Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, November 14, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, November 14, 2017

11:00 am in 345 Altgeld Hall,Tuesday, November 14, 2017

#### Notes on the margins of E-theory

###### Paul VanKoughnett (Northwestern University )

Abstract: The deformation space of a height n formal group over a finite field has an exact interpretation into homotopy theory, in the form of height n Morava E-theory. The K(t)-localizations of E-theory, for t < n, force us to contend with the margins of the deformation space, where the formal group's height is allowed to change. We present a modular interpretation of these marginal spaces, and discuss applications to homotopy theory.

12:00 pm in 243 Altgeld Hall,Tuesday, November 14, 2017

#### Skeins and Characters

###### Charles Frohman (U. Iowa)

Abstract: Skein theory is a $K$-theoretic like construction. Think of the underlying three- manifold as a ring, and a link in that manifold as a projective module. Crossings correspond to extensions of one module by another, and the skein relation says that the extension is equivalent to the direct sum of the two links that it extends. The skein module is the K-group from this relation. If the underlying three manifold is a cylinder over a surface, the links act like a category of bimodules, and the skein module is an algebra. In the talk, I will define the Kauffman bracket skein algebra and describe its properties.

1:00 pm in 347 Altgeld Hall,Tuesday, November 14, 2017

#### Almost sure scattering for the energy-critical NLS

###### Jason Murphy   [email] (Missouri S&T Math)

Abstract: We consider the defocusing energy-critical nonlinear Schrödinger equation in four space dimensions with radial (i.e. spherically-symmetric) initial data below the energy space. In this setting, the problem is known to be ill-posed. Nonetheless, we can show that for suitably randomized radial initial data, one obtains global well-posedness and scattering almost surely. This is joint work with R. Killip and M. Visan.

1:00 pm in 345 Altgeld Hall,Tuesday, November 14, 2017

#### Ax–Schanuel and Strong Minimality

###### Vahagn Aslanyan (Carnegie Mellon)

Abstract: Schanuel's conjecture is a transcendence conjecture on the complex exponential function. The Ax–Schanuel theorem is a differential analogue of that conjecture proven by J. Ax in 1971. It establishes a transcendence result for the solutions of the exponential differential equation $y' = yx'$ in a differential field. I will discuss Ax–Schanuel type results for other differential equations/functions, most importantly, for the third order non-linear differential equation of the modular $j$-invariant, due to J. Pila and J. Tsimerman. I will show how that kind of results can be used to show that certain sets in differentially closed fields are strongly minimal and geometrically trivial. As an application I will give a new proof for a theorem of J. Freitag and T. Scanlon establishing strong minimality and geometric triviality of the differential equation of the $j$-function (which is the first example of a strongly minimal set in $\mathrm{DCF}_0$ with trivial geometry which is not $\aleph_0$-categorical). If time permits, I will discuss Ax–Schanuel type conjectures for the Painleve equations based on the results of J. Nagloo and A. Pillay on strong minimality and geometric triviality of those equations.

2:00 pm in 347 Altgeld Hall,Tuesday, November 14, 2017

#### Estimation in Tournaments and Graphs with Monotonicity Constraints

###### Sabyasachi Chatterjee (Illinois Stat)

Abstract: We consider the problem of estimating the probability matrix governing a tournament or linkage in graphs from incomplete observations, under the assumption that the probability matrix satisfies natural monotonicity constraints after being permuted in both rows and columns by some latent permutation. We propose a natural estimator which bypasses the need to search over all possible latent permutations and hence is computationally tractable. We then derive asymptotic risk bounds for our estimator. Pertinently, we demonstrate an automatic adaptation property of our estimator for several sub classes of our parameter space which are of natural interest, including generalizations of the popular Bradley Terry Model in the Tournament case, the β model and Stochastic Block Model in the Graph case, and Hölder continuous matrices in the tournament and graph settings.

3:00 pm in 243 Altgeld Hall,Tuesday, November 14, 2017

#### Kodaira-Saito vanishing via Higgs bundles in positive characteristic

###### Donu Arapura (Purdue University)

Abstract: In 1990, Saito gave a strong generalization of Kodaira’s vanishing theorem using his theory of mixed Hodge modules. I want to explain the statement in the special case of a variation of Hodge structure on the complement of a divisor with normal crossings. Unlike Saito’s original proof, I will describe a proof using characteristic p methods.

3:00 pm in 241 Altgeld Hall,Tuesday, November 14, 2017

#### Sharp Thresholds for Random Constraint Satisfaction Problems

###### Felix Clemen (Illinois Math)

Abstract: One of the most intensively studied phenomena in probabilistic combinatorics is the threshold phenomenon, whereby a structure undergoes a rapid transformation as a result of a small change in a parameter responsible for its random nature. We will examine these phase transitions in constraint satisfaction problems (CSPs). A CSP consists of a set of variables, their domain and a collection of constraints determining which combinations of assignments of values to variables are restricted. The goal is to find an assignment of values to all variables that satisfies all constraints simultaneously. Through this, a variety of different problems in combinatorics, computer science, information theory, physics and operations research can be modeled.

4:00 pm in 245 Altgeld Hall,Tuesday, November 14, 2017

#### Nonlinear dispersive equations on large domains and wave turbulence

###### Zaher Hani   [email] (Georgia Institute of Technology)

Abstract: In this talk, we will be mainly concerned with the following question: Suppose we consider a nonlinear dispersive or wave equation on a large compact domain of characteristic size L. What is the effective dynamics when L is very large? This question is relevant for equations that are naturally posed on large domains (like the water waves equation on the ocean), and in turbulence theories for dispersive equations. It’s not hard to see that the answer is intimately related to the particular time scales at which we study the equation, as one often obtains different effective dynamics on different timescales. After discussing some relatively “trivial” time scales (and their corresponding effective dynamics), we shall attempt to access longer times scales and try to describe the effective equations that govern the dynamics there. The ultimate goal is to reach the so-called the “kinetic time scale” over which it is conjectured that the effective dynamics are described by a kinetic equation called the “wave kinetic equation”. This is the main claim of wave turbulence theory. We will discuss several results that are aimed at addressing the above problematic for the nonlinear Schrodinger equation. Recent results are joint works with Tristan Buckmaster, Pierre Germain, and Jalal Shatah.