Past Seminars

Abstract:

Over one-dimensional bases, Gabber and Beilinson proved theorems on the commutation of the nearby cycle functor and the vanishing cycle functor with duality. In this talk, I will explain a way to unify the two theorems, confirming a prediction of Deligne. I will also discuss the case of higher-dimensional bases and applications to local acyclicity, following suggestions of Illusie and Gabber. This is joint work with Qing Lu.

Abstract:

Free resolutions over the polynomial ring have a storied and active
record of study. However, resolutions over other rings are much more
mysterious; even simple examples can be infinite! In these cases, we
look to any combinatorics of the ring to glean information. This talk
will present a minimal free resolution of the ground field over the
semigroup ring arising from rational normal 2-scrolls, and (if time
permits) a computation of the Betti numbers of the ground field for all
rational normal k-scrolls.

Abstract:

In 1985, Shi found a generalization of the Robinson-Schensted algorithm to (extended) affine symmetric groups and described their Kazhdan-Lusztig cells in terms of combinatorics. Recently, Chmutov, Lewis, Pylyavskyy, and Yudovina developed its generalization, called the affine matrix-ball construction (abbreviated AMBC). It provides a bijection from an (extended) affine symmetric group to the set of triples (P,Q,?) where P and Q are row-standard Young tableaux of the same shape and ? is an integer vector satisfying certain inequalities. In this talk, I will briefly explain this algorithm, and discuss how this is related to representation of (extended) affine symmetric groups, especially the asymptotic Hecke algebras introduced by Lusztig. This work is joint with Pavlo Pylyavskyy.

Abstract:

In this talk we will outline the basic premise for the analogy between knots (in manifolds) and primes (in number fields). This analogy involves some rather heavy definitions; we will review the topological background as needed, while taking a more intuitive angle on the arithmetic machinery. Time permitting, we will briefly sketch an extension of the Frobenius automorphism which is a major tool in understanding the analogy between the Legendre symbol and the linking number.

Abstract:

We continue the introduction of the Garver-Patrias-Thomas paper by
exploring what the map from a quiver representations to its generic Jordan
form looks like in the case of a Type A quiver, and explain how this map is
related to bijections and generating functions for plane partitions.

Abstract:

Shimura varieties are a certain class of algebraic varieties over number fields with lots of symmetries, introduced by Shimura and Deligne nearly half a century ago. They have been playing a central role in number theory and other areas. Langlands proposed a program to compute the L-functions and cohomology of Shimura varieites in 1970s; this was refined by Langlands-Rapoport and Kottwitz in 1980s. I will review some old and recent results in this direction.

Abstract:

One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian Gr(k,n) is a quotient of the ring S of symmetric polynomials in k variables. More precisely, it is the quotient of S by the ideal generated by the k consecutive complete homogeneous symmetric polynomials hn?k+1,hn?k+2,...,hn. We propose and begin to study a deformation of this quotient, in which the ideal is instead generated by hn?k+1?a1,hn?k+2?a2,...,hn?ak for some k fixed elements a1,a2,...,ak of the base ring. This generalizes both the classical and the quantum cohomology rings of Gr(k,n). We find two bases for the new quotient, as well as an S3-symmetry of its structure constants, a "rim hook rule" for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the ai as signed indeterminate), which suggests a geometric or combinatorial meaning for the quotient. There are multiple open questions and opportunities for further research.

Abstract:

We explore some of the major results in the study of finite Hecke algebras and their character tables. These algebras are useful in the study of representations of finite Chevalley groups, and also appear in the study of quantum groups and knot/link invariants. We'll run through some equivalent definitions of this versatile object, and then talk about a couple approaches to its character theory. In particular, Starkey's rule is a combinatorial formula for the character table of the type A finite Hecke algebra. We'll briefly sketch a proof of this result and talk about the possibility for extension to other types. Starkey's rule allows us to calculate the weights of Ocneanu traces, which are important invariants in knot theory relating to type A Hecke algebras, and a type B version of Starkey's rule would give us insights into the type B analogue to Ocneanu traces.

Abstract:

We develop a variational method for nonlinear equations with a gradient flow structure. Such equations arise in applications of a wide range, such as porous median flows, material science, animal swarms, and chemotaxis. Our method builds on the JKO framework, which evolves the equation as a gradient flow with respect to the Wasserstein metric. As a result, our method has built-in positivity preserving, entropy decreasing properties, and overcomes stability issue due to the strong nonlinearity and degeneracy. Furthermore, our method is massively parallelizable, and thus extremely efficient in high dimensions. Upon discretization of the PDE constraint, we also prove the ??convergence of the fully discrete optimization towards the continuum JKO scheme.

Abstract:

In this talk, I'll discuss ongoing work with Alejandro Morales generalizing a 30-year old result of Jackson on permutation enumeration: we consider the enumeration of arbitrary factorizations of a Coxeter element in a well generated finite complex reflection group, keeping track of the fixed space dimension of the factors. As in the case of the symmetric group, the factorization counts are ugly, so the goal is to choose a basis for the generating function in which the answer is nice. In the case of the infinite families of monomial matrices, we accomplish this via combinatorial arguments; a notion of transitivity of a factorization appears for the "type D" group G(m, m, n). I'll also describe some puzzling partial results in the exceptional cases.

Abstract:

Target is rapidly growing into the online retail space. Behind that growth, there are a lot of advanced mathematical problems to solve. Some mathematics problems in supply chain are easy to spot (traveling salesman problem!), and some are less obvious. We’ll talk through at least 2 less-obvious problems which are currently being investigated by Target’s Supply Chain analytics teams. The goal of this talk is to give insight into the broad range of problems we are solving at Target. Additionally, I’ll spend a little time talking about solutions. What makes a good solution in Industry? What makes a good problem?

Samantha Schumacher has been working math problems at Target for the last 5 years. She currently leads an analytics and engineering team for the network planning & digital fulfillment of Target’s supply chain. She has her PhD in applied mathematics from University of Minnesota and an undergraduate degree in theater from Smith College. So, if all goes according to plan, she will use those theater skills to keep the talk engaging! She is also the long-time author of the blog: www.SocialMathematics.net which considers the interaction of mathematics and the modern world.

Abstract:

Postnikov's plabic graphs in a disk are used to parametrize totally positive Grassmannians. One of the key features of this theory is that if a plabic graph is reduced, the face weights can be uniquely recovered from boundary measurements. On surfaces more complicated than a disk this property is lost. In this talk, we investigate a certain semi-local transformation of weights for plabic networks on a cylinder that preserves boundary measurements. We call this a plabic R-matrix. We explore the properties of the plabic R-matrix, including the symmetric group action it induces on plabic networks and its underlying cluster algebra structure.

Abstract:

We continue with Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes, and this has to do with a description of the idempotents in the partition algebras. We use a fundamental relationship that goes back to Green, between idempotents an algebra, and idempotents in a factor algebra by an idempotent, and in the algebra obtained by cutting by that idempotent.

Abstract:

In my forthcoming book, “The Code of Capital: How Law Creates Wealth and Inequality”, I argue that capital is coded in law, or more specifically in core institutions of private law, including property, collateral, trust, corporate, bankruptcy as well as contract law. These “modules” of the code of capital bestow critical attributes on simple assets, such as objects or claims, and there by transform them from simple into a wealth generating assets. These attributes are priority, durability, convertibility and universality, and critically, they depend on effective state backing. Recently, law has met a potential competitor in the form of the digital code. Blockchain and similar technologies, in particular, makes it possible to encode commitments such that they will be executed irrespective of a change of heart, or circumstances. Decentralized verification of resources and rights instead of ex post vindication of legal title promises a world without courts and without coercive law enforcement. In short, “a new code” may be on the horizon and its relation to the legacy, or legal, code raises fundamental questions about how to organize economic relations in an uncertain world and about who gets to set the parameters for doing so.

Bio: Katharina Pistor is the Edwin B. Parker Professor of Comparative Law at Columbia Law School and director of the Law School’s Center on Global Legal Transformation. Her research and teaching spans corporate law, corporate governance, money and finance, property rights, comparative law and law and development. She has published widely in legal and interdisciplinary journals and is the author can co-author of several books. She is the recipient of the Max Planck Research Award (2012) and of several grants by, among others the Institute for New Economic Thinking, and the National Science Foundation. Her most recent book, “The Code of Capital: How the Law Creates Wealth and Inequality” is forthcoming at Princeton University Press.

Abstract:

In the last 50 years computing power has experienced an enormous development: every two years the number of transistors has doubled since the 1970s. However, even nowadays when we can perform very fast computations it is not clear a priori if one can obtain rigorous results based on the output of computer calculations. In this talk I will explain the basics of interval analysis and how it can be used to prove theorems in different contexts of PDE, ranging from fluid dynamics to spectral geometry.

Abstract:

An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of poset

Abstract:

In R^n, the Bakry-Émery theorem states that the sharp constant in various functional inequalities for uniformly log-concave measures is no worse than the sharp constant for the Gaussian measure. As a consequence, uniformly log-concave measures inherit certain nice features of the Gaussian, like good measure concentration properties.

In this talk, I'll discuss quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants. In particular, if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it must (almost) split off a Gaussian factor. As a corollary, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs combine Stein’s method, optimal transport, and simple variational arguments.

Joint work with Max Fathi.

Abstract:

We will start on Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). This will be preceded by a review of the set-up in this category, which is built up out of partition algebras. Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes.

Abstract:

To play the dollar game on a graph, start by assigning to each vertex a number of dollars of either wealth or debt. From this initial state, called a "divisor", the vertices lend and borrow with their neighbors according to chip-firing rules in an attempt to reach a debt-free state. The set of all possible winning states is the "complete linear system" of the divisor. We are interested in determining its cardinality. In the figure below, each vertex represents one of the 201 winning positions resulting from giving ten dollars to one vertex of the cycle graph on five vertices. This is joint work with Sarah Brauner and Forrest Glebe.

Abstract:

I will discuss stochastic interacting particle systems in the Kardar-Parisi-Zhang universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable cases, i.e., for which certain observables can be written in an exact form suitable for asymptotic analysis. Examples include a new continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, limit shapes of particles density can be described in an explicit way. Asymptotics of fluctuations around infinite traffic jams, also available explicitly, bring phase transitions of a novel nature.

Abstract:

Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.

Abstract:

We will start on Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). This will be preceded by a review of the set-up in this category, which is built up out of partition algebras. Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes.

Abstract:

I will review results on classification of quantum superintegrable systems on two-dimensional Euclidean space allowing separation of variables in Cartesian coordinates and possessing an extra integral of third or fourth order. The exotic quantum potential satisfy a nonlinear ODE and have been shown to exhibit the Painleve property. I will also present different constructions of higher order superintegrable Hamiltonians involving Painleve transcendents using four types of building blocks which consist of 1D Hamiltonians allowing operators of the type Abelian, Heisenberg, Conformal or Ladder. Their integrals generate finitely generated polynomial algebras and representations can be exploited to calculate the energy spectrum. I will point out that for certain cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. I will describe how other sets of integrals can be build and used to provide a complete solution.

Abstract:

Let $G=G(N,p)$ be an Erd\H{o}s--R\'enyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

Abstract:

Complex biological systems involve multiple space and time scales. To get an integrated understanding of these systems involves multiscale modeling, computation and analysis. In this talk, I will discuss two such examples in cell biology and illustrate how to use multiscale methods to explain experimental data. The first example is on chemotaxis of bacterial populations. I will present recent progress on embedding information of single cell dynamics into models of cell population dynamics. I will clarify the scope of validity of the well-known Patlak-Keller-Segel chemotaxis equation and discuss alternative models when it breaks down. The second example is on the axonal cytoskeleton dynamics in health and disease. I will present a stochastic multiscale model that gave the first mechanistic explanation for the cytoskeleton segregation phenomena observed in many neurodegenerative diseases.

Abstract:

The natural analog of Teichmuller theory for hyperbolic manifolds in dimension 3 or greater is trivialized by Mostow Rigidity, so mathematicians have worked to understand more general deformations. Two well studied examples, convex real projective structures and complex hyperbolic structures, have been investigated extensively and provide independently developed deformation theories. Here we will discuss a surprising connection between the these, and construct a one parameter family of geometries deforming complex hyperbolic space into a new geometry built out of real projective space and its dual. This connects the aforementioned deformation theories and provides geometric motivation for a representation-theoretic observation of Cooper, Long, and Thistlethwaite

Abstract:

I will continue Katy's talk from last week, and talk about chapter 9 of Diamond and Shurman's "A First Course in Modular Forms." We'll define Galois representations corresponding to both elliptic curves and modular forms. Then we'll state the modularity theorem, which asserts that every elliptic curve corresponds to a modular form, and their Galois representations are equal. If we have time, we'll relate this back to the version of the theorem from last week, which equated points on an elliptic curve to Fourier coefficients of a modular form. The proof of the theorem, however, will be left as an exercise to the listener.

Abstract:

We study the Cauchy problem for the 3D Gross-Pitaevskii equation. Global well-posedness in the natural energy space was proved by Gerard.
We prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
The crucial ingredients are new generalized Strichartz estimates and some new observed "NULL" structures of the Gross-Pitaevskii equation after some normal form type transform.
This is a joint work with Zaher Hani and Kenji Nakanishi.

Abstract:

We use topological methods to investigate the average size of n-Selmer groups of elliptic curves over F_q(t).
Loosely speaking, the n-Selmer group of an elliptic curve measures objects which look like the n-torsion of the elliptic curve.
We relate the question of computing the average size of the n-Selmer group to demonstrating
homological stability for a sequence of moduli spaces
of these n-Selmer elements.
Via monodromy arguments, we show the number of components of these moduli spaces stabilizes, which determines the
average size after taking a large q limit.

Abstract:

The convex hull of a set of points,C, serves to expose extremal properties of C and can help identify elements in C of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point’s relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be completed on reasonably large, high-dimensional data sets.

Lori Ziegelmeier received an A.S. and A.A. from Colby Community College in 2005. She completed her B.S. in Mathematics and B.A. in Liberal Arts and History, M.S. in Mathematics, and Ph.D. in Mathematics all at Colorado State University in 2007, 2009, and 2013, respectively. Since completing her doctoral degree, she has been in the Department of Mathematics, Statistics, and Computer Science at Macalester College and is currently an Assistant Professor. Her research is in the area of geometric and topological data analysis.

Abstract:

Data visualization is often overlooked by people working with data and/or modeling but can in fact reveal very useful insight into problems at hand. R is an open-source programming language for statistical computing and graphics with increasing popularity among practitioners in data science and machine learning in recent years. The "ggplot2" package in R, in particular, provides very powerful, intuitive and versatile tools for data visualization. An overview of these tools will be presented in this talk accompanied by a live demo.

Abstract:

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture. A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers.

Abstract:

Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three paradigmatic facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

Abstract:

We will complete the steps in the definition of Deligne's category described in section 2.2 of the paper, On blocks of Deligne's category Rep(S_t), by Comes and Ostrik.

Abstract:

The general chemistry program has been experimenting with several new pedagogies in order to improve student learning and successful outcomes. I’ll begin with our placement system and new adaptive courseware/learning analytics project, and will discuss any other project as time allows.

1. ALEKS Placement/Learning Modules
2. Adaptive Courseware (Smart Sparrow)
Using Adaptivity and Learning Analytics to Identify At-Risk Students and Implement Early Interventions
3. Absolute grading in 1061
Six Lower-Stakes Exams + Final
E-proctored Exams – available over a 12 hour window
4. Credit/No Credit Grading in Lab Courses + Specifications Grading

Abstract:

For as long as people have studied geometry, they have counted
geometric objects. For example, Euclid's Elements starts with the
postulate that there is exactly one line passing through two distinct
points in the plane. Since then, the kinds of counting problems we are
able to pose and to answer has grown. Today enumerative geometry is a
rich subject with connections to many fields, including combinatorics,
physics, representation theory, number theory and integrable systems.

In this talk, I will show how to solve several classical counting
questions. Then I will describe a more modern problem with roots in
string theory which has been the subject of intense study for the last
two decades, namely the study of the Gromov-Witten invariants of the
quintic threefold, a Calabi-Yau manifold. I will explain a recent
break-through in understanding the higher genus invariants that stems
from a seemingly unrelated problem related to the study of holomorphic
differentials on Riemann surfaces.

Abstract:

The Modularity Theorem (also known as the Taniyama-Shimura-Weil Conjecture) states, essentially, that every elliptic curve arises from a modular form. It is a special case of the Langlands correspondence and was a major part of Andrew Wiles' proof of Fermat's Last Theorem. In this talk, I will sketchily discuss the theorem in its various forms, leading up to the statement that the L-function of an elliptic curve agrees with the L-function of a modular form. Along the way, we will encounter some algebraic geometry words, e.g. "moduli space," "universal curve," and "sheaf."

Abstract:

Traveling waves arise in partial differential equations in a broad range of applications, for instance chemical reactions, propagation of electrical signals, or the dynamics of traffic jams. The notion of stability of a traveling wave solution concerns its resilience to small perturbations and can usually be inferred from an eigenvalue problem obtained by linearizing the PDE about the solution. In this talk, I will discuss these ideas in the context of the FitzHugh--Nagumo system, a simplified model of nerve impulse propagation. I will present existence and stability results for (multi)pulse solutions, and I will describe a phenomenon whereby unstable eigenvalues accumulate as a single pulse is continuously deformed into a double pulse

Abstract:

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop an approach by using the Hamiltonian structures of the linearized equation and an instability index formula to find the sharp stability conditions. We studied the flow with Sinus profile in details and found the sharp stability boundary in the whole parameter space, which corrected previous results in the fluid literature. The addition of the Coriolis force brings some fundamental changes to the stability of shear flows. Moreover, we also study the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping near the shear flows. This is joint work with Hao Zhu and Jincheng Yang.

Abstract:

We consider the one-dimensional linear free space Schrödinger equation on a bounded interval subject to homogeneous linear boundary conditions. We prove that, in the case of pseudoperiodic boundary conditions, the solution of the initial-boundary value problem exhibits the phenomenon of revival at specific ("rational'') times, meaning that it is a linear combination of a certain number of copies of the initial datum. Equivalently, the fundamental solution at these times is a finite linear combination of delta functions. At other ("irrational'') times, for suitably rough initial data, e.g., a step or more general piecewise constant function, the solution exhibits a continuous but fractal-like profile. Further, we express the solution for general homogeneous linear boundary conditions in terms of numerically computable eigenfunctions. (Joint work with Peter Olver and David Smith)

Abstract:

A nematic liquid crystal is essentially a liquid that exhibits some orientational order. The liquid crystalline state is typically observed in materials composed of highly asymmetric molecules for temperatures in the intermediate range when the material is neither a classical liquid nor a solid. In this talk we will be interested in describing a phase transition between the nematic and the isotopic state when the nematic order melts and the liquid crystal turns into a regular isotopic fluid. The experimental observations of this process indicate that the transition proceeds via evolution of interfaces separating different phases where the interfaces are generally not smooth. Our goal in this talk is to explain presence of these phase boundary singularities from a mathematical point of view .

The nematic-to-isotopic phase transition can be described within the so called Landau-de Genes variational theory for a tensor-valued order parameter. Because this theory is rather complex, in order to develop an initial understanding of transitions between the ordered and disordered states, we propose a simpler toy model based on the modified Ginzburg-Landau-type energy defined over vector fields on the plane. The corresponding variational model consists of anisotropic gradient terms and a potential that vanishes on two disconnected sets. While this model may not quantitatively describe the nematic state, the topology of the simplified target set retains the crucial features of the full problem. The principal observation from the study of the simplified model is that the phase boundary singularities can be explained by large disparity between the elastic constants that specify the gradient contribution to the energy. In the talk we will present a combination of rigorous analysis and numerics that leads to this conclusion.

Abstract:

The pentagram map is a discrete dynamical system on the space of plane polygons. Gekhtman, Shapiro, Tabachnikov, and Vainshtein used the combinatorics and Poisson geometry associated to certain networks/quivers on surfaces to prove that this map is integrable. Recently, Mari-Beffa and Felipe introduced a version of the pentagram map on Grassmannians, and found a Lax representation. I will generalize the techniques of Gekhtman et. al. to the Grassmann case, and show this gives a "formal" integrable system in a non-commutative algebra.

Abstract:

Following section 2.2 of the paper, On blocks of Deligne's category Rep(S_t), by Comes and Ostrik, we give the definition of this category and explore some of its first properties.

Abstract:

Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithm, I will present a few new applications of interpolative decomposition in numerical partial differential equations, quantum chemistry, and machine learning.

Abstract:

(joint work with A. Shepler and E. Sommers)
Complex reflection groups W are the finite subgroups of GL_n(C) with the following property: when they act on polynomials in n variables, their invariant ring is again a polynomial algebra. It is also known by
a result of Eagon and Hochster that, for any of their W-representations U, the U-isotypic polynomials form a _free_ module over the W-invariant subalgebra. In a few cases, we know an explicit basis for these U-isotypic polynomials. This talk will discuss a class of complex reflection groups W, sometimes called the "coincidental types", where conjecturally we know explicit bases for the U-isotypic component when U is any tensor product of the exterior powers of the reflection representation and its dual. This conjecture would explain pleasant product formulas for combinatorial objects, such as face numbers for finite type cluster complexes.

Abstract:

The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.

However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.

We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur

Abstract:

In this talk, we introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.

Abstract:

Network dynamical systems play an important role in many fields of science; whenever there are agents whose time evolution is linked through some interaction structure, we may view the system as a network and model it accordingly. However, despite their prevalence, network dynamical systems are in general not well understood. One can identify two reasons for this. First of all, many coordinate changes and other transformations from well-known dynamical systems techniques do not respect the underlying network structure. Second of all, despite this somewhat `ethereal' character, systems with a network structure often display behavior that is highly anomalous for general dynamical systems. Examples of this include very unusual bifurcation scenarios and high spectral degeneracies. As a possible explanation of this, it can be shown that a large class of network ODEs admit hidden symmetry, which may be discovered through the so-called fundamental network construction. In most cases, this underlying symmetry does not come from a group though, but rather from a more general algebraic structure such as a monoid or category. I will show how the fundamental network allows one to adapt techniques from dynamical systems theory to a network setting, and how some of the more unusual properties of networks may be explained. In doing so, I will mostly focus on spectral properties of linear network maps.

Abstract:

The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk I will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be presented.

Abstract:

TBA

Abstract:

The solution of large symmetric real (or Hermitian complex) eigenvalue
problems is central to applications ranging from electronic structure
calculations to the study of vibrations in mechanical systems. A few
of these applications require the computation of a large number of
eigenvalues and associated eigenvectors. For example, when dealing
with excited states in quantum mechanics, it is not uncommon to seek a
few tens of thousands of eigenvalues of matrices of sizes in the tens
of millions. In such situations it is imperative to resort to
`spectrum slicing' strategies, i.e., strategies that extract slices of
the spectrum independently. The presentation will discuss a few of
these techniques that are based on a combination of filtering
(polynomial, rational) and standard projection methods (Lanczos,
subspace iteration). The talk will also discuss our recently released
code named EVSL (for eigenvalues slicing library) that implements
these ideas.

Abstract:

One of the many interpretations of Catalan numbers is that Cn counts necklaces with n white beads and n+1 black beads. Motivated by this, we generalize the Catalan numbers and their q-analogues to many other necklaces. In this talk, we will discuss two surprising properties of these generalizations. First, they exhibit a q=-1 phenomenon with respect to necklace reflection, which can be extended to a cyclic sieving phenomenon for more exotic symmetries. Second, they conjecturally satisfy a "parity-unimodality" property, which in classical cases is a combinatorial shadow of a certain sl2 representation.

Abstract:

Seagate is the global leader in data storage solutions, developing amazing products that enable people and businesses around the world to create, share and preserve their most critical memories and business data. Over the years the amount of information stored has grown from megabytes all the way to geopbytes, confirming the need to successfully store and access huge amounts of data. As demand for storage technology grows the need for greater efficiency and more advanced capabilities continues to evolve. Today data storage is more than just archiving; it’s about providing ways to analyze information, understand patterns and behavior, to re-live experiences and memories. It’s about harnessing stored information for growth and innovation. Seagate is building on its heritage of storage leadership to solve the challenge of getting more out of the living information that’s produced everyday. What began with one storage innovation has morphed into many systems and solutions becoming faster, more reliable and expansive. These considerations will be explored in this lecture on the application of Advanced Dynamics System Analysis & Synthesis to provide Solutions for future Storage Challenges.

Bio
Dr. Raye A. Sosseh is a managing Principal engineer and leads the Advanced Dynamics Group in Seagate’s Advanced Product Development department. His background is in dynamics and controls of embedded systems with particular applications to Hard Disk Drives. Dr. Sosseh holds a Ph.D. in Mechanical Engineering and a Masters in Electrical Engineering from the Georgia Institute of Technology. He joined Seagate in 2001 and has contributed to the staging and development of numerous products in Seagate’s Mission Critical and Cloud solutions swimlanes. Dr. Sosseh has over 10 patents assigned and pending patents. Raye and his family enjoy the lakes and restaurants in the Minneapolis area.

Abstract:

We continue to discuss how the Riemann-zeta function plays a role in different areas of physics, from condensed matter theory to quantum mechanics

Abstract:

The Swift-Hohenberg equation is a PDE which models formation of stripe and spot patterns in many physical settings. I will study a modification in which pattern formation is triggered by a propagating interface, and discuss the bifurcation structure based on the interface speed. I will focus on analytical results in reduced equations, in particular a singular perturbation problem for a system of ODEs arising from a traveling wave ansatz. I will also present numerical results in the Swift-Hohenberg and reduced equations which organize the bifurcation structure into a two-dimensional surface we call the moduli space. This is joint work with Ryah Goh, Oscar Goodloe, Alex Milewski, and Arnd Scheel.

Abstract:

The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page for the sphere at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules.

Abstract:

Ujae Kang will present on the Federal Reserve over the years and its influence on the yield curve. Then, he will cover what to expect from the Federal Reserve in the coming years.

Bio: Ujae Kang is Director of Enterprise Risk Management at UnitedHealth Group. He is an Associate of the Society of Actuaries and has an Master of Financial Mathematics from the University of Minnesota's School of Mathematics. He also provides economic research and insights to UnitedHealth Group's Asset Liability Management Committee as well as to other investors. For more information on Ujae go to linkedin.com/in/ujaeaugustinekang

Abstract:

A group of us are developing educational technology as part of the
Distributed Open Education Network (Doenet). We'll describe our vision for
an open educational technology ecosystem and highlight some solutions that
we are currently implementing. Then, we'll open the floor to discuss your
dreams for technological innovations that will enable you to better help
students learn.

Abstract:

Affine Deligne--Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport--Zink spaces. Their irreducible components provide an interesting class of cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to count the number of F_q points as q goes to infinity; this boils down to computing a certain twisted orbital integral. After applying techniques from local harmonic analysis, we reduce to computing a particular coefficient of the matrix for the inverse Satake transform. Using an interpretation of this coefficient in terms of a q-analogue of Kostant's partition function, we are able to reduce the problem to the previously known special cases of the conjecture proved by Hamacher--Viehmann and Nie. This is joint work with Yihang Zhu.

Abstract:

This talk derives the contact point forces, namely the normal force and static friction, that act on a ball rolling without slipping on a horizontal plane. It is assumed that the ball is actuated by n point masses, which are free to move inside the ball. The dynamics of a disk and ball actuated by internal point masses are simulated numerically and the minimum coefficients of static friction required to prevent slippage are computed

Abstract:

Recently, many people, including Huang, Lewis, Morales, Reiner, and Stanton, have enumerated certain factorizations in GLnFq. Much of their results are essentially q-analogues of factorization enumerations coming from Sn in the q-->1 sense. I think this work constitutes a few significant puzzle pieces in an enormous, really difficult puzzle. I will talk about another piece (or fraction of a piece) of this puzzle that I can contribute, which is an enumeration of factorizations of the identity into a product of regular-elliptic elements. Ideally, this is q-analogous to a result of Stanley from the 1981, which enumerates factorizations of the identity into a product of n-cycles. Time permitting, I will talk about my approach and/or the geometry that corresponds to this work.

Abstract:

Have you ever heard that the symmetric group $\mathfrak{S}_n$ is “equal” to the finite general linear group $\mathrm{GL}_n \mathbb{F}_q$ in the limit $q \to 1$? This “equality” casts many mathematical shadows. I will discuss some of my favorites, which involve the (complex) characters of the two groups. I will include a primer on $\mathrm{GL}_n \mathbb{F}_q$ characters, and I will assume some foundational representation theory facts. Next, I will turn the tides and consider the limit $q \to \infty$. Mysteriously, some relationship between $\mathrm{GL}_n \mathbb{F}_q$ and $\mathfrak{S}_n$ as $q \to \infty$ is also casting shadows. Time permitting, I will discuss how character theory can be used to enumerate factorizations in order to segue into tomorrow’s seminar.

Abstract:

I plan to talk about some number theoretic implications of the Stone-von Neumann theorem. The Stone-von Neumann theorem is a uniqueness theorem about commutation relations between position and momentum operators. I will give a historical discussion about number theory results implied by Stone-von Neumann.

Abstract:

A \Gamma-category is a functor from the category of finite based sets and basepoint preserving functions \Gamma^op to Cat. We construct a model category structure on the category of \Gamma-categories, which is symmetric monoidal closed to the Day convolution product. The fibrant objects in this model category structure are those
\Gamma-categories which are often called special \Gamma-categories. The main objective of this research is to establish a Quillen equivalence between a natural model category structure on the category of (small) permutative categories and strict symmetric monoidal functors Perm and our model category structure on \GammaCat. The weak equivalences of the natural model category structure are equivalences of underlying categories. In the paper [1], Segal defined a functor from the category of (small) symmetric monoidal categories into \GammaCat which can be described as a nerve functor for symmetric monoidal categories. The right adjoint \bar{K} of our Quillen equivalence is a thickening of Segal's nerve functor. We construct a permutative category L called Leinster's category, having the universal property that each \Gamma-category extends uniquely to a symmetric monoidal functor along an inclusion functor \Gamma^op —> L. The left adjoint L of our Quillen equivalence is a composite functor composed of the symmetric monoidal extension functor indicated above followed by a homtopy colimit functor. In the paper [2], Mandell had shown that Segal's nerve functor (followed by the ordinary nerve functor) induces an equivalence between a homotopy category of Perm, obtained by inverting those strict symmetric monoidal functors which induce a weak homotopy equivalence of simplicial sets upon applying the nerve functor, and a homotopy category of \Gamma-spaces \GammaS obtained by inverting pre-stable equivalences which are those maps of \Gamma-categories which induce a degreewise weak homotopy equivalence of simplicial sets upon applying an E_\infty-completion functor. The objective of Mandell's work is to understand the relation between connective spectra and \Gamma-spaces obtained by applying the Segal's nerve functor to symmetric monoidal categories whereas our objective is to construct a model category of symmetric monoidal categories which is
symmetric monoidal closed.

References:
[1] G. Segal, Categories and cohomology theories, Topology 13 (1974) 293-312.
[2] M. A. Mandell, An Inverse K-theory functor, Doc. Math 15 (2010) 765-791.

Abstract:

Dynamic linear models (DLMs), a subset of state space models, describe the output of a dynamic system as a function of a non-observable state process affected by random errors. Because DLMs can be used either for traditional time series analysis tasks (making inferences on observed states or prediction future observations) or for feature generation in machine learning tasks, they are a very useful tool for any data scientist who works with time series data. As a data scientist on the Data Analytics team for 3M Finance, I work primarily with time series data from the general ledger. Our team both leads data science projects and assists in organizational development of internal capabilities around data science.
In this talk, I will first provide an overview of the Finance organization and describe the technical tasks our team addresses. Next, I will give an introduction on the mathematics of DLMs. Finally, I will conclude showing examples of how DLMs can be used on time series data in a Jupyter notebook demo.

Abstract:

For any finite group G and a fixed prime p, the topology of the poset of nontrivial elementary abelian p-subgroups contains significant information about the representations of G in characteristic p. For the symmetric group, this poset and its homology remain rather mysterious. A reasonable conjecture is that as we consider larger symmetric groups, the associated posets should get topologically more connected. I will rephrase this conjecture as a representation stability phenomenon ala Church-Ellenberg-Farb, and provide evidence for it by exhibiting high acyclicity of certain subposets.t:

Abstract:

According to a result of Farmer, the homology of this complex vanishes except in the bottom and top dimensions. This property is a consequence of shellability of the complex. We will discuss these matters, used in a proof of Nakaoka's theorem on stability of the corestriction map in the group homology of the symmetric groups.

Abstract:

We will begin with defining cryptographic multilinear maps, briefly discussing some of their applications, and referencing one such map from Boneh-Silverberg '03. After that, we will extend a problem involving isogenies of elliptic curves into an open problem of finding cryptographic invariant maps from Boneh, et al. '18.stract:

Abstract:

The Swift-Hohenberg equation is a widely studied partial differential equation which is known to support a variety of spatially localized structures. The one-dimensional equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, the related radially-symmetric spatially-localized solutions take on a significantly different snaking structure which consists of three major components. To understand this transition we apply a dimensional perturbation in an effort to use well-developed methods of perturbation theory and dynamical systems. In particular, we are able to identify key characteristics that lead to the segmentation of the snaking branch and therefore provide insight into how the bifurcation structure changes with the spatial dimension.

Abstract:

We apply methods of quantum mechanics to construct self-adjoint, purely discrete pseudo-differential operators which characterize a nuclear Frechet Schwartz space of automorphic forms.

Abstract:

Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations.

Abstract:

Many machine learning procedures aimed to extract information from data can be defined as precise mathematical objects that are constructed in terms of the data. It is often assumed that the data is “big” in complexity but also in quantity, and in this “large amount of data’’ setting, a basic mathematical concept that one can explore is that of closure of a given class of statistical procedures (i.e. what are the limiting procedures as the number of data points available goes to infinity.) In this talk, I will explore this notion in the context of graph-based methods. Examples of such methods include minimization of Cheeger cuts, spectral clustering, and graph-based bayesian semi-supervised learning, among others. I will introduce some of the mathematical ideas needed for the analysis, as well as show some of the implications of it: our results show statistical consistency of the methods, provide with quantitative information in the form of scaling of parameters and rates of convergence, imply qualitative properties at the discrete level, and suggest the use of appropriate algorithms.

Abstract:

We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n-by-n real matrices and an n-dimensional real vector, find a sequence of K matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the K matrices and the given vector. This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We propose a simple exact algorithm for this problem. The efficiency of our algorithm depends heavily on whether the given set of matrices has the oligo-vertex property, a concept we introduce for a finite set of matrices. The oligo-vertex property captures how the numbers of extreme points for a sequence of K convex polytopes related to the given matrices grow with respect to K. We derive several sufficient conditions for a set of matrices to have this property, and pose several open questions related to this property.

Abstract:

The smooth version of the 4-dimensional Poincare Conjecture (S4PC) states that every homotopy 4-sphere is diffeomorphic to the standard 4-sphere. One way to attack the S4PC is to examine a restricted class of 4-manifolds. For example, Gabai's proof of Property R implies that every homotopy 4-sphere built with one 2-handle and one 3-handle is standard. In this talk, we consider homotopy 4-spheres X built with two 2-handles and two 3-handles, which are uniquely determined by the attaching link L for the 2-handles in the 3-sphere. We prove that if one of the components of L is the connected sum of a torus knot T(p,2) and its mirror (a generalized square knot), then X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.

Abstract:

A theorem of Nakaoka states that the inclusion of symmetric groups induces an isomorphism of group homology provided that the degrees of the symmetric groups are large enough. This week I will continue with the proof of this, explaining the equivariant cohomology spectral sequence and the specific calculations that arise, trying to assume as little as I can.

Abstract:

Local rings are ubiquitous in commutative algebra and algebraic geometry. In this talk I will describe two avenues for computing in local rings with respect to prime ideals, first using the associated graded algebra and then using only Nakayama's lemma. Time permitting, I will demonstrate various examples and applications, such as computing the Hilbert-Samuel multiplicity, using Macaulay2.

Abstract:

Given a data set $\{x_i\}_{i=1}^n$ with labels $\{y_i\}_{i=1}^N$ on the first $N$ data points the goal of semi-supervised is to infer labels on the remaining $\{x_i\}_{i=N+1}^n$ data points. In this talk we use a random geometric graph model with connection radius $r(n)$. The framework is to consider objective functions which reward the regularity of the estimator function and impose or reward the agreement with the training data, more specifically we will consider discrete p-Laplacian and fractional Laplacian regularization.

The talk concerns the asymptotic behaviour in the limit where the number of unlabelled points increases while the number of training points remains fixed. The results are to uncover a delicate interplay between the regularizing nature of the functionals considered and the nonlocality inherent to the graph constructions. I will give almost optimal ranges on the scaling of $r(n)$ for asymptotic consistency to hold. Furthermore, I will setup the Bayesian interpretation of this problem.

This is joint work with Matt Dunlop (Caltech), Dejan Slepcev (CMU) and
Andrew Stuart (Caltech).

Mathew Thorpe is a Research Fellow at the Cantab Capital Institute for the Mathematics of Information, Department of Applied Mathematics and Theoretical Physics, University of Cambridge.

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Abstract:

The pulmonary airway system is a highly hierarchical tree-like 3D network in charge of transferring oxygen from the upper airways down to the alveolar region where the gas exchange occurs between air and blood. It is also the place of numerous particle and fluid transport processes. Given the complexity of the anatomy and of the physics involved, numerical simulation is a remarkable tool for investigating the properties of this organ seen as a "transport system", understanding its behavior and possible failures, and predicting the outcome of therapies. However, the range of scales represents here a huge challenge: from the meter scale of the entire organ size down to the micron scale of the mucus layer, about 5 to 6 orders of magnitude of length scale are crossed.

In this talk, we will present several examples of numerical models able to capture the various aspects of the lung airway system, from the simplest linear approach of gas transport to more advanced computational fluid dynamics simulation. We will show, in particular, how the range of scales involved and the difficulty to access the actual parameters in the patient imposes to use a hierarchy of models and a diversity of numerical techniques. We will explain how these models can be used either to reach a general understanding of the system or to design patient specific diagnosis and therapy.