This seminar series gives visitors and staff members the opportunity to explain the context and aims of their work. These research talks cover any field in the mathematical sciences, and should be presented in a way that is understandable and interesting to a broad audience. Seminar information and recordings can be found below and in the SMRI Seminar YouTube playlist.
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SMRI Seminar ‘Scalable Operators’
Nigel Higson, Pennsylvania State University
Thursday 12 September 2024
Abstract: The operation of integration is almost – but not quite – the inverse of differentiation. In higher dimensions, the problem of determining similar
almost-inverse relations, involving for instance the Laplace operator or the Dirac operator, very often on curved spaces, is commonly encountered, and the theory of
pseudodifferential operators was created to solve it. I shall give an introduction to the simple, elegant and geometric approach to pseudodifferential operators that was
recently developed by Erik van Erp and Robert Yuncken. It uses what I call scalable operators. The theory of scalable operators offers a convenient and coordinate-free
means of studying algebras of pseudodifferential operators on geometric spaces, as I shall try to illustrate using the example of symmetric spaces.
SMRI Seminar ‘Learning of, for, and by dynamical systems’
Nisha Chandramoorthy, The University of Chicago
Thursday 29 August 2024
Abstract: Across scientific and engineering applications, we are interested in sampling typical or observable states achieved by complex dynamical systems. To generate such samples, we are often given approximate numerical models and data at various fidelities and time/length scales. Our goals are to develop rigorous computations that can scale to high-dimensional settings like atmospheric and oceanic dynamics and aerospace systems to answer questions such as: i) how can we reduce the dimension of Bayesian sampling algorithms even in complex nonlinear systems? ii) how do small perturbations in the dynamical model/equations affect its long-term behavior? iii) how can we learn effective low-dimensional representations of the dynamics so as to predict desired quantities? We will look at low-dimensional examples of chaotic systems that illustrate why answering these questions is challenging and explore principled approaches to circumvent these challenges.
SMRI Seminar ‘Totality of rational points on modular curves’
Jun Yong Park (June Park), The University of Sydney
Thursday 15 August 2024
Abstract: People want to count elliptic curves over global fields such as the field Q of rational numbers or the field Fq(t) of rational functions over the finite field Fq. To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}. In this talk, I will explain the exact counting formula for all elliptic curves over Fq(t) along with an explanation for the geometric origin of lower order main terms, as well as basic generalities, relevant ideas and methods.
SMRI Seminar ‘The Missing Link: Establishing the Parallels Between Censored Covariate and Missing Data’
Tanya P. Garcia, UNC Chapel Hill
Thursday 27 June 2024
Abstract: For years, researchers have focused on outcomes where the exact timing is uncertain. Now, there is a growing interest in understanding factors that influence timing but are not fully known, termed covariates. Until now, these two scenarios have been treated separately, ignoring potential overlap in methodologies.
We address this gap by establishing connections between them, allowing us to identify similarities and determine when methods can be applied interchangeably. This connection has led to the development of five novel approaches for handling situations where some factors are known but their timing might depend on when data collection stopped, a phenomenon termed informative covariate censoring. We rigorously tested these methods to assess their robustness and efficiency under various assumptions, even when those assumptions may not be entirely accurate.
We also explored their asymptotic properties and proposed a hypothesis test for evaluating the informativeness of covariate censoring. To validate these approaches, we conducted empirical studies using data from a study on Huntington’s disease. Specifically, we examined cognitive decline leading up to clinical diagnosis, utilizing the newly developed methods to analyze the data. This real-world application allows us to assess the performance of these methods in a practical setting, demonstrating their robustness and efficiency in handling complex datasets.
Overall, our work not only introduces innovative techniques for addressing covariate-related challenges but also provides valuable insights into cognitive decline in Huntington’s disease, showcasing the practical relevance of our findings.
SMRI Seminar ‘Morse theory for eigenvalues of self-adjoint families’
Greg Berkolaiko, Texas A&M University
Thursday 16 May 2024
Abstract: The question of optimizing an eigenvalue of a family of self-adjoint operators that depends on a set of parameters arises in diverse areas of mathematical physics. Among the particular motivations for this talk are the Floquet-Bloch decomposition of the Schroedinger operator on a periodic structure, nodal count statistics of eigenfunctions of quantum graphs, conical points in potential energy surfaces in quantum chemistry and the minimal spectral partitions of domains. In each of these problems one seeks to identify and/or count the critical points of the eigenvalue with a given label (say, the third lowest) over the parameter space which is often known and simple, such as a torus.
Classical Morse theory is a set of tools connecting the number of critical points of a smooth function on a manifold to the topological invariants of this manifold. However, the eigenvalues are not smooth due to presence of eigenvalue multiplicities or diabolical points ”. We rectify this problem for eigenvalues of generic families of finite-dimensional operators. The diabolical contribution ” to the Morse indicies ” of the problematic points turns out to be universal: it depends only on the multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family. Using tools such as Clarke subdifferential and stratified Morse theory of Goresky–MacPherson, we express the diabolical contribution ” in terms of homology of Grassmannians of appropriate dimensions.
SMRI Seminar ‘Optimization for Spacial Design Problems’
Hung Phan, University of Massachusetts, Lowell
Thursday 16 May 2024
Abstract: In computer graphic applications or design problems, a three dimensional object is often represented by a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy spatial constraints that are imposed either by observation from the real world, or by concrete design specifications of the object. In this talk, we model this design problem as a convex optimization problem, then apply splitting algorithms to find optimal solutions.
SMRI Seminar ‘A bystander contemplating the local Langlands correspondence’
Uri Onn, The Australian National University
Thursday 9 May 2024
Abstract: In this informal survey talk I will focus on a finite analogue of the local Langlands correspondence proved by Macdonald in 1980, and possible ramifications.
SMRI Seminar ‘The generalized knot complement problem’
Dave Futer, Temple University, Philadelphia
Thursday 2 May 2024
Abstract: In 1908, Tietze posed the provocative question of whether a mathematical knot – a knotted rope, with its ends spliced together – is completely determined by the topological shape of the empty space that surrounds the knot. After being open for 80 years, this question was solved in 1988, and we know the answer is Yes. Since that time, mathematicians have studied the open question of whether the same property holds for knots in general 3-dimensional spaces. Given a 3-dimensional manifold M, is a knot in M entirely determined by the space that surrounds it in M? I will discuss some recent work on this question using hyperbolic (negatively curved) geometry. This is joint work with Jessica Purcell and Saul Schleimer.
SMRI Seminar ‘McKay correspondance and toric geometry’
Will Donovan, Tsinghua University
Thursday 11 April 2024
Abstract: Finite subgroups of the matrix group SU(2) may be studied algebraically via their representations. They may also be studied geometrically via two-dimensional complex manifolds naturally associated to them. The McKay correspondence is a general phenomenon which, in particular, explains how these two approaches relate. I’ll introduce this using diagramatics from toric geometry, indicate how the correspondence generalizes to higher dimensions, and discuss open questions and current projects.
SMRI Seminar ‘Derandomizing Algorithms via Spectral Graph Theory’
Salil Vadhan, Harvard University
Thursday 18 April 2024
Abstract: Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability. However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).
In this talk, I will describe an approach to proving the space (as opposed to time) part of this conjecture via spectral graph theory. Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree). If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved. These problems also lead us to explore new notions of what it means for two directed graphs to “spectrally approximate” each other, which may be of independent interest.
Joint works with Jack Murtagh, Omer Reingold, Aaron Sidford, AmirMadhi Ahmadinejad, Jon Kelner, John Peebles, and Ted Pyne. Watch the recording.
SMRI Seminar ‘Sperner’s Lemma: a generalization with surprising applications’
Francis Su, Harvey Mudd College
Thursday 14 March 2024
Abstract: Who doesn’t like one of these three: geometry, topology, and combinatorics? And even if you don’t, you will still love Sperner’s lemma, which is a combinatorial statement that is equivalent to the Brouwer fixed point theorem in topology. I’ll explain what it is, why it’s so amazing, give heartwarming old and new proofs, and present a recent generalization to polytopes that has surprised me with diverse applications: to the study of triangulations, to fair division problems, and the Game of Hex. Watch the recording.
SMRI Seminar ‘A Tutorial and Perspectives on Monte Carlo Simulation Optimization’
Shane G. Henderson, Cornell University
Thursday 7 March 2024
Abstract: I provide a tutorial and some perspectives on simulation optimization, in which one wishes to minimize an objective function that can only be evaluated with noise through a stochastic computer simulation. First, I’ll give a few examples and intuitively explain some central issues in the area. Second, I’ll explain why so-called sample-path functions can exhibit extremely complex behavior that is well worth understanding. Third, I’ll argue that more attention should be devoted to the finite-time performance of solvers than on ensuring convergence properties that may only arise in asymptotic time scales that may never be reached in practice. I’ll outline an approach for obtaining such results analytically (through Lyapunov functions) and introduce a framework and code for computational experiments that can further this goal. Fourth (if time permits, though I doubt it will), I’ll advocate the use of a layered approach to formulating and solving optimization problems, whereby a sequence of models are built and optimized, rather than first building a simulation model and only later “bolting on” optimization, partly through an example of my work involving bike sharing with the organization Citi Bike in New York city.
Watch the recording.
SMRI Seminar ‘State-space models as graphs’
Víctor Elvira, University of Edinburgh
Thursday 29 February 2024
Abstract: Modeling and inference in multivariate time series is central in statistics, signal processing, and machine learning. A fundamental question when analyzing multivariate sequences is the search for relationships between their entries (or the modeled hidden states), especially when the inherent structure is a directed (causal) graph. In such context, graphical modeling combined with sparsity constraints allows to limit the proliferation of parameters and enables a compact data representation which is easier to interpret in applications, e.g., in inferring causal relationships of physical processes in a Granger sense. In this talk, we present a novel perspective consisting on state-space models being interpreted as graphs. Then, we propose novel algorithms that exploit this new perspective for the estimation of the linear matrix operator and also the covariance matrix in the state equation of a linear-Gaussian state-space model. Finally, we discuss the extension of this perspective for the estimation of other model parameters in more complicated models. Watch the recording.
SMRI Seminar ‘Symmetry, old and new’
Daryl Cooper, The University of California, Santa Barbara
Thursday 22 February 2024
Abstract: I will discuss symmetry from a combinatorial perspective. Examples include wallpaper groups, 4-valent graphs, regular languages, molecules, Penrose tilings, and geometric 3-manifolds. It turns out that for each of these classes there is a finite universal geometric object that encodes all the possibilities. Watch the recording.
SMRI Seminar ‘State-space models as graphs’
Víctor Elvira, University of Edinburgh
Thursday 29 February 2024
Abstract: Modeling and inference in multivariate time series is central in statistics, signal processing, and machine learning. A fundamental question when analyzing multivariate sequences is the search for relationships between their entries (or the modeled hidden states), especially when the inherent structure is a directed (causal) graph. In such context, graphical modeling combined with sparsity constraints allows to limit the proliferation of parameters and enables a compact data representation which is easier to interpret in applications, e.g., in inferring causal relationships of physical processes in a Granger sense. In this talk, we present a novel perspective consisting on state-space models being interpreted as graphs. Then, we propose novel algorithms that exploit this new perspective for the estimation of the linear matrix operator and also the covariance matrix in the state equation of a linear-Gaussian state-space model. Finally, we discuss the extension of this perspective for the estimation of other model parameters in more complicated models. Watch the recording.