2014-15 Colloquia

Tuesdays at 4:10 in
Carver 204 or as announced

 

For more information, contact
Songting Luo
Derrick Stolee
Steve Butler (on leave)

 
 

Spring 2015

April 21: Thomas Hagstrom (Souther Methodist University) Towards the ultimate solver for wave equations in the time domain
(Host: Hailiang Liu)

April 28: Oleg Pikhurko (Warwick) Measurable circle squaring

 

Abstracts

April 28: Oleg Pikhurko (Warwick) Measurable circle squaring

In 1990 Laczkovich proved that, for any two sets A and B in R^n with the same non-zero Lebesgue measure and with boundary of box dimension less than n, there is a partition of A into finitely many parts that can be translated by some vectors to form a partition of B.
In particular, this resolved the long-standing Tarski's circle squaring problem.
We show that one can additionally require that all parts are Lebesgue measurable.

Joint work with András Máthé and Łukasz Grabowski.

April 21: Thomas Hagstrom (Souther Methodist University) Towards the ultimate solver for wave equations in the time domain

Ecient time-domain solvers for wave propagation problems must include three crucial components:
i. Radiation boundary conditions which provide arbitrary accuracy at small cost (spectral convergence, weak dependence on the simulation time
and wavelength)
ii. Robust high-resolution volume discretizations applicable in complex geometry (i.e. on grids that can be generated eciently) - we believe that
high-resolution methods enabling accurate simulations with minimal dofs-per-wavelength are necessary to solve dicult 3 + 1-dimensional
problems with the possibility of error control.
iii. Algorithms for directly propagating the solution to remote locations - avoid sampling the wave whenever possible.

In this talk we will discuss recent developments in all three areas, including our own work on the construction of complete radiation boundary
conditions (CRBCs), which are optimal local radiation conditions, and on high-order energy stable volume discretization methods.

April 14: Krishna Rajan (MSE, ISU) Mathematics of big data and materials science

This presentation will give an overview of possible areas where mathematics can drive the research of big data, especially in the context of materials science research.  Examples of where mathematics can drive both experimental and computational  work in materials science from a  data science perspective are given. Based on work in my group I will discuss our use of  computational homology, analysis of point cloud data and manifold learning methods.  The presentation will invite a discussion of where mathematics can intersect with materials informatics.

April 9: Daniel Kral (University of Warwick) Analytic approach to discrete problems

The recently emerged theory of combinatorial limits has opened new links between analysis, combinatorics, computer science, group theory and probability theory.
Combinatorial limits give an analytic way of representing large discrete objects.

In the talk, we focus on limits of permutations and (dense) graphs, and their applications in extremal combinatorics and theoretical computer science. We will particularly focus on limits that are uniquely determined by finitely many constraints and we will present counterexamples to two conjectures posed by Lovasz and Szegedy on the structure of the topological space of typical points of such limits.

The talk will be self-contained and no previous knowledge of the area is needed.

April 9: Michelle Guinn (Belmont University) Smoothing techniques to enhance stereoscopic imagery

The objective of my research is to design an algorithm to present enhanced stereoscopic imagery that is adapted to the viewing distance of the observer, with seamless transitions among stereo and hyperstereo levels. I will design an algorithm that use the image smoothing techniques to provide this enhancement. The research will improve images that Soldiers can use to perform several tasks and can potentially provided better situational awareness.

April 7: Johanna Franklin (Hofstra) Randomness and Birkhoff's ergodic theorem for measure-preserving transformations

A point in a probability space is algorithmically random if it has no rare measure-theoretic properties that are defined simply, and ergodic theorems describe regular measure-theoretic behavior. I will discuss Birkhoff’s ergodic theorem with respect to transformations that are measure-preserving but not necessarily ergodic in the context of a computable probability space. Then I will show that each point in such a space that is not Martin-Loef random fails to satisfy Birkhoff’s ergodic theorem with respect to every computable set and measure-preserving transformation.

This work is joint with Henry Towsner.

Friday April 3 @ 3:10 pm in Carver 204: Daniel Fresen (Yale) Concentration in stochastic geometry

We consider a random sample of n independent and identically distributed points in d dimensional Euclidean space, with common probability distribution u. We study properties of the random polytope p_n defined as the convex hull of the sample. The main focus will be on deviation inequalities (multivariate Gnedenko law of large numbers). Other topics that will be touched upon include functionals of random polytopes (volume, number of vertices etc), and the local limiting Poisson process near the boundary.

March 31: David Herzog (Drake) Stabilization by noise and the existence of optimal Lyapunov

We discuss certain, explosive ODEs in the plane that become stable under the addition of noise. In each equation, the process by which stabilization occurs is intuitively clear: Noise diverts the solution away from any instabilities in the underlying ODE. However, in many cases, proving rigorously this phenomenon occurs has thus far been difficult and the current methods used to do so are rather ad hoc. Here we present a general, novel approach to showing stabilization by noise and apply it to these examples. We will see that the methods used streamline existing arguments as well as produce optimal results, in the sense that they allow us to understand well the asymptotic behavior of the equilibrium measure at infinity.

March 24: Elisabeth Werner, IMA Associate Director/Case-Western:  Entropy inequalities for log concave functions

We discuss  entropy inequalities for log concave  functions. Among them are  reverse log Sobolev inequalities. This  leads naturally to  a concept 
of relative entropy and Renyi entropy  for  such  functions. 

Connections are given  to  the theory of convex  bodies.

March 11 PDE Candidate Xiang Xu (Purdue) Cubic instability in the Q-tensor theory of nematic liquid crystals

The Landau-de Gennes theory on the study of nematic liquid crystals was initiated by the Physics Nobel Prize winner P.G. de Gennes in the 1970's, and it began to attract the extensive attention of mathematicians in recent years. This theory is a phenomenological theory in which stable states of the liquid crystal material correspond to minimizers of a free energy. In its free energy form, there is however an unusual cubic term. In this talk we will discuss the dynamic effects induced by this cubic term by considering the $L^2$ gradient flow dynamics generated by the free energy. We work in  both two and three dimensions and focus on understanding the relations between the physicality of the initial data and global well-posedness of the system.

March 10: David Roberson: Unique vector colorings: Rigidity, graph products, and cores

A vector t-coloring of a graph is an assignment of unit vectors to the vertices of the graph such that the inner product of vectors assigned to adjacent vertices is less than or equal to -1/(t-1). The vector chromatic number of a graph is the minimum t for which it admits a vector t-coloring. An n-coloring of a graph can be converted to a vector n-coloring by mapping the vertices of each color class to the vertices of a regular (n-1)-simplex. This shows that the vector chromatic number is always at most the chromatic number, but it can sometimes be much smaller. In this talk we will focus on finding graphs with unique (up to isometry) optimal vector colorings. Our main tool for finding such graphs will be a result from rigidity theory.

We will also investigate vector colorings of categorical products of graphs. In particular, we consider the following three statements:

(A): If G and H are uniquely vector t-colorable, then the only vector t-colorings of G x H are the convex combinations of the vector t-colorings induced by the factors.

(B): If G is uniquely vector t-colorable and H has vector chromatic number greater than t, then G x H is uniquely vector t-colorable.

(C): The vector chromatic number of G x H is the minimum of the vector chromatic numbers of G and H.

These are vector coloring analogs of three statements considered by Duffus, Sands, and Woodrow. In particular, (C) is a vector analog of Hedetniemi's conjecture. We prove (C) and prove (A) and (B) for 1-walk-regular graphs. Lastly, we discuss a surprising connection between unique vector colorability and cores (graphs which do not admit homomorphisms to any proper subgraph)

March 9: PDE Candidate Pablo Stinga (University of Texas) How to understand fractional Laplacians: the language of semigroups. Some applications

In the past 10 years we observed an increasing interest in the PDE community on equations that involve fractional powers of the Laplace operator in the whole space. This operator is a classical object in Mathematical Physics, Harmonic Analysis, Functional Analysis, Probability and Potential Theory. Nevertheless, some fine tools needed in the study of the PDEs like Harnack inequalities and Schauder regularity estimates are not available from the general theory. In 2007 L. Caffarelli and L. Silvestre introduced a powerful tool to handle the fractional Laplacian known as the *extension problem*. This turned out to be a major breakthrough in the theory of fractional equations because thanks to it many nonlinear problems could be attacked by using known techniques from usual PDE theory.

We will explain a novel point of view to handle fractional operators:  the semigroup language approach. The method was introduced in my PhD thesis and it has been subsequently developed and exploited in several different situations. The first main feature of the language is that it clearly explains what a fractional Laplacian is. From here on we can generalize the Caffarelli-Silvestre extension problem to fractional powers of positive operators, that eventually allows us to treat nonlocal problems. Applications include nonlinear problems like the obstacle problem for the fractional Laplacian or for a nonlocal Monge-Ampére equation, semilinear nonlocal Neumann problems arising from the Keller-Segel model of chemotaxis, the fractional Laplacian on a Riemannian manifold, the fractional discrete Laplacian, and Harnack and Schauder estimates for nonlocal divergence form equations, among many others.

March 3: HHMI-STEM Initiative candidate Ashley Suominen: Algebra and secondary school mathematics: Identifying and classifying mathematical connections

Many stakeholders concur that secondary teacher preparation programs should include study of abstract algebraic structures, and most certification programs require an abstract algebra course for prospective mathematics teachers. However, research has shown that undergraduate students struggle to understand fundamental concepts and, upon completion of the course, were unable to articulate connections between abstract algebra and secondary school mathematics. This three-part study involved a textbook analysis, the creation of a comprehensive connection list, and a series of expert interviews. In the textbook analysis, I examined nine abstract algebra textbooks, identified any connections made in the text, and categorized them into five types of connections: alternate or equivalent representations, comparison through common features, generalization, hierarchical or inclusion, and real world application. I then interviewed 12 mathematicians and mathematics educators involved in abstract algebra teaching and research to understand how they describe connections between abstract algebra and secondary mathematics. Participants’ descriptions of connections reflected their experiences with the secondary curriculum and differed according to their individual conceptualizations of abstract algebra. That is, participants with views of abstract algebra based on axioms, solving equations, or geometry prioritized different sets of connections. The findings of this study identified the various mathematical connections between abstract algebra and secondary school mathematics and   provided the vocabulary to discuss such connections.    

March 3: Ning Su (Tsinghua University) Entropy solutions of degenerate parabolic-hyperbolic equations

We are interested in a class of nonlinear parabolic-hyperbolic equations arising from practical and engineering fields. Due to the strong degeneration of the equations, the framework of entropy solutions has been introduced and set up since 1970s.

I will review the developing of the notion of entropy solutions, discuss  existence and uniqueness of the solutions for initial-boundary value problems,
and show that under quite general conditions the entropy solutions possess L^1 contraction property and hence obey comparison principles. Moreover, in some circumstance we must consider the notion of   maximal and minimal entropy solutions.

Thursday, February 12 in TBA Carver: Stochastic candidate Oleksii Mostovyi (U Texas) Optimal investments in incomplete markets

A fundamental problem of mathematical finance is that of an economic agent who invests in a financial market so as to maximize the expected utility.

In a continuous-time framework, the problem was studied for the first time by Merton (1969), who derived a nonlinear partial differential equation (Bellman equation) for the value function of the optimization problem and produced a closed-form solution for the power, logarithmic, and exponential utility functions.

The modern approach to the problem of expected utility maximization is based on duality, which makes it possible to substantially relax the conditions on the utility function and the model of the market, such as the requirement of Markov state processes.

In this presentation, we will consider Merton’s portfolio problem in various formulations and methods of solutions. We will also provide an explicit first-order expansion formula for the power investor’s value function seen as a function of the underlying market price of risk process and quantify the second-order error. Our result yields a tool for approximation of the “less” tractable models by the “more” tractable ones that we will discuss as well.

Tuesday, February 10: Zdenek Dvorak: Towards dichotomy for planar boolean CSP

For relations {R_1,..., R_k} on a finite set D, the {R_1,...,R_k}-CSP is a computational problem specified as follows:

Input: a set of constraints C_1, ..., C_m for variables x_1, ..., x_n, where each constraint is of form R_i(x_{j_1}, x_{j_2}, ...) for some i in {1, ..., k}

Output: decide whether it is possible to assign values from D to all the variables so that all the constraints are satisfied.

The CSP problem is boolean when |D|=2.  Schaefer gave a sufficient condition on the relations in a boolean CSP problem guaranteeing its polynomial-time solvability, and proved that all other boolean CSP problems are NP-complete.

In the planar variant of the problem, we additionally restrict the inputs only to those whose incidence graph (with vertices C_1, ..., C_m, x_1, ..., x_n and edges joining the constraints with their variables) is planar.  It is known that the complexities of the planar and general variants of CSP do not always coincide. For example, let NAE={(0,0,1),(0,1,0),(1,0,0),(1,1,0),(1,0,1),(0,1,1)}.
Then {NAE}-CSP is NP-complete, while planar {NAE}-CSP is polynomial-time solvable.

We give some partial progress towards showing a characterization of the complexity of planar boolean CSP similar to Schaefer's dichotomy theorem.

Monday, February 9 in 290 Carver Stochastic candidate Tonci Antunovic (UCLA) Stochastic competition on finite and infinite networks

Two type Richardson model introduced in 90's by Haggstrom and Pemantle is a stochastic competition model in which two types of particles spread through a graph using the first passage percolation dynamics. While originally studied on the Euclidean lattice, recent attempts to model large real world networks (internet, social networks) raises natural questions about the behavior of the model on other kinds of graphs. We will present results about the behavior of the model on large random regular graphs and on a version of the model on lifts of general Cayley graphs. We will also present a related competition model on preferential attachment networks.
This is based on several papers written in collaboration with Yael Dekel, Elchanan Mossel, Yuval Peres, Eviatar Procaccia and Miklos Racz.

Thursday, February 5 in TBA Carver - Stochastic candidate Jack Hanson (Indiana)  Geodesics and fluctuations in first-passage percolation

In first-passage percolation (FPP), random weights are placed on the edges of a graph and used to define a random metric t(x,y). On the d-dimensionsal integer lattice Z^d, many questions remain about the large-scale behavior of the metric and its geodesics. In the 1990s, C. Newman conjectured that (for d = 2) infinite geodesics should have asymptotic direction, and that geodesics having the same direction should merge. There is also a longstanding claim by physicists that Var(t(0,x)) should be smaller than |x|^{1 - epsilon}, and some progress towards this was made in special cases by Benjamini-Kalai-Schramm and Benaim-Rossignol. I will discuss my work on these and related questions, including a proof of a version of Newman's conjecture (that geodesics are directed in sectors) and a proof that the sublinear variance phenomenon holds for general distributions.

Tuesday, February 3 in 204 Carver Stochastic candidate Lingjiong Zhu (NYU) Self-exciting point processes

Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other self-exciting point processes will also be discussed.

Thursday, January 29 in 268 Carver - PDE candidate Mimi Dai (UI-Chicago) Ill-posedness of the Navier-Stokes equation  

January 26 in 298 Carver PDE candidate Alden Waters(Ecole Normale Superieure) Gaussian beams and reconstruction methods in inverse problems

One of the central questions in inverse problems is the reconstruction of waves emitted from the boundary of an obstacle. The Gaussian beam Ansatz which accurately approximates waves in the high frequency regime has been useful to numerical analysts and geophysicists since the 1960s. We show that using these suitably localized approximations to solutions that new questions of reconstruction of materials can be addressed.

January 20 in 204 Carver- ALG candidate Alexandra Seceleanu (UNL) Symbolic versus ordinary powers for ideals of points    

The problem of describing the set of hypersurfaces passing through a finite set of points with given multiplicity leads to challenging mathematical questions. For example, one can ask what the minimum degree of such a hypersurface is or how many independent hypersurfaces there are of any given degree. The most general forms of these questions are still open and have given rise to longstanding conjectures in algebraic geometry.

Searching for structural reasons to explain some of the these conjectures, Harbourne and Huneke proposed an approach based on comparisons between the set of all polynomials vanishing at the points to a prescribed order, which is called a symbolic power ideal, and algebraically better understood counterparts, namely the ordinary powers of the ideal of base points. Two questions will be shown to be related: How tight can this comparison be made? Which arrangements of lines in the plane have no points where only two lines meet? I will answer these and many more questions while considering some special arrangements of lines with unexpected combinatorial and algebraic properties.

Wednesday, January 21 in 298 Carver- ALG candidate Jonas Hartwig (UC Riverside) Quantized enveloping algebras and generalized Weyl algebras

Quantized enveloping algebras originally arose in the context of quantum integrable systems in the early 1980's. Since then, it has gradually been understood that these algebras are intimately connected to many areas of mathematics and physics. In this  talk I will discuss some recent results about the structure and representation theory for quantized enveloping algebras, while highlighting the role played by various generalized Weyl algebras.

January 22 -January 22 - PDE candidate Tianling Jin (University of Chicago) The Nirenberg problem and its generalizations: A unified approach

The classical Nirenberg problem asks for which functions on the sphere arise as the scalar curvature of a metric that is conformal to the standard metric. In this talk, we will discuss similar questions for fractional Q-curvatures. This is equivalent to solving a family of nonlocal nonlinear equations of order less than n, where n is the dimension of the sphere. We will give a unified approach to establish existence and compactness of solutions. The main ingredient is the blow up analysis for nonlinear integral equations with critical Sobolev exponents. We will also discuss related topics including solutions with isolated singularities. This talk is based on joint works with L. Caffarelli, Y.Y. Li, Y. Sire and J. Xiong.   

 

January 23 in 298 Carver - ALG candidate Anna Marie Bohmann (Northwestern) Constructing equivariant cohomology

Many topological spaces come equipped with an action by a group of symmetries and we naturally want to understand phenomena preserved by these actions. In fact, analyzing such group actions can be fruitful even in contexts that initially appear nonequivariant. Algebraic topology uses tools such as cohomology and one of the challenges of equivariant homotopy theory is to develop cohomology theories adapted to the setting where groups act. In this talk, I will describe new work that allows us to construct equivariant cohomology theories. These methods take as their input algebraic data familiar to representation theorists, and provide better access to the underlying algebra of the cohomology theory than previous constructions.


Archived

Tuesday, January 20 in 204 Carver- ALG candidate Alexandra Seceleanu (UNL) Symbolic versus ordinary powers for ideals of points    
Poster

Wednesday, January 21 in 298 Carver- ALG candidate Jonas Hartwig (UC Riverside) Quantized enveloping algebras and generalized Weyl algebras
Poster

Thursday, January 22 in 232 Carver - PDE candidate Tianling Jin (University of Chicago) The Nirenberg problem and its generalizations: A unified approach    
Poster

Friday, January 23 in 298 Carver - ALG candidate Anna Marie Bohmann (Northwestern) Constructing equivariant cohomology  
Poster

Monday, January 26 in 298 Carver PDE candidate Alden Waters(Ecole Normale Superieure) Gaussian beams and reconstruction methods in inverse problems
Poster

Thursday, January 29 in 268 Carver - PDE candidate Mimi Dai (UI-Chicago) Ill-posedness of the Navier-Stokes equation  
Poster

Tuesday, February 3 in 204 Carver Stochastic candidate Lingjiong Zhu (U Minn) Self-exciting point processes
Poster

Thursday, February 5 in TBA Carver - Stochastic candidate Jack Hanson (Indiana) Geodesics and fluctuations in first-passage percolation

Monday, February 9 in 290 Carver Stochastic candidate Tonci Antunovic (UCLA) Stochastic competition on finite and infinite networks

Tuesday, February 10: Zdenek Dvorak: Towards dichotomy for planar boolean CSP

Thursday, February 12 in TBA Carver: Stochastic candidate Oleksii Mostovyi (U Texas) Optimal investments in incomplete markets

March 3: HHMI-STEM Initiative candidate presentation @ 3:10 in Carver TBA
Ashley Suominen (University of Georgia): Algebra and secondary school mathematics: Identifying and classifying mathematical connections
Host: Elgin Johnston  

March 3: Ning Su (Tsinghua University) Entropy solutions of degenerate parabolic-hyperbolic equations
(Host: Hailiang Liu)

March 24: Elisabeth Werner, IMA Associate Director/Case-Western Reserve University:  Entropy inequalities for log concave functions
(Host: Leslie Hogben)

March 31: David Herzog (Drake) Stabilization by noise and the existence of optimal Lyapunov functions
Host: Ananda Weerasinghe

Friday April 3 @ 3:10 pm in Carver 204: Daniel Fresen (Yale) Concentration in stochastic geometry
POSTER

April 7: Johanna Franklin (Hofstra) Randomness and Birkhoff's ergodic theorem for measure-preserving transformations
Host: Time McNicholl POSTER

April 9 in Carver 268 at 3:10 pm: Michelle Guinn (Belmont University) Smoothing techniques to enhance stereoscopic imagery
POSTER

April 9: Daniel Kral (University of Warwick) Analytic approach to discrete problems
POSTER

April 14: Krishna Rajan (MSE, ISU) Mathematics of big data and materials science
POSTER Host: Cliff Bergman