Past Seminars

Abstract:

We address two important subproblems of Structure from Motion Problem. The first subproblem is known as Multi-way Matching, where the input includes multiple sets, with the same number of objects and noisy measurements of fixed one-to-one correspondence maps between the objects of each pair of sets. Given only noisy measurements of the mutual correspondences, the Multi-way Matching problem asks to recover the correspondence maps between pairs of them. The desired output includes the original fixed correspondence maps between all pairs of sets. The second subproblem is called Multi-Perspective Simultaneous Embedding (MPSE). The input for MPSE assumes a set of pairwise distance matrices defined on the same set of objects and possibly along with the same number of projection operators. MPSE embeds points in 3D so that the pairwise distances are preserved under the corresponding projections. Our proposed algorithm for Multi-way Matching problem iteratively solves the associated non-convex optimization problem. We prove that for a specific noise model, if the initial point of our proposed iterative algorithm is good enough, the algorithm linearly converges to the unique solution. Numerical experiments demonstrate that our method is much faster and more accurate than the state-of-the-art methods. For MPSE, we propose a heuristic algorithm and provide an extensive quantitative evaluation with datasets of different sizes, as well as several examples that illustrate the quality of the resulting solutions.

I received my PhD in mathematics from the University of Minnesota in 2018, under the supervision of my advisor Prof. Gilad Lerman. Since 2018, I've been working as a Postdoctoral Research Associate at the Department of Mathematics of the University of Arizona.

Abstract:

Let $\alpha=(a,b,\ldots)$ be a composition. Consider the associated poset $F(\alpha)$, called a fence, whose covering relations are $$x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\.$$ We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$. These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. We show that if one of the parts of $\alpha$ is greater than the sum of the others, then the conjecture is true. We conjecture that $L(\alpha)$ enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call $d$-divided posets, generalizing work of Claussen and simplifying a construction of Gansner. This is joint work with Thomas McConville and Clifford Smyth.See seminar website for password: http://www-users.math.umn.edu/~ovenh001/seminar.html

Abstract:

I will discuss several applications of random walks to graph-based learning, both for theoretical analysis and algorithm development. Graph-based learning is a field within machine learning that uses similarities between datapoints to create efficient representations of high-dimensional data for tasks like semi-supervised classification, clustering and dimension reduction. Our first application will be to use the random walk interpretation of the graph Laplacian to characterize the lowest label rate at which Laplacian regularized semi-supervised learning is well-posed. Second, we will show how analysis via random walks leads to a new algorithm that we call Poisson learning for semi-supervised learning at very low label rates. Finally, we will show how stochastic coupling of random walks can be used to prove Lipschitz estimates for graph Laplacian eigenfunctions on random geometric graphs, leading to new spectral convergence results.

This talk will cover joint work with many people, including Brendan Cook (UMN), Nicolas-Garcia Trillos (Wisconsin-Madison), Marta Lewicka (Pittsburgh), Dejan Slepcev (CMU), Matthew Thorpe (University Manchester).

Abstract:

In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves in random media. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.

https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Abstract:

The event will have graduate students speaking about their experiences with their internships.

The facilitator of the event is Montie Averie.

The participants of the Graduate Student Internship Panel will be:

Carter Chain (interned at Travelers)

Brendan Cook (interned at Target)

Ty Frazier (interned at Securian Financial and Lawrence Berkeley national lab)

Jacob Hegna (interned at Google)

Sarah Milstein (interned at Ayasdi, 3M, IDA, Cargill and Smart Information Flow Technologies (SIFT))

Amber Yuan (interned at Argonne national lab,  ExxonMobil  and Activision Blizzard).

The event is organized by the SIAM Student Chapter at the University of Minnesota.

Abstract:

In water-limited regions, competition for water resources results in the formation of vegetation patterns; on sloped terrain, one finds that the vegetation typically aligns in stripes or arcs. The dynamics of these patterns can be modeled by reaction-diffusion PDEs describing the interplay of vegetation and water resources, where sloped terrain is modeled through advection terms representing the downhill flow of water. We focus on one such model in the 'large-advection' limit, and we prove the existence of traveling planar stripe patterns using analytical and geometric techniques. We also discuss implications for the stability of the resulting patterns, as well as the appearance of curved stripe solutions.

Abstract:

I'll present a heuristic for finding special families of partially ordered sets. The heuristic says that the posets with order polynomial product formulas are the same as the posets with good dynamical behavior. Here the order polynomial is a certain enumerative invariant of the poset. Meanwhile, the dynamics includes promotion of linear extensions, and rowmotion of order ideals and P-partitions. This talk includes joint work with Tri Lai, and with Martin Rubey.

see seminar website for password: http://www-users.math.umn.edu/~ovenh001/seminar.html

Abstract:

We will briefly review, through two examples, classic deterministic PDE models of population dynamics structured according to attributes such as age and/or size. First, we describe how the original McKendrick model was used to motivate China's one-child policy, and generalize it to include an imposed, finite interbirth refractory period. We quantify the effectiveness of this softer, staggered birth policy and discuss its predicted effectiveness. We then review sizer-timer-adder-type models used to quantify proliferating cell populations. Here, blow-up in mean cell sizes can arise, which represents a challenging numerical problem. Finally, we extend these classic deterministic models to allow for both demographic and growth-rate stochasticity by developing a fully kinetic theory. Marginalization of the full density functions results in a set of coupled kinetic models similar to the BBGKY hierarchy. We map out the different combinations of stochastic descriptions and show how the classic age-dependent population models are connected to this hierarchy, the lowest order of which is a master equation for the total stochastic population. Differences in the stochastic description of birth through budding or splitting are explored.

Abstract:

In this talk, we will derive a priori interior Hessian estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian phase. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data, on a uniformly convex, bounded domain in R^n.

Abstract:

The need for fairness, accountability, and transparency in computer models that make or inform decisions about people has become increasingly clear over the last several years. One application area where these topics are particularly important is criminal justice, as statistical models are being used to make or inform decisions that impact highly consequential decisions— those concerning an individual’s freedom. In this talk, I’ll highlight three threads of my own research into the use of machine learning and a statistical models in criminal justice models that demonstrate the importance of careful attention to fairness, accountability, and transparency. In particular, I’ll discuss how predictive policing has the potential to reinforce and amplify unfair policing practices of the past. I’ll also discuss some of the ways in which recidivism prediction models can fail to require the accountability and transparency necessary to prevent gaming.

Kristian Lum is on the faculty at University of Pennsylvania's School of Engineering and Applied Science and is the former Lead Statistician at the Human Rights Data Analysis Group (HRDAG). Kristian’s research primarily focuses on examining the uses of machine learning in the criminal justice system, including demonstrating the potential for predictive policing models to reinforce and amplify historical racial biases in law enforcement. She has also served on Research Advisory Councils for the New York City’s Mayor’s Office of Criminal Justice and Philadelphia’s First Judicial District tasked with advising on the development of fairer algorithmic pre-trial risk assessments.

Kristian holds an M.S. and Ph.D. from the Department of Statistical Science at Duke University and a B.A. in Mathematics and Statistics from Rice University.

Abstract:

The spread of epidemics is a very complex process, and stochastic diffusion models on networks have been found useful, especially when modeling their spread in large and heterogeneous populations, where individual and community level behaviors need to be represented. A fundamental problem in such models is to understand how to control the spread of an epidemic by interventions such as vaccination (which can be modeled as node removal) and social distancing (which can be modeled as edge removal). A number of heuristics have been studied, such as selecting nodes based on degree and eigenscore. However, rigorous algorithms with approximation guarantees are not well understood, and is the focus of this talk.

We will discuss two approaches for epidemic control from a network science perspective. The first involves reducing the spectral radius of the graph, motivated by a characterization that shows that epidemics in some models die out fast if the spectral radius is below a threshold. We discuss algorithms for this problem and its generalizations. The second approach involves stochastic optimization, using a sample average approximation combined with rounding. We show that this approach gives near optimal solutions in practice, and have interesting structural properties, which might be useful
in finding practical interventions.

Anil Vullikanti is a Professor in the Department. of Computer Science and the Biocomplexity Institute at the University of Virginia. His research interests are in the broad areas of network science, dynamical systems, combinatorial optimization, and distributed computing, and their applications to computational epidemiology and social networks.

Abstract:

As states begin to reopen, there is an urgent need for
COVID-19 projections that can guide decision-makers in relaxing
restrictions in a way that minimizes the chance of resurgence. Though
researchers have access to a wide variety of mathematical infectious
disease transmission models, limited data and uncertainties about
epidemiological features of COVID-19 make statistical calibration of
these models challenging. Futhermore, decision-makers are often more
interested in future projections under specified re-opening scenarios
than in estimated parameter values. In this presentation, I outline
mathematical and statistical approaches for projecting COVID-19
incidence, hospitalization, and deaths. I describe a simple class of
transmission models that balance parsimony with epidemiological
realism for the epidemic in Connecticut. Using unique access to data
from Connecticut, and information about the Governor's stated
reopening plans, we find that closure of schools and the statewide
“Stay Safe, Stay Home” order have effectively reduced COVID-19
transmission in Connecticut, with model projections estimating
incidence at about 1,500 new infections per day. If close
interpersonal contact increases quickly in Connecticut following
reopening on May 20, the state is at risk of a substantial increase
COVID-19 infections, hospitalizations, and deaths by late Summer 2020.
However, real-time metrics including case counts, hospitalizations,
and deaths may fail to provide enough advance warning to avoid
resurgence. Substantial uncertainty remains in our knowledge of
cumulative COVID-19 incidence, the proportion of infected individuals
who are asymptomatic, infectiousness of children, the effects of
testing and contact tracing on isolation of infected individuals, and
how contact patterns may change following reopening.

Abstract:

Abstract:

One of the most far reaching proofs of Cayley's formula, that the number n^{n-2} counts trees on n labeled vertices, is via Kirchhoff's Matrix Tree theorem. After Denes, Schaeffer, and many others, there is a well-exploited correspondence between trees and transitive factorizations in the symmetric group; in particular, the number n^{n-2} counts shortest factorizations of the long cycle (12..n) in transpositions. Furthermore, Burman and Zvonkine (and independently Alon and Kozma) have given a "higher-genus" formula that enumerates arbitrary length factorizations of long cycles, where each transposition (ij) is weighted by its own variable w_ij, and which has a product form involving the eigenvalues of the Laplacian L(K_n) of the complete graph.In joint work with Guillaume Chapuy, we consider a (partial) analog of the weighted Laplacian for complex reflection groups W. The weights are specified via any given flag of parabolic subgroups, generalizing the definition of Jucys-Murphy elements. We prove a product formula for the enumeration of weighted reflection factorizations of Coxeter elements, that subsumes the Chapuy-Stump formula and in part the Burman-Zvonkine formula. Its proof is based on an interesting fact that relates the exterior powers of the reflection representation with those W-characters that are non-zero on the Coxeter class. We present some further applications of these techniques, in particular, a uniform simple(r) way to produce the chain number h^n*n!/|W| of the noncrossing lattice NC(W). An extended abstract for this work was accepted for FPSAC 2020 and is available at my website (https://www.irif.fr/~douvr001/).

Abstract:

Two kinds of singular transformations of a rotationally invariant \alpha-stable process (x(t))_{t \ge 0} in a d-dimensional Euclidean space R^d are considered. They are both connected with the notion of a local time on a given surface S in R d for the process (x(t))_{t \ge 0} (it is supposed that \alpha \in (1, 2) and d \ge 2). The first transformation is determined by a given continuous function (p(x))_{x \in S} with non-negative values and it consists in killing the process (x(t))_{t \ge 0} at a point x \in S with “the intensity p(x)”. This kind of membranes can be called an elastic screen by analogy to that in the theory of diffusion processes. The second transformation is likewise determined by a given function (p(x))_{x \in S} with positive values and its result is the process (x(t))_{t \ge 0} for which any point x \in S is sticky with “the intensity r(x)”. It is shown that each one of these membranes is associated with some initial-boundary value problem for a pseudo-differential equation related to the process (x(t))_{t \ge 0}. These facts are established with the help of some generalization of classical theory of single-layer potentials for situations where, instead of differential, the pseudo-differential equation mentioned above is considered.

Abstract:

Testing django 2.2 upgrade

Abstract:

The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. However, the numerical solution of this problem remains extremely challenging and standard techniques can fail to compute the correct solution. Recently, several methods have been proposed that obtain the solution by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to a non-standard implicit boundary condition. Unfortunately, standard techniques for analyzing numerical methods for fully nonlinear elliptic equations fail in this setting. We introduce a modified PDE that couples the usual Monge-Ampere equation to a Hamilton-Jacobi equation that restricts the transportation of mass. This leads to a simple framework for guaranteeing that a numerical method will converge to the true solution, which applies to a large class of approximation schemes. We describe some simple examples. A range of challenging computational examples demonstrate the effectiveness of this method, including the recent application of these methods to problems in beam shaping and seismic inversion.https://umn.zoom.us/j/531493431  

Abstract:

In 2019, D'Adderio proved that if G(x;q) is a vertical-strip LLT polynomial, then G(x;q+1) is positive in the elementary symmetric functions basis. A conjectured formula
for the coefficients in this basis was given earlier in 2019 by Alexandersson. We give a new proof of D'Adderio's result which also proves the conjectured formula.

The problem of finding such an e-expansion is surprisingly similar to the still open problem of Shareshian-Wachs, regarding the e-expansion of chromatic polynomials associated with unit-interval graphs. We shall discuss this connection as well.

Abstract:

The {\em Vandermonde determinant} is ubiquitous in algebraic combinatorics and representation theory. One application of the Vandermonde is as a generator for a `harmonic' model of the coinvariant ring attached to the symmetric group which eschews the use of -- and computational issues involved with -- quotient rings. We present an extension of the Vandermonde determinant to {\em superspace} (a symmetric algebra tensor an exterior algebra) and use it to generate a variety of modules including the recently defined `Delta Conjecture coinvariant rings' of Haglund-Rhoades-Shimozono as well as (conjecturally) a trigraded module for the full Delta Conjecture. We use superspace Vandermondes to build bigraded superspace quotients tied to the geometry of {\em spanning configurations} studied by Pawlowski-Rhoades which satisfy a superspace version of Poincar\'e Duality and (conjecturally) exhibit unimodality properties which suggest a superspace version of Hard Lefschetz. Joint with Andy Wilson.

Abstract:

The box-ball system is a cellular automaton in which a sequence of balls moves along a row of boxes. An interesting feature of this automaton is its soliton behavior: regardless of the initial state, the balls in the system eventually form themselves into connected blocks (solitons) which remain together for the rest of time.

In 2014, T. Lam, P. Pylyavskyy, and R. Sakamoto conjectured a formula which describes the solitons resulting from an initial state of the box-ball system in terms of the tropicalization of certain polynomials they called cylindric loop Schur functions. In this talk, I will describe the various ingredients of this conjecture and discuss its proof.

Abstract:

Slender body theory (SBT) facilitates computational simulations of thin filaments in a 3D viscous fluid by approximating the hydrodynamic effect of each fiber as the flow due to a line force density along a 1D curve. Despite the popularity of SBT in computational models, there had been no rigorous analysis of the error in using SBT to approximate the interaction of a thin fiber with fluid. In this talk, we develop a PDE framework for analyzing the error introduced by this approximation. In particular, given a 1D force along the fiber centerline, we define a notion of ‘true’ solution to the full 3D slender body problem and obtain an error estimate for SBT in terms of the fiber radius. This places slender body theory on firm theoretical footing. We also present similar estimates in case of free-ended and rigid filaments.

Abstract:

Bio: https://www.linkedin.com/in/shaearmstrong/

Abstract:

A formula of Cayley (1889) says that the number of trees on vertex set [n]:={1,2,...,n} is n^{n-2}. Among its many proofs, my favorite is a gorgeous bijection due to Andre Joyal in 1981. One can also view Cayley's formula as asserting that there are n^{n-1} vertex-rooted trees on [n], or equivalently n^{n-1} eventually constant self-maps on [n].

This talk will review Joyal's proof, and its recent revisitation by Tom Leinster in arXiv:1912.12562. Leinster gives a beautiful q-analogue of the proof, that proves a q-analogous theorem of Fine and Herstein (1958). The latter theorem counts those linear self-maps of an n-dimensional vector space over a finite field F_q which are eventually constant, that is, nilpotent as linear maps.

Abstract:

In this talk, we'll discuss the problem of constructing meaningful distances between probability distributions given only finite samples from each distribution. We approach this through the use of data-adaptive and localized kernels, and in a variety of contexts. First, we construct locally adaptive kernels to define fast pairwise distances between distributions, with applications to unsupervised clustering. Then, we construct localized kernels to determine a statistical framework for determining where two distributions differ, with applications to measure detection for generative models. Finally, we'll begin to address the question of measure detection without a priori known labels of which distribution a point came from. This is addressed through active learning, in which one can choose a small number of points at which to query a label. This is ongoing work with Xiuyuan Cheng (Duke) and Hrushikesh Mhaskar (CGU), among others.

Alex Cloninger is an Assistant Professor in the Mathematics Department and the Hal?c?o?lu Data Science Institute at UCSD. He received his PhD in Applied Mathematics and Scientific Computation from the University of Maryland in 2014, and was then an NSF Postdoc and Gibbs Assistant Professor of Mathematics at Yale University until 2017, when he joined UCSD. Alex researches problems around the analysis of high dimensional data. He focuses on approaches that model the data as being locally lower dimensional, including data concentrated near manifolds or subspaces. These types of problems arise in a number of scientific disciplines, including imaging, medicine, and artificial intelligence, and the techniques developed relate to a number of machine learning and statistical algorithms, including deep learning, network analysis, and measuring distances between probability distributions.

Abstract:

We discuss geometry-based statistical learning techniques for performing model reduction and modeling of certain classes of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of a system, e.g. from molecular dynamics, and we estimate, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of then system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain, via eigenfunctions of an empirical Fokker-Planck equation (constructed from data), reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas.

In the second setting we only have access to a (large number of expensive) simulators that can return short paths of the stochastic system, and introduce a statistical learning framework for estimating local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by effective stochastic diffusion-like equations.

Abstract:

TBA

Abstract:

The -Grothendieck polynomials are simultaneous generalizations of Schubert and Grothendieck polynomials that arise in the study of the connective K-theory of the flag variety. They can be calculated as a generating function of combinatorial objects known as pipe dreams, as well as recursively via geometrically-motivated divided difference operators. We combine these two points of view by defining a chromatic lattice model whose partition function is a -Grothendieck polynomial. This is joint work-in-progress with Ben Brubaker, Claire Frechette, Andy Hardt, and Emily Tibor.

Abstract:

The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced by Jacot et al., where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in many other recent papers. In the overparametrization regime, a fully-trained deep neural network is indeed equivalent to the kernel regression predictor using the limiting NTK. And the gradient descent achieves zero training loss for a deep overparameterized neural network. However, it was observed by Arora et al. that there is a performance gap between the kernel regression using the limiting NTK and the deep neural networks. This performance gap is likely to originate from the change of the NTK along training due to the finite width effect.  The change of the NTK along the training is central to describe the generalization features of deep neural networks.In the work, we study the dynamic of the NTK for finite width deep fully-connected neural networks. We derive an infinite hierarchy of ordinary differential equations, the neural tangent hierarchy (NTH) which captures the gradient descent dynamic of the deep neural network. Moreover, under certain conditions on the neural network width and the data set dimension, we prove that the truncated hierarchy of NTH approximates the dynamic of the NTK up to arbitrary precision. This is a joint work with Horng-Tzer Yau.

Abstract:

The experimental and mathematical study of the motion of bodies immersed in fluids with variable concentration fields (e.g. temperature or salinity) is a problem of great interest in many applications, including delivery of chemicals in laminar micro-channels, or in the distribution
of matter in the ocean. In this lecture we present some recent experimental and mathematical advances we have made for several such problems. First, we review results on how the shape of a tube can be used to sculpt the profile of chemical delivery in pressure driven laminar shear flows. Then, we explore recent results for the behavior of matter trapped vertically in a variable density water column. For this second problem, we experimentally observe and mathematically model a new attractive mechanism we have found in our laboratory by which particles suspended within stratification may self-assemble and form large aggregates without need for short range binding effects (adhesion). This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and the geometry of the aggregate, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles. We experimentally observe that many particles demonstrate a collective motion revealing a system which self-assembles, appearing to solve jigsaw-like puzzles on its way to organizing into a large scale disc-like shape, with the effective force increasing as the collective disc radius grows. We overview our modeling and simulation campaign towards understanding this intriguing dynamics,
which may play an important role in the formation of particle clusters in lakes and oceans.

Abstract:

Contact homology is an invariant of the contact structure, which is an odd-dimensional counterpart of a symplectic structure. It was proposed by Eliashberg, Givental and Hofer in 2000. The application of contact homology and its variants include distinguishing contact structures, knot invariants, the Weinstein conjecture and generalization, and calculating Gromov-Witten invariants. In this talk, I will start with the notion of contact structures, then give a heuristic definition of the contact homology as an infinite dimensional Morse homology, and finally explain the major difficulties to make the definition rigorous. This is a joint work with Ko Honda.

Abstract:

I will talk about the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the unique steady state as time goes to infinity (equilibration), with an explicit algebraic rate. The proof is based on a uniform-in-time bound on the first moment of the density distribution, combined with an energy dissipation rate estimate. This is the first result on the equilibration of aggregation-diffusion equations for a general class of weakly confining potentials $W(r)$: those satisfying $\lim_{r\rightarrow\infty}W(r)<\infty$.

Abstract:

TBA

Abstract:

If you can't measure it, you can't manage it - Using math tricks in measuring ESG performance.
We will look at the science of scoring, focusing on quant techniques that enable investors to make choices based on meaningful ESG data.

Bio: https://www.linkedin.com/in/michaelzhan/

Abstract:

It is a classical result that in the N-body problem with positive energy, all solutions are unbounded in both forward and backward time. If all of the mutual distances between the particles tend to infinity with nonzero speed, the solution in called purely hyperbolic. In this case there is a well-defined asymptotic shape of the configuration of N points. We consider the scattering problem for solutions which are purely hyperbolic in both forward and backward time: given an initial shape at time minus infinity, which final shapes at time plus infinity can be reached via purely hyperbolic motions ? I will describe some recent work on this problem using a variation on McGehee's blow-up technique. After a change of coordinates and timescale we obtain a well-defined limiting flow at infinity and use it to get Chazy-type asymptotic estimates on the positions of the bodies and to study scattering solutions near infinity. This is joint work with G. Yu, R. Montgomery and N. Duignan.

Abstract:

 Two teams of Financial Mathematics graduate students will present the results of their projects completed over an intensive 10-day winter workshop.  The first presentation will be on Data Analysis, Visualization and Statistical/Machine Learning Modeling for Mortgage Prepayment and Delinquency Rates. The Industry Mentor for this project , in attendance for the talk, was: He LuThe second presentation will be on Inflation Rate Curve Modeling.  This project was led by Industry Mentor Matt Abroe.  

Abstract:

Graph cuts have been studied for decades in the mathematics and computer science communities. For example, a celebrated result in optimization relates the cut minimization problem (under some membership constraints) with a maximum flow problem via the well known max flow-min cut duality theorem. Another very important problem formulated in the computer science community that uses graph cuts is motivated by data clustering: while direct minimization of a graph cut is reasonable as it penalizes the size of interfaces, the optimization is not able to rule out partitions of data into groups that are highly asymmetric in terms of size. In order to avoid trivial partitions, and provide a more reasonable clustering approach, the original optimization of graph cuts is modified by adding an extra balancing term to the objective function in either additive or multiplicative form. A canonical example, with historical motivation, is the so called Cheeger cut problem. Minimization of Cheeger cuts for data clustering is on the one hand intuitively motivated, but on the other, is a highly non-convex optimization problem with a pessimistic NP hard label tamped on it (at least in a worst case scenario setting). Nevertheless, in the past decade or so, several algorithmic improvements made the minimization of Cheeger cuts more feasible, and at the same time there was a renewed interest in studying statistical properties of Cheeger cuts. New analytical ideas have provided new tools to attack problems that were elusive using classical approaches from statistics and statistical learning theory. Despite the advances, several questions remain unanswered.

The purpose of this talk is to present some of these theoretical developments, with emphasis on new results where, for the first time, high probability converge rates of Cheeger cuts of proximity graphs over data clouds are deduced. These quantitative convergence rates are obtained by building bridges between the original clustering problem and another field within the mathematical analysis community that has seen enormous advancements in the past few years: quantitative isoperimetric inequalities. This connection serves as a metaphor for how the mathematical analyst may be able to contribute to answer theoretical questions in machine learning, and how one may be able to deduce statistical properties of solutions to learning optimization problems that have a continuum counterpart.

Abstract:

I will talk about the two papers Eisenbud-Fløystad-Schreyer-2003 and Eisenbud-Erman-Schreyer-2015 in which they introduce Tate resolutions for projective spaces and products of projective spaces. As an application, I will talk about Horrocks' criterion for vector bundles in those settings.

Abstract:

I will start with the basics of line bundles and vector bundles in commutative algebra, specifically over the projective space. This is an introduction for the talk on Friday at the Commutative Algebra Seminar about the Tate resolutions for projective spaces and products of projective spaces.

Abstract:

Moduli spaces of Riemann surfaces are a fundamental object in algebraic geometry. Their geometry is rich and holds many outstanding mysteries. One way to probe genus g Riemann surfaces is to understand the maps they admit to the simplest Riemann surface, the Riemann sphere. In my talk, I will describe one facet of this approach, a formula for the double ramification cycle (joint work with R. Pandharipande, A. Pixton and D. Zvonkine). Along the way, we will see connections to combinatorics, number theory and symplectic geometry.

Abstract:

Let E be an elliptic curve over the rationals. Divisibility of the set of rational points on E by some integer m can occur locally or globally. If E has global divisibility by m, then E has local divisibility by m; however, work of Katz shows that the converse is only guaranteed up to isogeny. Cullinan and Voight showed that the probability than an elliptic curve has global divisibility by an integer m is non-zero for all integers m allowed by Mazur's classification of rational torsion on elliptic curves. In this talk, I will discuss the probability that E has global divisibility by 5 or 7, given that E has local divisibility by 5 or 7, respectively.

Abstract:

 In this talk, I'll discuss the contractual freight business, in which a large shipper makes a contract with a company like CH Robinson to procure carriers (trucks) for their goods over the course of a year for a given rate, as opposed to using the volatile "spot" or transactional market. Because these year-long contracts aren't legally binding, some shippers treat them more like an American option on the underlying price of freight, but this has game-theoretic economic consequences for the shipper! Dr. Taipale, Data Scientist at C.H. Robinson will also talk about the data science and mathematical skills that are important for her job at C.H. Robinson

Bio :  https://www.linkedin.com/in/kaisa-taipale-2630256/detail/contact-info/

Abstract:

For a local ring (A,m) of dimension n,we study the natural map from the n-th Koszul cohomology on a minimal set of generators of m to the top local cohomology of A supported at m. We construct complete normal domains for which this map is zero, thus answering a question of Dutta in the negative. If time permits, we present precise information on the kernel of this map for a large class of Stanley-Reisner rings.

Abstract:

Understanding the spectral stability of solutions to partial differential equations is an important step in predicting long-time dynamics. Recently, it has been shown that a topological invariant known as the Maslov Index can play an important role in determining spectral stability for systems that have a symplectic structure. In addition, related ideas have lead to a suggested generalization of the notion of spatial dynamics to general, multidimensional spatial domains. In this talk, the notions of spectral stability, the Maslov Index, and spatial dynamics will be introduced and an overview of recent results will be given.

Abstract:

One hundred years ago, when the theory of self-adjoint operators was
just being developed, Hilbert and Polya independently speculated on
the possibility of using self-adjoint operators (the fact that all
eigenvalues are real) to prove the Riemann Hypothesis. In 1977, a
flawed numerical computation of eigenvalues of the invariant
Laplace-Beltrami operator on the modular curve seemed to indicate that
zeros of zeta appeared as spectral parameters.

Abstract:

(joint work with Zach Hamaker; arXiv:1809.1057) Monotone triangles are combinatorial objects in bijection with alternating sign matrices, a fascinating generalization of permutation matrices. We will review this connection, and the fact that strong Bruhat order on permutations has a natural extension to monotone triangles. We will then explain an analogous extension of the weak Bruhat order on permutations to monotone triangles. This comes from extending the notions of descents in permutations and the "bubble-sorting" action of the 0-Hecke algebra on permutations to monotone triangles. We will also explain one of our motivations: to give a natural family of shellings for Terwilliger's recently defined order on subsets.

Abstract:

We consider (volume-)frozen percolation on the triangular lattice. The model can be described informally as follows. Fix a large integer $N$. Initially, all vertices are vacant. We let clusters grow (vertices become occupied) as long as their volume is strictly smaller than $N$, and they stop growing (they "freeze") when their volume becomes at least $N$. A vertex $v$ is frozen if it belongs to an occupied cluster with volume at least $N$.

In this model, there exists a sequence of "exceptional scales" $(m_k(N))$: roughly speaking, if we consider frozen percolation in a box of side length $m_k(N)$, then as $N\to\infty$, the probability that $0$ is frozen in the final configuration is bounded away from $0$; while if we consider the process in a box of side length that is far from $m_k(N)$ and $m_{k+1}(N)$ (but between them), then as $N\to\infty$, the corresponding probability will go to $0$. The limiting exception scale, $m_\infty(N)$, is not studied and almost nothing is known. In an ongoing project with Pierre Nolin, we show that if we consider the process in a box of side length $m_\infty(N)$, then there are "avalanches" of freezings: the number of frozen circuits surrounding the origin divided by $\log\log{N}$ converges to an explicit constant in probability. If time allows, I will also talk about the analogous result in the forest fire process.

Abstract:

Renewable energy sources - solar and wind - are inherently variable, and unlike a traditional power plant, energy generation can't be 'turned up' or 'turned down' at will. In the past several years, the rapidly falling price of Lithium Ion batteries and related technology has made it more feasible than ever to build solar or wind farms coupled with energy storage capabilities to mitigate some of the variability of the renewable resource and provide more control over the energy output of the farm. In this talk, I will give an overview of working at NextEra Analytics and give examples of how we are applying math to operate energy storage projects as well as evaluate potential new renewable + energy storage hybrid projects.

Madeline graduated in May from the University of Minnesota with a PhD in Mathematics co-advised by Drs. Gilad Lerman and Wei-Kuo Chen. She started in the Applied Math Group at NextEra Analytics in June and works primarily on math related to energy storage.

Abstract:

The vast collection of detailed personal data has enabled machine learning to have a tremendous impact on society. Algorithms now provide predictions and insights that are used to make or inform consequential decisions on people. Concerns have been raised that our heavy reliance on personal data and machine learning might compromise people’s privacy, produce new forms of discrimination, and violate other kinds of social norms. My research seeks to address this emerging tension between machine learning and society by focusing on two interconnected questions: 1) how to make machine learning better aligned with societal values, especially privacy and fairness, and 2) how to make machine learning methods more reliable and robust in social and economic dynamics. In this talk, I will provide an overview of my research and highlight some of my recent work on fairness in machine learning and differentially private synthetic data generation.

Steven Wu is an Assistant Professor of Computer Science and Engineering at the University of Minnesota. His research interests are in algorithms and machine learning, with a focus on privacy-preserving data analysis, algorithmic fairness, and algorithmic economics. From 2017 to 2018, he was a post-doc researcher at Microsoft Research-New York City in the Machine Learning and Algorithmic Economics groups. In 2017, he received his Ph.D. in computer science under the supervision of Michael Kearns and Aaron Roth at the University of Pennsylvania, where his doctoral dissertation received Penn’s Morris and Dorothy Rubinoff Award for best thesis. His research is supported by an Amazon Research Award, a Facebook Research Award, a Mozilla research grant, a Google Faculty Research Award, a J.P. Morgan Research Faculty Award, and the National Science Foundation.

Abstract:

One of the interests of low-dimensional topologists is
understanding which smooth 4-manifolds can bound a given 3-manifold, or,
as a special case, understanding the set of 3-manifolds up to so-called
homology cobordism (to be defined in the talk). This question turns out
to have applications to the study of triangulations of high-dimensional
manifolds, and is a natural proving ground for Floer-theoretic
techniques of studying 3-manifolds. In this talk, we will give some
structure theorems about the homology cobordism group, and show that
there are three-manifolds that are very far from having any of the seven
non-hyperbolic Thurston geometries. This talk includes joint work with
I. Dai, K. Hendricks, J. Hom, L. Truong, and I. Zemke.

Abstract:

If X is a smooth projective variety over a number field, the Hodge and Tate conjectures describe how information about the subvarieties of X is encoded in the cohomology of X. We explore the role that certain automorphic representations, called algebraic Hecke characters, can play in understanding which cohomology classes of X arise from subvarieties. We use this to deduce the Hodge and Tate conjectures for certain self-products of varieties, including some self-products of K3 surfaces. This is joint work with J. Lang.

Abstract:

Optimal Transport is concerned with the question of how to best move one measure to another (this could be sand on a beach or products from a warehouse to consumers). I will explain the basic definition of Wasserstein distance and then describe how it can be used as a tool to say interesting things in other fields. (1) How to get new regularity statements for classical objects in number theory almost for free (irrational rotations on the torus, quadratic residues in finite fields). (2) How to best distribute coffee shops over downtown Minneapolis. (3) Finally, how to obtain higher dimensional analogues of classical Sturm-Liouville theory: simply put, Sturm-Liouville theory says that eigenfunctions of the operator Ly = -y''(x) +p(x)y(x) (think of sin(kx) and cos(kx)) cannot have an arbitrary number of roots; we present a generalization to higher dimensions that is based on a simple (geometric) inequality.

Abstract:

Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches.
In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs.
In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.

Abstract:

Classically, a Siegel modular form is said to be singular or distinguished if many of its Fourier coefficients are 0, in a precise sense. I will explain the construction of singular and distinguished modular forms on the exceptional groups E_6, E_7, E_8. Moreover, time permitting, I will also explain an analogue of the Saito-Kurokawa lift, which produces cusp forms on Spin(8) and the exceptional group G_2 out of holomorphic Siegel modular forms on Sp(4).

Abstract:

In this talk we will first give an overview of the known
Gibbsian line ensembles in the KPZ universality class. Then we will
construct the discrete log-gamma line ensemble, which is associated
with the log-gamma polymers. This log-gamma line ensemble enjoys a
random walk Gibbs resampling invariance that follows from the
integrable nature of the log-gamma gamma polymer model via the
geometric RSK correspondence. By exploiting such resampling
invariance, we show the tightness of this log-gamma line ensemble
under the weak noise scaling. Furthermore, a Gibbs property, as
enjoyed by the KPZ line ensemble, holds for all subsequential limits.

Abstract:

Personalized recommendations are useful to assist users in their choices in web-based market places, entertainment engines, information providers. Learning individual preferences is an important step. Users express their preferences by rating, ranking, (possibly inconsistently) comparing, liking and clicking items. Such data contain information about the individual user’s ranking of the items. Click-through data can be seen as (consistent) pair comparisons. The Mallows rank model allows to analyse rank data, but its computational complexity has limited its use to a particular form, based on Kendall’s distance. We developed new computationally tractable methods for Bayesian inference in Mallows models that work with any right-invariant distance. Our method performs inference on the latent consensus ranking of all items and on the individual latent rankings by Bayesian augmentation. Current popular recommendation algorithms are based on matrix factorisations, have high accuracy and achieve good clickthrough rates. However diversity of the recommended items is often poor and most algorithms do not produce interpretable uncertainty quantifications of the recommendations. With a simulation study and real life data examples, we demonstrate that compared to matrix factorisation approaches, our Bayesian Mallows method makes personalized recommendations mpared to matrix factorisation approaches, our Bayesian Mallows method makes personalized recommendations.

Arnoldo Frigessi is professor of statistics at the University of Oslo, leads the Oslo Center for Biostatistics and Epidemiology and is director of BigInsight. BigInsight is a centre of excellence for research-based innovation, a consortium of industry, business, public actors and academia, developing model based machine learning methodologies for big data. Originally from Italy, where he had positions in Rome and Venice, he moved to Norway in 2019 as a researcher at the Norwegian Computing Centre, before he became professor at the University of Oslo.

Frigessi has developed statistical methodology motivated by specific problems in science, technology and industry. He has designed stochastic models to study principles, dynamics and patterns of complex dependence. Inference is usually based on computationally intensive stochastic algorithms. Currently, he has research collaborations in genomics, personalised therapy in cancer, infectious disease models, eHealth research, personalised and viral marketing, s

Abstract:

When G is a reductive (non-compact) Lie group, one can consider automorphic forms for G. These are functions on the locally symmetric space X_G associated to G that satisfy some sort of nice differential equation. When X_G has the structure of a complex manifold, the _modular forms_ for the group G are those automorphic forms that correspond to holomorphic functions on X_G. They possess close ties to arithmetic and algebraic geometry. For certain exceptional Lie groups G, the locally symmetric space X_G is not a complex manifold, yet nevertheless possesses a very special class of automorphic functions that behave similarly to the holomorphic modular forms above. Building upon work of Gan, Gross, Savin, and Wallach, I will define these modular forms and explain what is known about them.

Abstract:

Though Einstein's fundamental theory of general relativity has already celebrated its one hundredth birthday, there are still many outstanding unsolved problems. The Kerr stability conjecture is one of the most important open problems, which posits that the Kerr metrics are stable solutions of the vacuum Einstein equation. Over the past decade, there have been huge advances towards this conjecture based on the study of wave equations in black hole spacetimes and structures in the Einstein equation. In this talk, I will discuss the recent progress in the stability problems with special focus on the wave gauge.

Abstract:

In this talk we will describe how one can detect regularity in Jordan curves through analysis of associated geometric square functions. We will particularly focus on the resolution to a conjecture of L. Carleson. Joint work with Xavier Tolsa and Michele Villa (https://arxiv.org/abs/1909.08581).

Abstract:

In this talk, I will introduce a new unsupervised learning framework for data points that lie on or close to a smooth manifold naturally equipped with a group action. In many applications, such as cryo-electron microscopy image analysis and shape analysis, the dataset of interest consists of images or shapes of potentially high spatial resolution, and admits a natural group action that plays the role of a nuisance or latent variable that needs to be quotient out before useful information is revealed. We define the pairwise group-invariant distance and the corresponding optimal alignment. We construct a graph from the dataset, where each vertex represents a data point and the edges connect points with small group-invariant distance. In addition, each edge is associated with the estimated optimal alignment group. Inspired by the vector diffusion maps proposed by Singer and Wu, we explore the cycle consistency of the group transformations under multiple irreducible representations to define new similarity measures for the data. Utilizing the representation theoretic mechanism, multiple associated vector bundles can be constructed over the orbit space, providing multiple views for learning the geometry of the underlying base manifold from noisy observations. I will introduce three approaches to systematically combine the information from different representations, and show that by exploring the redundancy created artificially across irreducible representations of the transformation group, we can get drastically improved nearest neighbor identification, when a large portion of the true edges are corrupted. I will also show the application in cryo-electron microscopy image analysis.

Abstract:

Current personalized cancer treatment is based on biomarkers which allow assigning each patient to a subtype of the disease, for which treatment has been established. Such stratifiedpatient treatments represent a first important step away from one-size-fits-all treatment.However, the accuracy of disease classification comes short in the granularity of thepersonalization: it assigns patients to one of a few classes, within which heterogeneity inresponse to therapy usually is still very large. In addition, the combinatorial explosivequantity of combinations of cancer drugs, doses and regimens, makes clinical testingimpossible. We propose a new strategy for personalised cancer therapy, based on producing acopy of the patient’s tumour in a computer, and to expose this synthetic copy to multiplepotential therapies. We show how mechanistic mathematical modelling, patient specificinference and simulation can be used to predict the effect of combination therapies in a breastcancer. The model accounts for complex interactions at the cellular and molecular level, andis able of bridging multiple spatial and temporal scales. The model is a combination ofordinary and partial differential equations, cellular automata and stochastic elements. Themodel is personalised by estimating multiple parameters from individual patient data,routinely acquired, including histopathology, imaging and molecular profiling. The resultsshow that mathematical models can be personalized to predict the effect of therapies in eachspecific patient. The approach is tested with data from five breast tumours collected in arecent neoadjuvant clinical phase II trial. The model predicted correctly the outcome after 12weeks treatment and showed by simulation how alternative treatment protocols would haveproduced different, and some times better, outcomes. This study is possibly the first onetowards personalized computer simulation of breast cancer treatment incorporating relevantbiologically-specific mechanisms and multi-type individual patient data in a mechanistic andmultiscale manner: a first step towards virtual treatment comparison.Xiaoran Lai, Oliver Geier, Thomas Fleischer, Øystein Garred, Elin Borgen, Simon Funke,Surendra Kumar, Marie Rognes, Therese Seierstad, Anne-Lise Børressen-Dale, VesselaKristensen, Olav Engebråten, Alvaro Köhn-Luque, and Arnoldo Frigessi, Tow