Motivic integration is a powerful technique in algebraic geometry developed by Kontsevich, Denef-Loeser, and others. For example, several interesting invariants can be attached to an isolated hypersurface singularity using motivic integration. In the Euler characteristic limit, these invariants are all related in a straightforward way, but the relationships between the motivic versions are more difficult to understand. In the first half of this talk, I will give an introduction to motivic integration. In the second part, I will discuss the aforementioned singularity invariants in more detail, including connections to knot Floer homology, Hilbert schemes, and the Igusa zeta function. The second part of the talk is based on joint work with Oblomkov and Wyss.
Markov processes have a wide range of applications, and in many situations the underlying process is a non-homogeneous Markov (nhm) process. Here the transition probabilities and the initial distribution comprise the parameters. In scenarios where intermittent observations regarding the state occupancy are available, the observations provide only the partial information (the exact transition times are not observed) of the underlying process. In this talk, we will consider a 2-state nhm process {X(t), t ≥ 0} with the state space S = {1, 2}. When the process is observed only intermittently, the likelihood function is expressed in terms of the transition probabilities and numerical algorithms may be needed for the purpose of estimation. We will discuss a data augmentation approach based on Poisson processes for the estimation of the parameters. We will also describe characterisation of the transition intensities and extension of the proposed approach to two correlated processes. This work is motivated by the partial observations of treatment responses in patients of neovascular Age-related Macular Degeneration (nAMD). The characteristic feature of nAMD is the fluid accumulated under the macula due to leakage from abnormal blood vessels and the target of the treatment is to reduce the fluid.
The "Byzantine generals problem" is an allegory describing a distributed system aiming for consensus in the presence of unreliable/malicious elements. A widely known classical approach to Byzantine fault tolerance shows the impossibility of dealing with one-third or more faulty elements, i.e., only f-faulty elements are enough to corrupt a system with 3f or fewer components. In this talk, I will overview fault tolerance in a toy model and examine it as a spin glass model, a magnetic state characterized by randomness in interactions in spin systems. Finally, I will discuss the threshold for faulty elements through a mean-field solution.
For certain PDEs, it is not always possible to find smooth solutions; hence traditionally, we tend to relax the requirement of smoothness to finding weak (distributional) solutions to those PDEs, using the theory of Sobolev spaces. In dealing with Sobolev spaces, especially for unbounded regions, we encounter functions that are locally integrable but are not globally integrable, but have globally finite Sobolev energy. Such functions are called Dirichlet-Sobolev functions. Constant functions are certainly of this kind, with zero energy. It is natural to ask whether such functions are always, after subtracting a suitable constant that may depend on the function, globally integrable. The focus of this talk is to present results related to this question in the context of complete metric measure spaces (including Riemannian manifolds and Carnot-Carathéodory spaces) equipped with a locally doubing measure supporting a local Poincaré type inequality.
Maxwell's equations describe electromagnetic wave propagation and the most dominant force in our world. These fundamental equations can be handled in a natural way by a hypercomplex algebra due to W. K. Clifford, a contemporary of J. C. Maxwell. We shall survey this kind of non-commutative complex analysis, that is useful in solving Maxwell's equations. Making use of a Cauchy integral formula for time-harmonic Maxwell's equations advocated by Alan McIntosh, we describe a recent boundary integral equation reformulation for Maxwell scattering. In recent joint work with Johan Helsing and Anders Karlsson, Lund, we have achieved an efficient solver, with almost down to machine precision, for computationally challenging plasmonic and eddy current electromagnetic scattering problems.
In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = −∆ − b + V, we introduce a family of discrete time random walks in the flow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrödinger semigroup {e^{−tA}}_{t≥0} is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a finite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence. This talk is based on a joint work with Satoshi Ishiwata (Yamagata University) and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9 (online-first article in Mathematische Annalen).
By 2016, the American Mathematical Monthly had published roughly fifty papers on the Γ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on Γ. We also discuss a surprising blank spot: there had been very little published work on the functional inverse of this function. This omission has been rectified somewhat, since.
Pfaffian functions are real or complex analytic functions that satisfy triangular systems of first-order partial differential equations with polynomial coefficients. Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry to the robustness of neural networks.
Most decision-making under uncertainty problems found in industry and studied by the scientific community can be framed as a two-stage stochastic program. In the past decades, the standard framework to address this class of mathematical programming problems follows a sequential two-step process, usually referred to as estimate-then-optimize, in which a predictive distribution of the uncertain parameters is firstly estimated, based on some machine/statistical learning (M/SL) method, and, then, a decision is prescribed by solving the two-stage stochastic program using the estimated distribution. In this context, most M/SL methods typically focus only on minimizing the prediction error of the uncertain parameters, not accounting for its impact on the downstream decision problem. However, practitioners argue that their main interest is to obtain near-optimal solutions from the available data with minimum decision error rather than a least-error prediction. Therefore, in this talk, we discuss the new framework referred to as task-based learning in which the M/SL training function also accounts for the downstream decision problem. As the M/SL method, we focus on decision trees, and study decision-making problems framed as a two-stage linear program. We present an exact Mixed-Integer Linear Program (MILP) formulation for the task-based learning method and construct two efficient recursive-partitioning Heuristic Strategies for the MILP. We conclude the talk by analyzing a set of numerical experiments illustrating the capability and effectiveness of the task-based prescriptive tree learning framework, benchmarking against the standard estimate-then-optimize framework, and discussing the computational capability of the constructed heuristic strategies vis-à-vis the MILP formulation.
In distributed data storage, information is distributed across multiple servers with redundancy, in such a way that multiple users can access it at the same time. The access requests that a distributed data storage system can support are described by a convex polytope, called the service rate region of the system. This talk is about the properties of the service rate region, and about how the algebra of the system determines the geometry of the corresponding polytope.
The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix G and a target stationary distribution μ^, construct a minimum norm perturbation, ∆, such that G^ = G + ∆ is also stochastic and has the prescribed target stationary distribution, μ^. We first consider rank-1 perturbations ∆ and show how to efficiently minimize their norm when such a solution is feasible. But sparsity and/or connectivity of the graph of G + ∆ may then get lost. We then impose a constraint on the support of ∆, that is, on the set of non-zero entries of ∆. This is particularly meaningful in practice since one cannot typically modify all entries of G. We first show how to construct a feasible solution G^ that has essentially the same support as the matrix G. Then we show how to compute globally optimal and sparse solutions using the component-wise l_1 norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to 10^5 × 10^5 in a few minutes.
I will discuss the inverse conductivity problem on recovering interior information about the electrical conductivity of a body from exterior electrical measurements. I will show how one can reconstruct the exact shape and position (called obstacles/inclusions) of inhomogeneities, using energy-comparisons with Neumann-to-Dirichlet maps. The method is at the same time both simple and surprisingly general, allowing inhomogeneities with parts that are finite positive and negative perturbations, parts that are superconducting or insulating, and parts originating from a Muckenhoupt weight (leading to degenerate/singular problems). The method can also recover collections of cracks in the form of hypersurfaces. If time permits it, I will also discuss how to handle more practical electrode models and noisy measurements in a rigorous way.
Strategic decision-making is challenging due to multiple strategic objectives and long planning horizons that make it difficult to assess the future impacts of proposed strategic actions with respect to these objectives. Moreover, strategy work often requires a balance between preparing for alternative scenarios for the future (i.e., developing a robust strategy), and trying to steer the course of change towards a desirable direction (i.e., developing a market-shaping strategy). We present a model-based framework for supporting the development of a robust, market-shaping strategy. For the purposes of this framework, we develop a new portfolio decision analytic model and algorithms to help generate decision recommendations for selecting strategic actions, when (i) the actions scenario- and objective-specific impacts, the baseline values for these impacts, as well as preferences between strategic objectives are incompletely specified, and (ii) information regarding scenario likelihoods is incomplete and may depend on the selected actions. This framework is applied in a high-impact case on supporting the strategy process at the payments unit of Nordea Bank Abp, the largest retail bank in the Nordic countries.
In his talk I will explain how similarities between the integers and the polynomial ring over a field allow us to find, and prove, connections between objects that at first glance seem to be quite different. As an example of this I will show how Lagranges interpolation theorem is nothing else but a particular case of the Chinese reminder theorem. If time allows I will show how a simple result on polynomials leads to the statement of the famous ABC conjecture.
Springer theory for reductive algebraic groups plays an important role in determining irreducible characters of finite groups of Lie type. We discuss its generalisation to the setting of graded Lie algebras. We explain how level-rank dualities arise from unipotent irreducible characters and their connections with the graded Springer theory. If time permits, we discuss a conjectural realisation of these dualities using affine Springer fibers.
In many scientific computing and machine learning problems, we expend the majority of our computational resources in solving large-scale linear algebra problems, typically linear systems Ax = b, eigenvalue problems Ax = λx, or the singular value decomposition A = USV'. Numerical linear algebra (NLA) is a research field that attempts to devise practical algorithms for solving these problems. Broadly, methods in NLA can be divided into three categories: direct, iterative, and randomized. In this talk I will give a whistle-stop tour of these classes of methods, highlighting the incredible robustness of classical (direct) methods, and the exciting speed and advances in randomized methods.
In this talk, we explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can look at some of these random structures here (https://skfb.ly/6VINC) and start making some guesses about their topological behavior.
Page content by: webmaster-math [at] list [dot] aalto [dot] fi