Departmental Colloquium

	Date	Speaker	Talk Information
View details	January 10	Ricardo Alonso PUC-Rio	Kinetic theory is a powerful tool for modeling. It connects apparently dissimilar topics in science and, if properly used, produces models having remarkable agreement with experimentation in an ample range of scales. We navigate over a series of examples where modern kinetic theory plays a central role in the understanding of the phenomenon in question. They will allow us to introduce key ideas of today’s kinetic theory. Viscoelastic materials, fiber networks, rod alignment, and wave propagation with peaked scattering are in the Menu!
View details	January 12	Martina Hofmanova Technische Universität Berlin	In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.
View details	January 12	Dane Taylor University of North Carolina, Chapel Hill	Many datasets are best represented by multilayer networks in which layers encode different types of connections such as categorical social ties, interconnected infrastructures, or a network at different instances in time. Because most techniques to analyze networks have been developed for single-layer networks, extending theory to the multilayer and temporal settings is an important pursuit in applied mathematics, statistics, and the myriad applications that involve networks. I will describe my recent research in this area, surveying several network-analysis techniques including manifold learning, community detection, and centrality analysis. To provide narrative, I will focus on describing the tradeoffs between two strategies that account for multiple layers: they can be aggregated (yielding a single network) or coupled together (giving rise to inter-layer and intra-layer edges). I will show that the limit of strong inter-layer coupling effectively aggregates layers, providing an important bridge between these two strategies.
View details	January 17	Wenjing Liao Johns Hopkins University	High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning. We model data sets as samples from a probability measure in R^D. When D is large, the well-known curse of dimensionality implies that an enormous amount of training data are required to achieve certain accuracy in statistical learning, making many tasks unfeasible. Fortunately many data sets in applications exhibit a low-dimensional structure, for example, a low-dimensional manifold. We are interested in building efficient representations of such data for the purpose of compression and inference. In this talk, I will present a multiscale algorithm that yields a data-driven dictionary, together with a fast transform mapping data into low-dimensional coefficients, and an inverse of such a map. Our algorithm offers a tool for the dimension reduction of manifold data, and we can further use the low-dimensional coefficients for manifold inference. I will include several numerical experiments on both synthetic and real data, confirming our theoretical results on finite-sample analysis and demonstrating the effectiveness of our algorithm.
View details	January 19	Andrew Snowden University of Michigan	Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area.
View details	January 26	Brandon Levin University of Chicago	The Langlands program is a far-reaching set of conjectural connections between analytic objects (e.g., modular forms) and arithmetic objects (e.g., elliptic curves). In 1987, Serre made a bold conjecture about modular forms in the spirit of a characteristic p Langlands program. Serre's conjecture (now a Theorem due to Khare-Wintenberger and Kisin) has a number of interesting consequences including Fermat's Last Theorem. This talk will begin with overview of Serre's original conjecture (the two dimensional case). There are now a number of generalizations of this conjecture to higher dimensions. After introducing these higher dimensional analogues, I will describe recent progress towards the weight part of these conjectures.
View details	January 31	Michele Coti Zelati University of Maryland	The process of mixing of a scalar quantity into a homogenous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers - a theory that dates back to the 1830's and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).
View details	February 2	Dan Shen University of South Florida	High dimensionality has become a common feature of "big data” encountered in many divergent fields, such as neuroimaging and genetic analysis, which provides modern challenges for statistical analysis. To cope with the high dimensionality, dimension reduction becomes necessary. Principal component analysis (PCA) is arguably the most popular classical dimension reduction technique, which uses a few principal components (PCs) to explain most of the data variation. We introduce Multiscale Weighted Principal Component Regression (MWPCR), a new variation of PCA, for neuroimaging analysis. MWPCR introduces two sets of novel weights, including global and local spatial weights, to enable a selective treatment of individual features and incorporation of class label information as well as spatial pattern within neuroimaging data. Simulation studies and real data analysis show that MWPCR outperforms several competing methods.
View details	March 9	Fabrizio Catanese Universität Bayreuth	Many important questions in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling the classical configurations (Pappus, Desargues, Fano, Hesse) and some new ones, I shall describe simple equations for such surfaces, as the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, related to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such examples and applications, by several authors, I shall concentrate on a recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing new results and quite general open questions, concerning rigid compact complex manifolds, and projective classifying spaces.
View details	March 22	Nick Trefethen University of Oxford	Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the textbooks? It seems to be not so, and indeed, one of the few treatments to be found in the textbooks, by Feynman, is incorrect. The shielding effect turns out to be not as simple as one might expect: it depends on the wires having finite radius. Nor is it as strong as one might imagine: it improves only linearly as the wire spacing decreases. This talk will present results on electrostatic Faraday shielding by Jon Chapman, Dave Hewett and myself (SIAM Review, 2015). Mathematically, this is a problem of harmonic measure. Physically, Faraday shielding cage can be regarded as electrostatic induction by a surface of limited capacitance.
View details	March 23	Peter Mucha University of North Carolina at Chapel Hill	We consider idealized network dynamics of individuals changing their opinions and connections in a coevolving process driving towards consensus. Discordant edges connecting disagreeing nodes are resolved either through one individual changing opinion to match the other or through a rewiring process, the details of which can lead to dramatically different results. We compare and contrast results for two opinions with two different rewiring systems: a "rewire-to-same" rule where individuals form new connections only with others who already hold the same opinion, and a "rewire-to-random" rule where no constraints are made in the rewiring. We investigate this latter system in the presence of more than two opinions. We also consider modifications that reinforce local clustering in the rewiring. Throughout this work, we identify and test model systems of differential equations for describing the system states. This talk represents joint work with Feng Shi, Nishant Malik, Hsuan-Wei Lee, Rick Durrett, James Gleeson, Alun Lloyd, David Sivakoff, Josh Socolar and Chris Varghese.
View details	March 30	Chun Liu Pennsylvania State University	Almost all biological activities involve transport and distribution of ions and charged particles. The complicated coupling and competition between different ionic solutions in various biological environments give the intricate specificity and selectivity in these systems. In this talk, I will introduce several extended general diffusion systems motivated by the study of ion channels and ionic solutions in biological cells. In particular, I will focus on the interactions between different species, the boundary effects and in many cases, the thermal effects.
View details	April 3	Paris Perdikaris; MIT	The analysis of complex physical and biological systems necessitates the accurate resolution of in- teractions across multiple spatio-temporal scales, the consistent propagation of information between concurrently coupled multi-physics processes, and the effective quantification of model error and para- metric uncertainty. Addressing these grand challenges is a multi-faceted problem that poses the need for a highly sophisticated arsenal of tools in stochastic modeling, high-performance scientific comput- ing, and probabilistic machine learning. Through the lens of three realistic large-scale applications, this talk aims to demonstrate how the compositional synthesis of such tools is introducing a new paradigm in scientific discovery. First, we present multi-scale blood flow simulations in the human brain, and show how high-order methods, massively parallel computing, and concurrent coupling of multi-physics solvers can uncover intrinsic physiological mechanisms in health and disease. We will demonstrate how the introduction of probabilistic machine learning techniques, and the key concept of multi-fidelity modeling, provide a scalable platform for information fusion and lead to significant computational ex- pediency gains. The second application involves an environmental study that illustrates how machine learning tools enable the synergistic combination of simulations, noisy measurements and empirical models towards quantifying the anthropogenic effect in the increasing acidification of coastal waters, and developing a cost-effective monitoring and prediction mechanism. Lastly, we consider the shape optimization of super-cavitating hydrofoils of an ultrafast marine vessel for special naval operations. Specifically, we show how the combination of turbulent multi-phase flow simulations and the concept of multi-fidelity Bayesian optimization allows us to tackle complex engineering design problems in which a rigorous assessment of uncertainty and risk becomes critical in policy and decision making.
View details	April 13	Colin Adams Williams College	Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed.
View details	April 20	Lek-Heng Lim The University of Chicago	This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. A d-dimensional hypermatrix may be viewed as a coordinate representation of an order-d tensor. The simplest type of interactions between two spaces are linear and thus described by 2-tensors (e.g., linear operators, bilinear forms); when there are d > 2 spaces, then the simplest type of interactions are multilinear and these are described by d-tensors. We will see that some of most fundamental notions in computations, mathematics, physics, and statistics are inherently multilinear; higher order tensors and hypermatrices are thus unavoidable in furthering our understanding of these notions. We will discuss how tools like rank, norms, determinant, eigen and singular values can be generalized to hypermatrices, and far from being artificial constructs, they appear naturally in a wide range of phenomena and are enormously useful.

Date

Speaker

Talk Information

January 10

Kinetic theory is a powerful tool for modeling. It connects apparently dissimilar topics in science and, if properly used, produces models having remarkable agreement with experimentation in an ample range of scales. We navigate over a series of examples where modern kinetic theory plays a central role in the understanding of the phenomenon in question. They will allow us to introduce key ideas of today’s kinetic theory. Viscoelastic materials, fiber networks, rod alignment, and wave propagation with peaked scattering are in the Menu!

View details

January 12

Martina Hofmanova
Technische Universität Berlin

In this talk, I will consider quasilinear parabolic PDEs subject to stochastic or rough perturbation and explain how various assumptions on coefficients and roughness of the noise naturally ask for different notions of solution with different regularity properties and different techniques of the proofs. On the one hand, the problems under consideration will be stochastic second order parabolic PDEs with noise smooth in space, either with a possible degeneracy in the leading order operator, where only low regularity holds true, or under the uniform ellipticity assumption, where arbitrarily high regularity can be proved under suitable assumptions on the coefficients. On the other hand, I will discuss a rough pathwise approach towards these problems based on tools from paracontrolled calculus.

View details

January 12

Dane Taylor
University of North Carolina, Chapel Hill

Many datasets are best represented by multilayer networks in which layers encode different types of connections such as categorical social ties, interconnected infrastructures, or a network at different instances in time. Because most techniques to analyze networks have been developed for single-layer networks, extending theory to the multilayer and temporal settings is an important pursuit in applied mathematics, statistics, and the myriad applications that involve networks. I will describe my recent research in this area, surveying several network-analysis techniques including manifold learning, community detection, and centrality analysis. To provide narrative, I will focus on describing the tradeoffs between two strategies that account for multiple layers: they can be aggregated (yielding a single network) or coupled together (giving rise to inter-layer and intra-layer edges). I will show that the limit of strong inter-layer coupling effectively aggregates layers, providing an important bridge between these two strategies.

View details

January 17

Wenjing Liao
Johns Hopkins University

High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning. We model data sets as samples from a probability measure in R^D. When D is large, the well-known curse of dimensionality implies that an enormous amount of training data are required to achieve certain accuracy in statistical learning, making many tasks unfeasible. Fortunately many data sets in applications exhibit a low-dimensional structure, for example, a low-dimensional manifold. We are interested in building efficient representations of such data for the purpose of compression and inference. In this talk, I will present a multiscale algorithm that yields a data-driven dictionary, together with a fast transform mapping data into low-dimensional coefficients, and an inverse of such a map. Our algorithm offers a tool for the dimension reduction of manifold data, and we can further use the low-dimensional coefficients for manifold inference. I will include several numerical experiments on both synthetic and real data, confirming our theoretical results on finite-sample analysis and demonstrating the effectiveness of our algorithm.

View details

January 19

Andrew Snowden
University of Michigan

Representation stability is a relatively new field that studies somewhat exotic algebraic structures and exploits their properties to prove results (often asymptotic in nature) about objects of interest. I will describe some of the algebraic structures that appear (and state some important results about them), give a sampling of some notable applications (in group theory, topology, and algebraic geometry), and mention some open problems in the area.

View details

January 26

Brandon Levin
University of Chicago

The Langlands program is a far-reaching set of conjectural connections between analytic objects (e.g., modular forms) and arithmetic objects (e.g., elliptic curves). In 1987, Serre made a bold conjecture about modular forms in the spirit of a characteristic p Langlands program. Serre's conjecture (now a Theorem due to Khare-Wintenberger and Kisin) has a number of interesting consequences including Fermat's Last Theorem. This talk will begin with overview of Serre's original conjecture (the two dimensional case). There are now a number of generalizations of this conjecture to higher dimensions. After introducing these higher dimensional analogues, I will describe recent progress towards the weight part of these conjectures.

View details

January 31

Michele Coti Zelati
University of Maryland

The process of mixing of a scalar quantity into a homogenous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers - a theory that dates back to the 1830's and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).

View details

February 2

Dan Shen
University of South Florida

High dimensionality has become a common feature of "big data” encountered in many divergent fields, such as neuroimaging and genetic analysis, which provides modern challenges for statistical analysis. To cope with the high dimensionality, dimension reduction becomes necessary. Principal component analysis (PCA) is arguably the most popular classical dimension reduction technique, which uses a few principal components (PCs) to explain most of the data variation. We introduce Multiscale Weighted Principal Component Regression (MWPCR), a new variation of PCA, for neuroimaging analysis. MWPCR introduces two sets of novel weights, including global and local spatial weights, to enable a selective treatment of individual features and incorporation of class label information as well as spatial pattern within neuroimaging data. Simulation studies and real data analysis show that MWPCR outperforms several competing methods.

View details

March 9

Fabrizio Catanese
Universität Bayreuth

Many important questions in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling the classical configurations (Pappus, Desargues, Fano, Hesse) and some new ones, I shall describe simple equations for such surfaces, as the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, related to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such examples and applications, by several authors, I shall concentrate on a recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing new results and quite general open questions, concerning rigid compact complex manifolds, and projective classifying spaces.

View details

March 22

Nick Trefethen
University of Oxford

Everybody has heard of the Faraday cage effect, in which a wire mesh does a good job of blocking electric fields and electromagnetic waves. Surely the mathematics of such a famous and useful phenomenon has been long ago worked out and written up in the textbooks? It seems to be not so, and indeed, one of the few treatments to be found in the textbooks, by Feynman, is incorrect. The shielding effect turns out to be not as simple as one might expect: it depends on the wires having finite radius. Nor is it as strong as one might imagine: it improves only linearly as the wire spacing decreases. This talk will present results on electrostatic Faraday shielding by Jon Chapman, Dave Hewett and myself (SIAM Review, 2015). Mathematically, this is a problem of harmonic measure. Physically, Faraday shielding cage can be regarded as electrostatic induction by a surface of limited capacitance.

View details

March 23

Peter Mucha
University of North Carolina at Chapel Hill

We consider idealized network dynamics of individuals changing their opinions and connections in a coevolving process driving towards consensus. Discordant edges connecting disagreeing nodes are resolved either through one individual changing opinion to match the other or through a rewiring process, the details of which can lead to dramatically different results. We compare and contrast results for two opinions with two different rewiring systems: a "rewire-to-same" rule where individuals form new connections only with others who already hold the same opinion, and a "rewire-to-random" rule where no constraints are made in the rewiring. We investigate this latter system in the presence of more than two opinions. We also consider modifications that reinforce local clustering in the rewiring. Throughout this work, we identify and test model systems of differential equations for describing the system states. This talk represents joint work with Feng Shi, Nishant Malik, Hsuan-Wei Lee, Rick Durrett, James Gleeson, Alun Lloyd, David Sivakoff, Josh Socolar and Chris Varghese.

View details

March 30

Chun Liu
Pennsylvania State University

Almost all biological activities involve transport and distribution of ions and charged particles. The complicated coupling and competition between different ionic solutions in various biological environments give the intricate specificity and selectivity in these systems. In this talk, I will introduce several extended general diffusion systems motivated by the study of ion channels and ionic solutions in biological cells. In particular, I will focus on the interactions between different species, the boundary effects and in many cases, the thermal effects.

View details

April 3

Paris Perdikaris;
MIT

The analysis of complex physical and biological systems necessitates the accurate resolution of in- teractions across multiple spatio-temporal scales, the consistent propagation of information between concurrently coupled multi-physics processes, and the effective quantification of model error and para- metric uncertainty. Addressing these grand challenges is a multi-faceted problem that poses the need for a highly sophisticated arsenal of tools in stochastic modeling, high-performance scientific comput- ing, and probabilistic machine learning. Through the lens of three realistic large-scale applications, this talk aims to demonstrate how the compositional synthesis of such tools is introducing a new paradigm in scientific discovery. First, we present multi-scale blood flow simulations in the human brain, and show how high-order methods, massively parallel computing, and concurrent coupling of multi-physics solvers can uncover intrinsic physiological mechanisms in health and disease. We will demonstrate how the introduction of probabilistic machine learning techniques, and the key concept of multi-fidelity modeling, provide a scalable platform for information fusion and lead to significant computational ex- pediency gains. The second application involves an environmental study that illustrates how machine learning tools enable the synergistic combination of simulations, noisy measurements and empirical models towards quantifying the anthropogenic effect in the increasing acidification of coastal waters, and developing a cost-effective monitoring and prediction mechanism. Lastly, we consider the shape optimization of super-cavitating hydrofoils of an ultrafast marine vessel for special naval operations. Specifically, we show how the combination of turbulent multi-phase flow simulations and the concept of multi-fidelity Bayesian optimization allows us to tackle complex engineering design problems in which a rigorous assessment of uncertainty and risk becomes critical in policy and decision making.

View details

April 13

Colin Adams
Williams College

Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed.

View details

April 20

Lek-Heng Lim
The University of Chicago

This talk is intended for those who, like the speaker, have at some point wondered whether there is a theory of three- or higher-dimensional matrices that parallels matrix theory. A d-dimensional hypermatrix may be viewed as a coordinate representation of an order-d tensor. The simplest type of interactions between two spaces are linear and thus described by 2-tensors (e.g., linear operators, bilinear forms); when there are d > 2 spaces, then the simplest type of interactions are multilinear and these are described by d-tensors. We will see that some of most fundamental notions in computations, mathematics, physics, and statistics are inherently multilinear; higher order tensors and hypermatrices are thus unavoidable in furthering our understanding of these notions. We will discuss how tools like rank, norms, determinant, eigen and singular values can be generalized to hypermatrices, and far from being artificial constructs, they appear naturally in a wide range of phenomena and are enormously useful.

Department of Mathematics

College of Science

Spring 2017