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Spring 2015
March 3: HHMISTEM 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 (Host: Hailiang Liu)
March 10: David Roberson (Host: Leslie Hogben)
March 17: Spring break
March 24: Elisabeth Werner, IMA Associate Director CaseWestern (Host: Leslie Hogben)
March 31:
April 7: Johanna Franklin (Host: Time McNicholl)
April 14: Krishna Rajan (MSE, ISU, Host: Cliff Bergman)
April 21: Thomas Hagstrom (Host: Hailiang Liu)
April 28:OPEN
Abstracts
March 3: HHMISTEM 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 threepart 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 (Host: Hailiang Liu)
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 continuoustime 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 closedform 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 firstorder expansion formula for the power investor’s value function seen as a function of the underlying market price of risk process and quantify the secondorder 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 polynomialtime solvability, and proved that all other boolean CSP problems are NPcomplete.
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 NPcomplete, while planar {NAE}CSP is polynomialtime 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 firstpassage percolation
In firstpassage percolation (FPP), random weights are placed on the edges of a graph and used to define a random metric t(x,y). On the ddimensionsal integer lattice Z^d, many questions remain about the largescale 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 BenjaminiKalaiSchramm and BenaimRossignol. 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) Selfexciting point processes
Selfexciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, selfexciting 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 selfexciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other selfexciting point processes will also be discussed.
Thursday, January 29 in 268 Carver  PDE candidate Mimi Dai (UIChicago) Illposedness of the NavierStokes 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 Qcurvatures. 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 (UIChicago) Illposedness of the NavierStokes equation
Poster
Tuesday, February 3 in 204 Carver Stochastic candidate Lingjiong Zhu (U Minn) Selfexciting point processes
Poster
Thursday, February 5 in TBA Carver  Stochastic candidate Jack Hanson (Indiana) Geodesics and fluctuations in firstpassage 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