarchical organization of the brain network. I will focus on a specific type of computation – discrimination of partially occluded objects— carried out by a small cortical circuitry composed of an intermediate visual cortical area V4 and its efferent prefrontal cortex. I will explore how distinct feedforward and feedback signals promote robust encoding of visual stimuli by leveraging predictive coding, a Bayesian inference theory of cortical computation which has been proposed as a method to create efficient neural codes. We implement a predictive coding model of V4 and prefrontal cortex to investigate possible computational roles of feedback signals in the visual system and their potential significance in robust encoding of nosy visual stimuli.

In sum, our results reveal the close link between structural complexity and computational versatility found in brain networks, which may be useful for developing more efficient artificial neural networks and neuromorphic devices.

Multiscale analysis of sets and measures

Series
Job Candidate Talk
Time
Tuesday, November 19, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben JayeClemson University

In this talk I will give an introduction to certain aspects of geometric Littlewood-Paley theory, which is an area of harmonic analysis concerned with deriving regularity properties of sets and measures from the analytic behavior of associated operators. The work we shall describe has been carried out in collaboration with Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa, and Michele Villa.

Translation and Systems Biology: Mathematical and computational modeling at the frontier of biomedical research

Series
Job Candidate Talk
Time
Thursday, February 7, 2019 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gesham MagombedzeBaylor Institute for Immunology Research

A major challenge in clinical and biomedical research is on translating in-vitro and in- vivo model findings to humans. Translation success rate of all new compounds going through different clinical trial phases is generally about 10%. (i) This field is challenged by a lack of robust methods that can be used to translate model findings to humans (or interpret preclinical finds to accurately design successful patient regimens), hence providing a platform to evaluate a plethora of agents before they are channeled in clinical trials. Using set theory principles of mapping morphisms, we recently developed a novel translational framework that can faithfully map experimental results to clinical patient results. This talk will demonstrate how this method was used to predict outcomes of anti-TB drug clinical trials. (ii) Translation failure is deeply rooted in the dissimilarities between humans and experimental models used; wide pathogen isolates variation, patient population genetic diversities and geographic heterogeneities. In TB, bacteria phenotypic heterogeneity shapes differential antibiotic susceptibility patterns in patients. This talk will also demonstrate the application of dynamical systems in Systems Biology to model (a) gene regulatory networks and how gene programs influence Mycobacterium tuberculosis bacteria metabolic/phenotypic plasticity. (b) And then illustrate how different bacteria phenotypic subpopulations influence treatment outcomes and the translation of preclinical TB therapeutic regimens. In general, this talk will strongly showcase how mathematical modeling can be used to critically analyze experimental and patient data.

The SQG equation

Series
Job Candidate Talk
Time
Thursday, January 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Javier Gómez-SerranoPrinceton University
There has been high scientific interest to understand the behavior of the surface quasi-geostrophic (SQG) equation because it is a possible model to explain the formation of fronts of hot and cold air and because it also exhibits analogies with the 3D incompressible Euler equations. It is not known at this moment if this equation can produce singularities or if solutions exist globally. In this talk I will discuss some recent works on the existence of global solutions.

Chaotic regimes for random dynamical systems

Series
Job Candidate Talk
Time
Friday, January 18, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alex BlumenthalUniv. of Maryland

It is anticipated that chaotic regimes (characterized by, e.g., sensitivity with respect to initial conditions and loss of memory) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way.

Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

Matrix Estimation with Latent Permutations

Series
Job Candidate Talk
Time
Thursday, January 17, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cheng MaoYale University
A wide variety of applied tasks, such as ranking, clustering, graph matching and network reconstruction, can be formulated as a matrix estimation problem where the rows and columns of the matrix are shuffled by a latent permutation. The combinatorial nature of the unknown permutation and the non-convexity of the parameter space result in both statistical and algorithmic challenges. I will present recent developments of average-case models and efficient algorithms, primarily for the problems of ranking from comparisons and statistical seriation. On the statistical side, imposing shape constraints on the underlying matrix extends traditional parametric approaches, allowing for more robust and adaptive estimation. On the algorithmic front, I discuss efficient local algorithms with provable guarantees, one of which tightens a conjectured statistical-computational gap for a stochastically transitive ranking model.

Network data: Modeling and Statistical Analysis

Series
Job Candidate Talk
Time
Thursday, January 10, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Subhabrata SenMIT
Network data arises frequently in modern scientific applications. These networks often have specific characteristics such as edge sparsity, heavy-tailed degree distribution etc. Some broad challenges arising in the analysis of such datasets include (i) developing flexible, interpretable models for network datasets, (ii) testing for goodness of fit, (iii) provably recovering latent structure from such data.In this talk, we will discuss recent progress in addressing very specific instantiations of these challenges. In particular, we will1. Interpret the Caron-Fox model using notions of graph sub-sampling, 2. Study model misspecification due to rare, highly “influential” nodes, 3. Discuss recovery of community structure, given additional covariates.

Pages