### Mathematics Colloquium

#### Spring 2024

###### Location and Time: Minard 210 at 3:00 PM (Refreshments at 2:30 in Minard 404)

***Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.**

##### Tuesday, January 30

###### Christian Parkinson, University of Arizona

Optimal Path Planning in the Hamilton-Jacobi Formulation

Abstract: We present a partial-differential-equation-based optimal path planning framework. This formulation relies on optimal control theory, dynamic programming, and Hamilton-Jacobi-Bellman equations, and thus provides an interpretable alternative to black-box machine learning algorithms. We briefly discuss grid-based numerical methods used to resolve the solution to the Hamilton-Jacobi-Bellman equation and generate optimal trajectories, and then describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be used to solve similar problems in higher dimensions and in nearly real-time. We demonstrate all of our methods with several examples.

##### Thursday, February 1

###### Yang Zhang, University of Washington

Microlocal methods in inverse problems arising in medical imaging

Abstract: Microlocal analysis provides a framework to study distributions (generalized functions) and operators from a phase space point of view. It is rooted in Fourier analysis and further developed by functional analysis and symplectic geometry. A fundamental concept in this field, singularities, refer to points in phase space where a distribution is not smooth. In this talk, we will discuss the study of several inverse problems using microlocal analysis, especially from the perspective of singularities. The first inverse problem is to reconstruct a density function of an object from its integral transform over cone surfaces, which arises in Compton camera imaging. Using microlocal analysis, we describe which features (singularities) of the object can be reconstructed in a stable way from local data measurements. Next, we will talk about inverse problems of recovering parameters in partial differential equations from boundary measurements, especially addressing nonlinear wave equations in ultrasound imaging with damping effects. We will explain how the analysis of new singularities, produced by the nonlinear interactions of waves, enables the determination of these parameters. In particular, we will address the challenges in inverse boundary value problems and the techniques to handle them.

##### Tuesday, February 6

###### David Smith, Yale-NUS College

Fokas diagonalization

Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

##### Tuesday, February 13

###### Jiaxin Jin, Ohio State University

Biochemical reaction networks

Abstract: I will discuss mathematical models of biochemical reaction networks. First, I will introduce mathematical models of biochemical reaction networks under the classical assumption of mass action kinetics. Then we restrict our attention to toric dynamical systems, which appear naturally as models of reaction networks and have very robust and stable dynamical properties. We then extend these models to disguised toric dynamical systems, which are generated by a reaction network N, such that they have toric realization with respect to some other network N’. This allows us to greatly extend the applicability of the theory of toric systems. Then, by studying input-output networks, I will describe how the network structure can be connected to the property of homeostasis in biology. As the main result, we show all homeostasis patterns supported by a given input-output network in terms of a combinatorial structure associated to the input-output network.

##### Thursday, February 15

###### Ke Chen, University of Maryland

Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms

Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in scientific computing. This talk delves into efficient training for PDE operator learning in both the forward and inverse PDE settings. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), and compare its numerical performance with baseline models on several inverse problems, including optical tomography and inverse scattering. We will briefly mention some future works at the end, focusing on the regularization of inverse problems in the context of operator learning.

##### Tuesday, February 20

###### Colin Defant, Harvard

Bender--Knuth Billiards in Coxeter Groups

Abstract: Let (W,S) be a Coxeter system, and write S={$s_i$ : i is in I}, where I is a finite index set. Consider a nonempty finite convex subset L of W. If W is a symmetric group, then L is the set of linear extensions of a poset, and there are important *Bender--Knuth involutions* $BK_i$ (indexed by I) defined on L. For arbitrary W and for each i in I, we introduce an operator $\tau_i$ on W that we call a *noninvertible Bender--Knuth toggle*; this operator restricts to an involution on L that coincides with $BK_i$ when W is a symmetric group. Given an ordering $i_1,...,i_n$ of I and a starting element u_0 of W, we can repeatedly apply the toggles in the order $\tau_{i_1},...,\tau_{i_n},\tau_{i_1},...,\tau_{i_n},....$ This produces a sequence of elements of W that can be viewed in terms of a beam of light that bounces around in an arrangement of transparent windows and one-way mirrors. Our central questions concern whether or not the beam of light eventually ends up in the convex set L. We will discuss several situations where this occurs and several situations where it does not. This is based on joint work with Grant Barkley, Eliot Hodges, Noah Kravitz, and Mitchell Lee.

##### Tuesday, April 9

*Special Tri-College Colloquium at Concordia

Talk: Integrated Science Center 301 (Refreshments: Integrated Science Center 362)

###### Dogan Çömez, NDSU

*Title:* Existence of local ergodic Hilbert transform of superadditive processes

Abstract: The study of (ergodic) Hilbert transform is strongly connected to the study of singular integrals and on the approach regions for the existence of such integrals. When T is a linear operator induced by an invertible measure preserving transformation, the a.e. existence of the ergodic Hilbert transform was shown by M. Cotlar. This result has been extended to several settings ever since, including its local version for invertible measure preserving flows. In this talk, we will discuss extension of this local version to the setting of the class of bounded symmetric admissible processes relative to invertible measure preserving flows.

##### Tuesday, April 16

###### Bartłomiej Zawalski, Kent State.

On star-convex bodies with rotationally invariant sections

Abstract: We will outline the proof that an origin-symmetric star-convex body K with sufficiently smooth boundary and such that every hyperplane section of K passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to a recent question asked by G. Bor, L. Hernandez-Lamoneda, V. Jimenez de Santiago, and L. Montejano-Peimbert, though with slightly different prerequisites. Our argument is built mainly upon the tools of differential geometry and linear algebra, but occasionally we will need to use some more involved facts from other fields like algebraic topology or commutative algebra.

##### Tuesday, April 23

*Special Tri-College Colloquium at NDSU

###### Ashok Aryal, MSUM

*Title: *An approach for recovering initial temperature via bounded linear time sampling

Abstract: Heat equations represent a cornerstone in the field of partial differential equations (PDEs). These equations offer invaluable insights into the dynamics of diffusion across various scientific domains. We have studied the recovery of an initial temperature profile from a finite-time observation taken at a fixed point of a thin, uniform, one-dimensional rod. In this talk, we begin with a comprehensive review of the heat equation, including its formulation and solution, to establish a

foundational understanding. Then, we will discuss how we constructed a linearly growing finite set of future times within a bounded interval and how we approximated the initial temperature profile with the desired accuracy, assuming the initial temperature is within a suitable closed L^{2}-subspace.

##### Tuesday, April 30

###### Indranil SenGupta, Florida International University

Finance-informed machine learning and portfolio management

Abstract: We will begin by showcasing a few outcomes related to stochastic modeling in financial markets. There are certain limitations that are widely recognized in the existing stochastic models in the literature. A refined stochastic model will be proposed using various machine/deep learning methods. By employing data science methodologies, we can refine the stochastic model and extract a deterministic parameter from the financial data set. Applying the analysis to the commodity market will yield some intriguing new findings. The results will also contribute to enhancing portfolio optimization and hedging strategies. We also explore how this research can be applied to environmental finance.