The colloquium is preceded by tea and cookies at 3:30 in MONT 201 and usually followed by dinner with the speaker.

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Title: Theory of matrix weights and convex body domination

Speaker: Sergei Treil (Brown University)

Faculty Sponsor: Guozhen Lu

Time: Thursday, February 22, 2018 at 4:00 pm

Place: MONT 214Abstract: Weighted estimates with matrix weights appear naturally in theory of stationary random processes and perturbation theory, and gained some new attention recently. After describing some motivations for such weighted estimates, I'll discuss the method of estimating vector-valued singular integral operators by the so called sparse operators. In the scalar case the sparse domination is a recently introduced powerful tool of harmonic analysis, and generalizing it to the vector valued case helps in many problems. Of course, trivial generalization does not work: the target space of our sparse operator is the set of convex-body-valued functions. It looks complicated, but the weighted estimated of such operators is an easy task. As an application we were able to get a new easy proof of the weighted estimates of the vector Calderon-Zygmund operators with matrix Muckenhoupt weights. Some intriguing open problems will also be discussed.

Title: Bernhard Riemann and his moduli space

Speaker: Lizhen Ji (University of Michigan)

Faculty Sponsor: Damin Wu

Time: Thursday, March 8, 2018 at 4:00 pm

Place: MONT 214Abstract: Though Riemann only published a few papers in his lifetime, he introduced many notions which have had long lasting impact on mathematics. One is the concept of Riemann surfaces, and another is the related notion of moduli space of Riemann surfaces. The road to formulate precisely the moduli space M_g of compact Riemann surfaces of genus g and to understand its meaning is long and complicated, and mathematicians are still working hard to understand its structures and properties from the perspectives of geometry, topology, analysis etc. Its analogy and connection with locally symmetric spaces have provided an effective way to study these problems. In this talk, I will describe some historical aspects which may not be so well-known and some recent results on the interaction between moduli spaces and locally symmetric spaces.

Title: spring break

Speaker:

Time: Thursday, March 15, 2018 at 4:00 pm

Place: MONT 214

Title: CANCELED
(Oligopoly Mean Field Games & Energy Production)

Speaker: Ronnie Sircar (Princeton University)

Faculty Sponsor: Bin Zhu

Time: Thursday, April 5, 2018 at 4:00 pm

Place: MONT 214Abstract: We discuss oligopoly games with a continuum of players that have mean field structure. These may be of Bertrand (price setting) or Cournot (quantity setting) type and may apply to analysis of consumer goods or energy markets respectively. Key advantages over finite player nonzero sum differential games are analytical and numerical tractability of the associated PDEs. Models for energy markets with competition between producers with heterogeneous costs (fossils vs. renewables) are presented as motivation. Sources of uncertainty in the stochastic version of the problem include controlled random discovery of reserves, and uncertain demand environments.

Title: Number Theory and Dynamics

Speaker: Joseph Silverman (Brown University)

Faculty Sponsor: Alvaro Lozano-Robledo

Time: Wednesday, April 11, 2018 at 4:30 pm

Place: MONT 104Abstract: Discrete dynamics is the study of iteration.ï¿½ A primary objective is the classification of points in a set $S$ according to their orbits under repeated applicaion of a self-map $f:S\to S$.ï¿½ Classically $S$ would be $\mathbb{R}^n$ or $\mathbb{C}^n$, and real and complex dynamics are mature and thriving fields of study.ï¿½ But for a number theorist, it is natural to take $S$ to be a set of arithmetic interest, for example $\mathbb{Q}$ or the field $\mathbb{F}_p$ of integers modulo $p$. The past 25 years has seen the development of the field of *Arithmetic Dynamics*, in which one studies dynamical analogues of classical results in number theory. Here are two illustrative problems that I will discuss: (1) Let $f(z)\in\mathbb{Q}[z]$ be a polynomial. There are always infinitely many complex numbers with finite forward orbit under iteration of $f$, but how many of those complex numbers can be rational numbers? (2) Let $f(z)\in\mathbb{F}_p[z]$ be a polynomial. How many mod $p$ starting points return to themselves under iteration of $f$? As is typical in number theory, questions tend to be easy to state and difficult to solve!Comments: A reception preceding the talk is in Monteith 201.

Title: Entropy, symmetry, and rigidity

Speaker: David Constantine (Wesleyan)

Time: Wednesday, April 25, 2018 at 4:00 pm

Place: MONT 104Abstract: Suppose you are given a large collection of nodes and are allowed to create a network, or graph, by making a specified overall number of connections between them. Which connections should you choose in order to maximize the number of different paths of length 10 or 100 or 1,000 which can be drawn in your graph? It turns out that the best way to form such a graph is to make it maximally symmetric. Counting such paths is related to the entropy of the graph, and this strategy of maximizing entropy by seeking symmetry works in a number of contexts. In this talk I'll explain what I mean by entropy, and why this concept is useful. Then I'll sketch a proof of the result mentioned above and say some things about how entropy rigidity theorems like this one appear in the fields of geometry and dynamics. The only background necessary for the talk is a bit of linear algebra.Comments: This talk is part of the math department's Awards Day ceremony.

Title: Poisson stochastic process and basic Schauder and Sobolev estimates in the theory of parabolic
equations

Speaker: Nicolai Krylov (University of Minnesota)

Faculty Sponsor: Oleksii Mostovyi

Time: Thursday, April 26, 2018 at 4:00 pm

Place: MONT 214Abstract: We show among other things how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other methods are available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. Joint work with E. Priola.

Organizer: Ralf Schiffler