feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
or Akil Narayan (akil (at) sci.utah.edu)Monday, February 10 at 4pm. In-person LCB 222
Speaker: Leonid Kunyansky
Department of Mathematics, University of Arizona
Title: Hybrid imaging modalities and the range of the spherical means transform
Abstract: The last two decades saw proliferation of novel coupled-physics imaging
modalities. A variety of sensitive but safe and inexpensive medical imaging
methods has been developed, that hold great promise for the breast imaging for
cancer, detection of ischemia, hemorrhaging, blood clots, etc. These imaging
techniques work by combining high-resolution ultrasound with electromagnetic
fields that are highly sensitive to the features of interest. In the first
part of my talk I will overview the most interesting of these modalities, with
emphasis on the underlying mathematics.
Inevitably, the introduction of the new techniques has posed a variety of new
and exciting inverse problems. In particular, a prominent role in this field is
played by the spherical means operator, that maps a function into a collection
of integrals over spheres with centers lying on a given surface. The problem
I will discuss is the description of the range of this operator. As it
happens, all of the classical results are suboptimal, in that they use twice
the amount of needed data. I will present a novel range description, that
overcomes this issue.
(This is a joint work with Peter Kuchment, Texas A&M University)
Monday, February 24 at 10am. In-person JTB 110 (Joint seminar with SIAM Northern States Section)
Speaker: Gautam Iyer
Department of Mathematical Sciences, Carnegie Mellon University
Title: Enhanced Dissipation and Mixing
Abstract: Consider a drop of dye in water. If left alone it takes a very
long time to mix. If you stir it a little, it mixes quickly. This is due
to enhanced dissipation, a phenomenon where the combined effect
of convection and diffusion increase the rate of energy dissipation.
This effect can be analyzed mathematically by using PDE methods to study
the advection diffusion equation, or using probabilistic methods to
study the mixing time of the associated Itô diffusion. This talk will be
an overview several recent results in this field, and present the main
ideas from both PDE and probabilistic perspectives.
Monday, February 24 at 4pm. In-person LCB 222
Speaker: Robyn Brooks
Science Research Initiative, University of Utah
Title: Computing Invariants of Multi Parameter Persistence Modules
Abstract: Persistent Homology is a tool of Computation Topology which is used to determine the topological features of a space from a sample of data points. In this talk, I will introduce the persistence pipeline and some cool applications of persistent homology. I will then present some basic tools from Discrete Morse Theory which can be used to better understand the multi-parameter persistence module of a filtration. In particular, the addition of a discrete gradient vector field consistent with a multi-filtration allows one to exploit the information contained in the critical cells of that vector field as a means of enhancing geometrical understanding of multi-parameter persistence. Results from joint work with Claudia Landi, Asilata Bapat, Barbara Mahler, and CeliaHacker, show that the rank invariant for nD persistence modules can be computed by selecting a small number of values in the parameter space determined by the critical cells of the discrete gradient vector field. These values may be used to reconstruct the rank invariant for all other possible values in the parameter space. I will then present some of my current research in other related invariants for multi-parameter persistence modules.
Monday, March 17 at 4pm. In-person LCB 222
Speaker: Jesse Chan
Department of Computational Applied Mathematics and Operations Research, Rice University
Title: An artificial viscosity approach to entropy stable high order DG methods
Abstract: Entropy stable discontinuous Galerkin (DG) methods improve the robustness of high order DG simulations of nonlinear conservation laws. These methods yield a semi-discrete entropy inequality, and rely on an algebraic flux differencing formulation which involves both summation-by-parts (SBP) discretization matrices and entropy conservative two-point finite volume fluxes. However, explicit expressions for such two-point finite volume fluxes may not be available for all systems, or may be computationally expensive to compute.
We propose an alternative approach to constructing entropy stable DG methods using an artificial viscosity coefficient based on the local violation of a cell entropy inequality and a local entropy dissipation estimate. The resulting method yields the same global semi-discrete entropy inequality satisfied by entropy stable flux differencing DG methods. The artificial viscosity coefficients are parameter-free and locally computable over each cell. The resulting artificial viscosity preserves high order accuracy, improves linear stability, and does not result in a more restrictive maximum stable time-step size under explicit time-stepping.
Monday, March 24 at 4pm. In-person LCB 222
Speaker: Pascal Steinke
Institute for Applied Mathematics, University of Bonn
Title: Homogenization of Interaction Energy of Dislocations
Abstract: Abstract: Dislocations are a type of material defects in the
crystallographic structure of metals. They form due to external forces
and play a fundamental role in determining the elasticity and
brittleness properties of the metal.
We consider the total interaction energy of dislocation loops placed on a
homogenization-lattice, then we let the lattice spacing tend to zero.
Under a well-separatedness condition, we will deduce that the asymptotic
behaviour is similar to that of dipoles on the same lattice. Inspired
by a previous work of R. James and S. Müller, using the notion of
H-Measures introduced by L. Tartar, we will derive a limiting
representation of the total interaction energy and establish its
Gamma-convergence to -infinity.
Joint work with Stefan Müller.
Monday, 31 at 10am. In-person JTB 110 (Joint seminar with SIAM Northern States Section)
Speaker: Katie Newhall
Department of Mathematics, UNC Chapel Hill
Title: Disordered Network Metamaterials: Hyperuniformity and Electrical Transport
Abstract: Advancements in materials design and manufacturing have allowed for the production of ordered and disordered metamaterials with diverse and novel properties. We seek to develop and 3D print disordered network metamaterials that have both low strength to weight ratios as well as the desirable physical properties of hyperuniform two-phase heterogeneous materials. Hyperuniform systems anomalously suppress density fluctuations on large length scales compared to typical disordered systems. In this talk, I will discuss the extent to which disordered network metamaterials formed from different tessellations of hyperuniform point clouds inherit hyperuniformity themselves. Defining a new network-based characterization, I show the extent to which hyperuniformity is inherited is a function of the network topology alone. I will also discuss characterizing the electric transport of such 2D structures, both experimentally and theoretically from the combinatorial weighted graph Laplacian. I seek to explain surprising sensitivity of the effective resistance to anisotropy and global network topology.
Monday, 31 at 4pm. In-person LCB 222
Speaker: Mauro Maggioni
Department of Applied Mathematics and Statistics, Johns Hopkins University
Title: On exploiting compositional structure in machine learning to avoid the curse of dimensionality: inference of interaction kernels for particle systems on networks, and the nonlinear single-index model for high-dimensional functions
Abstract:
We consider stochastic systems of interacting particles in Euclidean spaces and on networks, which are commonly used in models throughout the sciences. While these systems have very high-dimensional state spaces, the laws of interaction between the agents may be quite simple: for example, they may depend only on pairwise interactions, and perhaps only on the pairwise distance of the pair of states in each interaction. We focus on the case of interacting particles on an unknown network, and we estimate both the interaction laws and the network. We cast this as an inverse problem, discuss when it is well-posed, and construct estimators with provably good statistical and computational properties, avoiding the curse of dimensionality of the state space of these systems, by maximally exploiting the symmetries and the compositional structure of the problems. This is joint work with Q. Lang (Duke), F. Lu (JHU), S. Tang (UCSB), X. Wang (JHU) , M. Zhong (UH).
In the second part of the talk, I will discuss a new, simple model of high-dimensional functions that generalizes the single-index model f(x)=g(<v,x>), to the case where the linear inner function projecting onto the one-dimensional line spanned by the unknown index vector v is replaced by a nonlinear counterpart, given by projection onto a(n unknown) one-dimensional curve. We construct estimators for the regression function that, under suitable assumptions, defeat the curse of dimensionality, achieve nearly optimal learning rates, and can be computed efficiently. This is joint work with Y. Wu (JHU).
April 21 at 4pm. In-person LCB 222
Speaker: Yiping Lu
Industrial Engineering & Management Science, Northwestern University
Title: Two Tales, One Resolution: Physics-Informed Test Time Scaling and Precondition
Abstract: In this talk, I will introduce a novel framework for physics-informed debiasing of machine learning estimators, which we call Simulation-Calibrated Scientific Machine Learning (SCaSML). This approach leverages the structure of physical models to achieve unbiased prediction and overcome the curse of dimensionality.
The SCaSML paradigm integrates a (potentially) biased machine learning algorithm with a de-biasing procedure that is rigorously designed using numerical analysis and stochastic simulation. Our methodology aligns with recent advances in inference-time computation¿similar to those seen in the large language model literature¿demonstrating that additional computation can enhance ML estimates.
Furthermore, we establish a surprising equivalence between our framework and another research direction that utilizes approximate (linearized) solvers to precondition iterative methods. This connection not only bridges two distinct areas of study but also offers new insights into improving estimation accuracy in complex, high-dimensional (PDE) settings.
April 23 at 4pm. In-person LCB 225 **Special Seminar: note
unusual time and location**
Speaker: Mike Novack
Department of Mathematics, Louisiana State University
Title: A variational model for 3D features in films/foams
Abstract: Area minimization among a suitable class of 2D surfaces is the basic variational model describing the interfaces in films/foams. In this talk we will discuss a modification of this paradigm in which surfaces are replaced with regions of small but positive volume. The model captures features of real films/foams, such as Plateau borders, that cannot be described by zero volume objects.
July 1 at 4pm. In-person LCB 215
Speaker: Kaitlin Harrison
School of Oceanography, University of Washington
Title: Mathematics and Astrobiology
Abstract: Astrobiology, or the study of the origin, evolution, and distribution of life in the universe, is a highly interdisciplinary field that seeks to address many of the fundamental questions about life: How did life form on earth? Are we alone in the universe? What are the limits of life, on our planet and beyond? I will present an overview of the field, followed by a few vignettes describing current uses of applied mathematics to address these questions in our solar system and beyond.
feldman (at) math.utah.edu),
Victoria Kala (victoria.kala (at) utah.edu),
and
Akil Narayan (akil (at) sci.utah.edu).
Past lectures: Fall 2024, Spring 2024, Fall 2023, Spring 2023, Fall 2022, Spring 2022, Fall 2021, Spring 2021, Fall 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.