Speaker: Cory Hauck, Oak Ridge National Lab and Univ of Tennessee

Title: Discontinuous Galerkin Methods for Transport Equations and the Diffusion Limit

Date: 09/01/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

In this talk, I will review the use of the discontinuous Galerkin (DG) method in capturing the diffusion limit of kinetic transport equations. One of the main drawbacks of this method is the large memory footprint. To address this issue, we have developed a new, low-memory approach that, among other things, includes a hybrid DG-finite volume scheme. Initial numerical analysis and preliminary numerical results will be presented to show the potential of the new approach.

Title: Optimal transportation between unequal dimensions

Date: 09/01/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Over the last few decades, the theory of optimal transportation has blossomed
into a powerful tool for exploring applications both within and outside mathematics
Its impact is felt in such far flung areas as geometry, analysis, dynamics, partial differential
equations, economics, machine learning, weather prediction, and computer vision.
The basic problem is to transport one probability density onto other, while minimizing a given cost
c(x,y) per unit transported. In the vast majority of applications, the probability densities
live on spaces with the same (finite) dimension. After briefly surveying a few highlights
from this theory, we focus our attention on what can be said when the densities instead live on
spaces with two different (yet finite) dimensions. Although the answer can still be
characterized as the solution to a fully nonlinear differential equation, it now becomes
badly nonlocal in general. Remarkably however, one can identify conditions under which
the equation becomes local, elliptic, and amenable to further analysis.

Title: Riemannian metrics on real projective spaces with all geodesics closed.

Date: 09/08/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

I'll describe a proof of the fact that when, n is not 3, the only Riemannian metrics on the n-dimensional real projective space with all geodesics closed are the standard constant curvature metrics. This is perhaps interesting because the same is not true for metrics on spheres.
Based on joint work with Samuel Lin.

Title: A shortcut to becoming an applied mathematician: sharp Poincar\'e inequalities on the Hamming cube

Date: 09/12/2016

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

A shortcut to becoming an applied mathematician: sharp Poincar\'e inequalities on the Hamming cube is the joint work with Paata Ivanisvili.
We show that certain classical isoperimetric inequalities (such as Beckner-Sobolev inequalities) can be improved, moreover we show that the improved inequalities are in fact follow from new isoperimetric inequalities on the Hamming cube or arbitrary dimension.

Title: Knot concordance and homology sphere groups

Date: 09/12/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We introduce the knot concordance group and various homology sphere groups constructed from 3-manifolds, and discuss two homomorphisms to the rational homology sphere group. If B denotes the homomorphism from the concordance group defined by taking double branched covers of knots, we
show that the kernel of B is infinitely generated. We also study the inclusion homomorphism P from the integral homology sphere group. Using work of Lisca we show that the image of P intersects trivially with the subgroup generated by lens spaces. This is joint work with Paolo Aceto.

Title: Face Numbers of Simplicial Polytopes and Simplicial Complexes

Date: 09/13/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The f-vector records the number of faces of a polytope or simplicial complex. An overview of what is known about f-vectors of simplicial polytopes and simplicial complexes will be given. We plan to cover the Kruskal-Katona theorem, the upper bound theorem, the g-theorem, and the g-conjecture.

Title: Non-Positive Curvature: A Journey of the Mind

Date: 09/15/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

I will spend some time motivating the notion of non-positive
immersions by talking about different kinds of non-positive curvature, leading up to the theorem of Helfer-Wise/Louder-Wilton (2014) which states that one-relator groups have non-positive immersions.

The piecewise linear objects appearing in tropical geometry are shadows, or skeletons, of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces.

Title: Variational Image Segmentation, Inpainting and Denoising

Date: 09/16/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Variational methods have attracted much attention in the past decade. With rigorous mathematical analysis and computational methods, variational minimization models can handle many practical problems arising in image processing, such as image segmentation and image restoration. We propose a two-stage image segmentation approach for color images, in the first stage, the primal-dual algorithm is applied to efficiently solve the proposed minimization problem for a smoothed image solution without irrelevant and trivial information, then in the second stage, we adopt the hill-climbing procedure to segment the smoothed image into reasonable parts. And we also improve the non-local total variation model. More precisely, we add an extra term to impose regularity to the graph formed by the weights between pixels. Thin structures in the image can benefit from this regularization term, because it allows to adapt the weights value from the global point of view, thus thin features will not be overlooked, like in the conventional non-local models. Since now the non-local total variation term has two variables, the image u and weights v, and it is concave with respect to v, the proximal alternating linearized minimization algorithm is naturally applied with variable metrics to solve the non-convex model. Numerical experiments demonstrate that, on inpainting and denoising problems, the new model better recovers thin structures.

In this set of lectures, we will discuss equivariant (mostly Z_2, with occasional digressions into other groups) versions of several theories collected on the term 'Floer homology.' The first two lectures will focus on equivariant Lagrangian Floer cohomology. We will first do a quick review of ordinary equivariant cohomology, give an abbreviated introduction to Lagrangian Floer cohomology, and discuss some of the technical issues involved in constructing equivariant versions of same. We will then go over two major approaches to resolving these technical difficulties, with some examples and applications of each to such theories as Heegaard Floer homology and symplectic Khovanov homology. The third lecture will take a slightly different tack and focus on equivariant versions of Seiberg-Witten Floer homology (and, eventually, its analog Heegaard Floer homology) with applications to the homology cobordism groups. We will talk briefly about the structure of the integer homology cobordism group, and then discuss Manolescu's use of a Pin(2)-equivariant version of Seiberg-Witten Floer homology to resolve the Triangulations Conjecture. Finally, we will then discuss additional recent applications of the same ideas, including the use of an involution on Heegaard Floer homology to construct an 'involutive' theory analogous to Z_4-Seiberg Witten Floer homology.

I will talk about a topological notion of volume called simplicial volume. The definition may seem to be purely topological, but it has geometric implications for hyperbolic manifolds. In fact, Gromov used properties of simplicial volume to give a new proof for Mostow Rigidity Theorem for hyperbolic manifolds. After giving basic definitions and properties, I will sketch the computation
of simplicial volume for hyperbolic manifolds given by Gromov and Thurston.

Title: Face Numbers of Simplicial Polytopes and Simplicial Complexes

Date: 09/20/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The f-vector records the number of faces of a polytope or simplicial complex. An overview of what is known about f-vectors of simplicial polytopes and simplicial complexes will be given. We plan to cover the Kruskal-Katona theorem, the upper bound theorem, the g-theorem, and the g-conjecture.

Speaker: Alek Vainshtein, University of Haifa, Israel

Title: Exotic cluster structures on SL_n

Date: 09/21/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Back in 2005, Berenstein, Fomin and Zelevinsky discovered a cluster structure in the ring of regular functions on a
double Bruhat cell in a semisimple Lie group, in particular, SL_n. This structure can be easily extended to the whole group.
The compatible Poisson bracket is given by the standard r-matrix Poisson-Lie structure on SL_n. The latter is a particular
case of Poisson-Lie structures corresponding to quasi-triangular Lie bialgebras. Such structures where classified in 1982
by Belavin and Drinfeld. In 2012, we have conjectured that each Poisson-Lie structure on SL_n gives rise to a cluster structure, and
gave several examples of exotic cluster structures corresponding to Poisson-Lie structures distinct from the standard one.
In my talk I will tell about the progress in the proof of this conjecture and its modifications.
Joint with M.Gekhtman and M.Shapiro.

The volume density of a hyperbolic link K is vol(K)/c(K) where vol(K) is the hyperbolic volume of S^3 - K, and c(K) is the crossing number. The determinant density of a link K is 2 pi log(det(K))/c(K) where det(K) is the determinant of K. We show that the set of volume densities is dense in [0, v_8] where v_8 is the volume of a regular ideal octahedron. We further show that the set of determinant densities is dense in [0, v_8]. We also discuss
stronger relations between volume and determinant of 2-bridge knots.

We will use a tessellation of the Poincaré disk, a model of hyperbolic geometry, as an invitation to learn about non-Euclidean geometry, group theory, and applications to other areas of mathematics.

Title: Babich-like asymptotic ansatz for three-dimensional point-source Maxwell's equations in an inhomogeneous medium at high frequencies

Date: 09/23/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

In this talk, I will first present a formulation using the geometrical-optics (GO) ansatz to solve the point-source Maxwell’s equations in inhomogeneous medium at high frequencies. A drawback of the GO ansatz is it is inaccurate near the point source as well as along certain directions. To address this issue, I will then present a new formulation based on a novel Babich-like asymptotic ansatz by using spherical Hankel functions. A PDE-based Eulerian approach has been developed to compute the asymptotic solution. Several numerical examples will be given to show that the new ansatz yields a uniformly accurate solution in the region of space containing a point source but no other caustics.

Title: Stability for the multidimensional rigid body and singular curves

Date: 09/26/2016

Time: 2:30 PM - 3:20 PM

Place: C304 Wells Hall

A classical result of Euler says that the rotation of a torque-free 3-dimensional rigid body about the short or the long axis is stable, while the rotation about the middle axis is unstable. I will present a multidimensional generalization of this result and explain how it can be proved by studying line bundles over singular algebraic curves.

Title: Boundary value problem and the Ehrhard inequality.

Date: 09/26/2016

Time: 4:02 PM - 5:00 PM

Place: C517 Wells Hall

I will present a new proof of the Ehrhard inequality. In fact I will talk about a more general result and the Ehrhard inequality will be consequence of it. The idea of the method is similar to Brascamp-Lieb's approach to Prekopa-Leindler inequality via sharp reverse Young's inequality for convolutions. Indeed, we shall rewrite essential supremum as a limit of Lp norms but with very specially chosen test functions and measures. Next rewriting Lp norm by duality as a scalar product the question boils down to an estimate of double integral of compositions of test functions by the mass of these functions. To verify the last estimate which looks like Jensen's inequality we will use a subtle inequality, a ``modified Jensen's inequality', which in its turn boils down to the fact that a corresponding quadratic form has a definite sign, and this is the main technical part of the method. If time allows we will show that in the class of even probability measures with smooth strictly positive density Gaussian measure is the only one which satisfies the functional form of the Ehrhard inequality on the real line with their own distribution function.

Title: Affine Cluster Variables As Generalized Minors

Date: 09/27/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

A fundamental result in the theory of cluster algebras due to Berenstein, Fomin, and Zelevinsky is the existence of (upper) cluster algebra structures on the coordinate rings of the double Bruhat cells in semi-simple algebraic groups, later this was extended to cover all Kac-Moody groups by Williams. In every case the initial cluster consists of a collection of generalized minors and generalizations of the classic Jacobi-Desnanot identity for minors of a matrix provide many of the exchange relations. However, most non-initial cluster variables are yet undocumented functions on the group and, in particular, they are usually expected not to be generalized minors. In this talk I will describe a special class of double Bruhat cells where all cluster variables turn out to be generalized minors and then discuss new identities among generalized minors resulting from these observations.

Speaker: Qing Nie, University of California, Irvine

Title: Data-Driven multiscale modeling of cell fate dynamics

Date: 09/27/2016

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Fates of cells are not preordained. Cells make fate decisions in response to different
and dynamic environmental and pathological stimuli. Recently, there
has been an explosion of experimental data at various biological scales, including
gene expression and epigenetic measurements at the single cell level,
lineage tracing, and live imaging. While such data provide tremendous detail
on individual elements, many gaps remain in our knowledge and understanding
of how cells make their dynamic decisions in complex environments. In
addition to developing new models to analyze data at each scale, we are working
on multiscale modeling challenges in analyzing single-cell molecular data
(data-rich scale) and their connections with spatial tissue dynamics (datapoor
scale). Our approach requires new and challenging mathematical and
computational tools in machine learning, stochastic analysis and simulations,
and PDEs with moving boundaries. We then use our novel data-driven multiscale
modeling approach to uncover new principles for cell fate dynamics in
development, regeneration, and disease.

Let p = p_1 ... p_n be a permutation in the symmetric group S_n written as a sequence. The descent set of p is the set of indices i such that p_i > p_{i+1}. A classic result of MacMahon states the the number of permutations in S_n with a given descent set is a polynomial in n. But little work seems to have been done concerning the properties of these polynomials. The peak set of p is the set of indices i such that p_{i+1} < p_i > p_{i+1}. Recently Billey, Burdzy, and Sagan proved that the number of permutations in S_n with given peak set is a polynomial in n times a power of two. I will survey what is known about these two polynomials, including their degrees, roots, coefficients, and analogues for other Coxeter groups.

Title: The Riemann-Roch theorem and its application to group structures of a non-singular cubic curve in P^2

Date: 09/28/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

A non-singular projective curve of degree 3 in
$\mathbb{P}^2_k$ and its point of inflection determines a unique additive group structure on the curve. The Riemann-Roch theorem
relates the dimension of the space of meromorphic functions on the curve with prescribed zeroes and poles to the genus of the curve.
Among many useful applications of the Riemann-Roch theorem, one can see the law of associativity using the degree-genus formula.

In this talk we will be discussing the knot concordance invariant Upsilon, derived from knot Floer homology. After introducing the definition and basic properties, we will discuss how the work of Hedden and Van Cott on tau-invariant of cable knots can be used to obtain a similar result in Upsilon, which is an inequality relating the Upsilon invariant of a knot and that of its cable. We will also see some applications of this result.

Title: Computing without subtracting (and/or dividing)

Date: 09/29/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Algebraic complexity of a rational function can be defined as the minimal number of arithmetic operations required to compute it. Can restricting the set of allowed arithmetic operations dramatically increase the complexity of a given function (assuming it is still computable in the restricted model)? In particular, what can happen if we disallow subtraction and/or division? This is joint work with D.
Grigoriev and G. Koshevoy.