## Spring 2017

__Colloquium__

https://math.columbian.gwu.edu**Organizer**: Valentina Harizanov

Regular colloquium meets on Friday 1:00-2:00 in Rome 204, January 17 to April 28, 2017

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**Title:** How a deterministic life gets influenced by a bunch of stochastic shocks

**Speaker:** Speaker: Annie Millet (University Paris 1 Panthéon-Sorbonne)

http://samm.univ-paris1.fr/-An

**Date and Time:** Friday, May 5, 1:00–2:00pm

**Place:** Rome 204

**Abstract:** We will present some results on parabolic (heat-type) and hyperbolic (wave-type) nonlinear partial differential equations subject to a random perturbation. Such «noise» models the sum of infinitesimal shocks in the environment. Besides well-posedness, some properties of the solution will be discussed, such as discretization schemes, concentration of the distribution when the intensity of the noise approaches 0, and long time behavior.

**Short bio:** Annie Millet is currently a professor at the University Paris 1 Panthéon-Sorbonne. She graduated from Paris 6 under the supervision of Antoine Brunel and has held positions in the University of Poitiers, the Ohio State University, the University of Angers and in the University of Paris 10-Nanterre. Her research topics are mainly in infinite dimensional stochastic calculus. Besides being on research frontiers, she has also been elected Research Vice-President and head of the Scientific Council of the University at Paris 1 Panthéon-Sorbonne from 2012 to 2016.

**Title:** Delta-matroids and Vassiliev invariants, Opening talk of Knots in Washington XLIV

http://at.yorku.ca/cgi-bin/

**Speaker:** Sergei Lando, Higher School of Economics, Skolkovo Institute of Science and Technology, Russia

**Date and Time:** Friday, April 28, 1:00–2:00pm

**Place:** Rome 351

**Abstract: **Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. There is also a natural way to define 4-term relations for abstract graphs, and graph invariants satisfying these relations produce weight systems: to each chord diagram its intersection graph is associated.

The notion of weight system can be extended from chord diagrams, which are orientable embedded graphs with a single vertex, to embedded graphs with arbitrary number of vertices that can well be nonorientable. These embedded graphs are a tool to describe finite order invariants of links: the vertices of a graph are in one-to-one correspondence with the link components.

We are going to describe two approaches to constructing analogues of intersection graphs for embedded graphs with arbitrary number of vertices. One approach, due to V. Kleptsyn and E. Smirnov, assigns to an embedded graph a Lagrangian subspace in the relative first homology of a 2-dimensional surface associated to this graph. Another approach, due to S. Lando and V. Zhukov, replaces the embedded graph with the corresponding delta-matroid, as suggested by A. Bouchet in 1980's. In both cases, 4-term relations are written out, and Hopf algebras are constructed.

Vyacheslav Zhukov proved recently that the two approaches coincide.

__Special Colloquium/Logic Seminar__

**Title:** *The age of cohesive powers*

**Speaker:** Rumen Dimitrov, Western Illinois University

http://www.wiu.edu/users/

**Date and Time**: Friday, April 28, 10:30–11:30a.m.

**Place:** Rome Hall (801 22^{nd} Street), Room 204

**Abstract:** Fraïssé defined the age of a structure *A* to be the class of all structures isomorphic to the finitely generated substructures of *A*. He then used this term to refer to one structure as being younger than another. In this talk, I will consider the age of the rationals *Q* as a dense linear ordering without endpoints and as a field, and will discuss connections with the earlier notion of the cohesive powers of *Q*. I will also establish different model-theoretic properties of cohesive powers.

**Title:** Dynamics of complex singularities and wavebreaking in 2D hydrodynamics with free surface

**Speaker:** Pavel Lushnikov, University of New Mexico and Landau Institute for Theoretical Physics

http://math.unm.edu/~plushnik/

http://www.itp.ac.ru/en/person

**Date and Time:** Friday, April 21, 1:00–2:00pm

**Place:** Rome 204

**Abstract: **We consider 2D hydrodynamics of ideal fluid with free surface. A time-dependent conformal transformation is used which maps a free fluid surface into the real line with fluid domain mapped into the lower complex half-plane. The fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. The initially flat surface with the pole in the complex velocity turns over arbitrary small time into the branch cut connecting two square root branch points. Without gravity one of these branch points approaches the fluid surface with the approximate exponential law corresponding to the formation of the fluid jet. The addition of gravity results in wavebreaking in the form of plunging of the jet into the water surface. The use of the additional conformal transformation to resolve the dynamics near branch points allows to analyze wavebreaking in details. The formation of multiple Crapper capillarysolutions is observed during overturning of the wave contributing to the turbulence of surface wave. Another possible way for thewavebreaking is the slow increase of Stokes wave amplitude through nonlinear interactions until the limiting Stokes wave forms with subsequent wavebreaking. For non-limiting Stokes wave the only singularity in the physical sheet of Riemann surface is the square-root branch point located. The corresponding branch cut defines the second sheet of the Riemann surface if one crosses the branch cut. The infinite number of pairs of square root singularities is found corresponding to infinite number of non-physical sheets of Riemann surface. Each pair belongs to its own non-physical sheet of Riemann surface. Increase of the steepness of the Stokes wave means that all these singularities simultaneously approach the real line from different sheets of Riemann surface and merge together forming 2/3 power law singularity of the limitingStokes wave. It is conjectured that non-limiting Stokes wave at the leading order consists of the infinite product of nested square root singularities which form the infinite number of sheets of Riemann surface. The conjecture is also supported by high precision simulations, where a quad (32 digits) and a variable precision (up to 200 digits) were used to reliably recover the structure of square root branch cuts in multiple sheets of Riemann surface

**Joint Colloquium/Applie Math/Analysis talk**

**Title: **Singularity Formation in Nonlinear Derivative Schrödinger Equations

**Speaker: ** Gideon Simpson, Drexel University

**Date and Time: April 19****, 2017**, Wednesday,2:30 pm-3:30pm

**Place:** Rome 771

**Abstract: **Direct numerical simulation of an $L^2$ supercritical variant of the derivative nonlinear Schrödinger equation suggests that there is a finite time singularity. Subsequent exploration with the dynamic rescaling method provided more detail about the blowup and a recent refined asymptotic analysis of the blowup solution gives predictions of the blowup rates. Due to the mixed hyperbolic-dispersive nature of the equation, these methods have limited the proximity to the blowup time. Using a locally adaptive meshing method, we are able to overcome these difficulties.

**Title:** Rotational symmetries of knots

**Speaker:** Swatee Naik, University of Nevada and NSF

http://wolfweb.unr.edu/

**Date and Time:** Friday, April 14, 1:00–2:00pm

**Place:** Rome 204

**Abstract:**Knots are circles embedded in the three dimensional sphere. Periodic knots, such as the overhand or trefoil knot, are invariant under a rotation, and this symmetry can be easily illustrated in a knot diagram drawn in the plane. It so happens that the orbit space is also a three-sphere, in which the image of a periodic knot is called a quotient knot. Many properties of periodic knots are a direct consequence of the branched covering set up that occurs between various three-manifolds that are naturally associated with the periodic knot and the quotient knot, respectively.

In this talk we will begin with definitions and examples, introduce the basics of the theory, and demonstrate how properties of periodic knots can be used to detect knots that are not periodic. Our tools will include knot polynomials, homology of branched covers, and Heegaard-Floer correction terms.

**Short Bio:**Dr. Swatee Naik is a Professor at the University of Nevada, Reno and currently a program director at the National Science Foundation. Her area of research is knot theory and low dimensional topology. The draft of a book on Classical Knot Concordance is work in progress with Charles Livingston, and we welcome feedback. Swatee has served in administrative roles including vice chair and chair of the department and chair of the university faculty senate. At NSF, her main duties are in Topology and Geometric Analysis. She is involved in many other programs, such as Research Experience for Undergraduates, Graduate Research Fellowships, Postdoctoral Fellowships, Enriched Doctoral Training and NSF Research Traineeships.

**Title:**The shortest path poset of Bruhat intervals

**Speaker:**Saul Blanco Rodriguez, Indiana University Bloomington

https://www.soic.indiana.edu/

**Date and Time:** Friday, April 7, 1:00-2:00pm

**Place: Rome 204**

**Abstract:** A Coxeter group W is a group generated by reflections; examples are the symmetric group and the hyperoctahedral group. These groups have many interesting combinatorial properties. For instance, one can define a partial order, called the Bruhat order, on the elements of W . If [u,v] is an interval in the Bruhat order, its Bruhat graph, B(u,v) includes the Hasse diagram of the poset [u,v] with edges directed upwards, as well as other edges that I will describe in the talk. While the longest u-v paths in B(u,v) are well-understood (they form a face poset of a regular cell decomposition of a sphere), not much is known about the other u-v paths in B(u,v). In this talk, I will describe what is known of the shortest u-v paths and point out connections to other areas.

**Special Colloquium**

**Title:**Just so stories a la carte around Geometry, Dynamics, and PDEs.

**Speaker:** Dmitri Burago, Penn State University

**Date and Time:** Tuesday, April 4, 2:20-3:20pm

**Place: Rome 351**

**Abstract:** The lecture consists of several mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some (hardly all) of the following topics, based on the input from the audience:

**Short Bio: **Dmitri Burago is a Distinguished Professor of Mathematics at Penn State University. His research interests include dynamical systems, algorithmic complexity, Finsler geometry, combinatorial group theory, and partial differential equations. In 1997, Burago was the recipient of an Alfred P. Sloan Research fellowship and, in 1995, he received Penn State's Faculty Scholar Medal for Outstanding Achievements. He has been a member of the St. Petersburg Mathematical Society since 1992. Before joining the Eberly College of Science faculty at Penn State in 1994, Burago was a faculty member at the University of Pennsylvania, a researcher at the St. Petersburg Institute for Informatics and Automation, and an assistant professor in the Department of Mathematics and Mechanics at St. Petersburg State University in Russia. He received doctoral and master's degrees from St. Petersburg State University in 1992 and 1986, respectively.

__Distinguished Speculative First of April Talk__

(with Center for Quantum Computing, Information, Logic, and Topology)

**Title: **Skein modules and Quantum Computing in 3-manifolds

**Speaker:** Uwe Kaiser, Boise State University

http://diamond.boisestate.edu/

**Date and Time:** Friday, March 31, 1:00-2:00pm

**Place: Rome 204**

**Abstract:** The mathematical model of anyons (quasi-particles used in the theory of topological quantum computation) are Unitary Modular Tensor Categories. These are also used to construct topological quantum field theories (TQFT) in dimension 2+1, and in particular define invariants of framed links in 3-manifolds. The invariants can be thought of as quantum amplitudes of anyon world lines in a 3-dimensional spacetime. They also satisfy skein relations, which can be used to define skein modules of framed links in 3-manifolds. Thus quantum amplitudes, which we think of as the measurement results of “quantum computing in a 3-manifold,” are functionals on skein modules. These modules are hard to compute in general, and it is not very well-known how they relate to the geometric topology of 3-manifolds. This, of course, is also true for the quantum invariants. Thus it is interesting to study the relation between these concepts. We describe some results known around these problems and speculate about relations to finite type invariants (Kontsevich integral) and categorifications (link homologies).

**Short Bio:** Dr. Uwe Kaiser received his PhD and habilitation from the University of Siegen in Germany. Additionally he also holds a K12-teaching certificate for Mathematics and Physics in Germany. Currently he is Associate Professor in the Mathematics Department at Boise State University, where he also serves as Associate Chair. His research interests are in Geometric and Algebraic Topology, more recently in Low Dimensional Topology and Quantum Topology. He is also working on several projects and grants related to STEM Education at Boise State University.

__Graduate Student Seminar followed by Friday’s Colloquium__

**Title:** Mapping space methods for skein modules

**Speaker:** Uwe Kaiser, Boise State University

http://diamond.boisestate.edu/~kaiser/

**Date and Time:** Thursday, March 30, 5:15-6:30pm

**Place:** Rome 352

**Special Colloquium**

**Title:**Making nonelementary classes more elementary

**Speaker:** William Boney, Harvard University

**Date and Time:** Friday, February 24, 1:30-2:30pm

**Place: MPA 305**

**Abstract: **Classification theory seeks to organize classes of structures (such as algebraically closed fields, random graphs, dense linear orders) along dividing lines that separate classes into well-behaved on one side and chaotic on the other. Since its beginning, classification theory has discovered a plethora of dividing lines for classes axiomatizable in first-order logic and has been applied to solve problems in algebraic geometry, topological dynamics, and more.

However, when looking at examining nonelementary classes (such as rank 1 valued fields or pseudoexponentiation), the lack of compactness is a serious impediment to developing this theory. In the past decade, Grossberg and VanDieren have isolated the notion of tameness. Tameness can be seen as a fragment of compactness that is strong enough to allow the construction of classification theory, but weak enough to be enjoyed by many natural examples. We will discuss the motivation for classification theory in nonelementary classes and some recent results using tameness, focusing on illuminating examples. No logic background will be assumed.

**Special Colloquium**

**Title:**Model theory and Painlevé equations

**Speaker:** James Freitag, University of Illinois at Chicago

**Date and Time:** Thursday, February 23, 1:30-2:30pm

**Place: Phillips 217**

**Abstract: **The Painlevé equations are six families of nonlinear order two differential equations with complex parameters. Around the start of the last century, the equations were isolated for foundational reasons in analysis, but the equations have since arisen naturally in various mathematical contexts. In this talk, we will discuss how to use model theory, a part of mathematical logic, to answer several open questions about Painlevé equations. We will also describe several other applications of model theory and differential algebraic equations to number theory.

**Special Colloquium**

**Title:**q-analogues of factorization problems in the symmetric group

**Speaker:** Joel Lewis, University of Minnesota

**Date and Time:** Friday, February 17, 2:15-3:15pm

**Place: Rome 351**

**Abstract: **Given a nice piece of combinatorics for the symmetric group S_n, there is often a corresponding nice piece of combinatorics for the general linear group GL_n(F_q) over a finite field F_q, called a q-analogue. In this talk, we'll describe an example of this phenomenon coming from the enumeration of factorizations. In S_n, the number of ways to write an n-cycle as a product of n - 1 transpositions is Cayley's number n^(n - 2). In GL_n(F_q), the corresponding problem is to write a Singer cycle as a product of n reflections. We show that the number of such factorizations is (q^n - 1)^(n - 1), and give some extensions. Mysteriously, the second answer is closely related to the first as q approaches 1. Our proofs do not provide an explanation for this relationship; instead, they proceed by exploiting the (complex) representation theory of GL_n(F_q).

**Special Colloquium**

**Title:** Connectivity and structure in matroids

**Speaker:** Stefan van Zwam, Louisiana State University

**Date and Time:** Wednesday, February 15, 11:00am-12:00pm

**Place:** Rome 459

**Abstract: **A general theme in matroid structure theory is that highly connected matroids exhibit more structure than matroids with low-order separations. We will discuss several examples of this phenomenon, as well as an application to the theory of error-correcting codes.

**Special Colloquium**

**Title:** Gaussian measures on infinite dimensional spaces and applications

**Speaker:** Nathan Totz, University of Massachusetts Amherst

**Date and Time:** Monday, February 13, 1:45-2:45pm

**Place:** Rome 459

**Abstract: **We review the classical extension of the Gaussian probability measure from finite dimensional spaces to infinite dimensional spaces. Such Gaussian measures (along with their weighted relatives) play an important role as invariants of flows defined on infinite dimensional spaces. As an application of this idea, we employ Gaussian measures to address the question of the long time existence of a flow corresponding to a family of modified surface quasigeostrophic equations, regarded as a flow on a space of Fourier coefficients. We present recent results (joint with Andrea Nahmod, Natasha Pavlovic, and Gigliola Staffilani) showing that such flows are global in time on a subset of a rough Sobolev space of full measure.

**Special Colloquium**

**Title:** Shapes of polynomial Julia sets

**Speaker:** Kathryn Lindsey, University of Chicago

**Date and Time:** Friday, February 10, 2:15-3:15pm

**Place:** Rome 351

**Abstract: **The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. W. Thurston asked “What are the possible shapes of polynomial Julia sets?” For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is “yes.” I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes.

**Title:** Some almost sharp scattering results for the cubic nonlinear wave equation

**Speaker:** Benjamin Dodson, John Hopkins University

**Date and Time:** Tuesday, January 31, 2017, 3:00-4:00pm

**Place:** Rome 771

**Abstract: **In this talk we will discuss some scattering results for the cubic nonlinear wave equation. We will prove these results for radial data nearly lying in the critical Sobolev space. We prove this using hyperbolic coordinates.

**Short Bio:** Prof. Ben Dodson is an assistant professor of mathematics at John Hopkins University who made a breakthrough in the scattering theory of global solutions in the nonlinear wave-like equations when he was at UC-Riverside for a year after graduating from the University of North Carolina-Chapel Hill in 2009. His did his postdoctoral studies at UC-Berkeley.

**Title:** Examples of Linear Algebra over Division Algebras

**Speaker:** Salahoddin Shokranian, University of Brasilia

**Date and Time:** Friday, January 27, 1:00-2:00pm

**Place:** Rome 204

**Abstract: **Matrices over some division algebras are considered. In the case of finite fields, applications are in coding theory, and in the case of non‐commutative division algebras, Hermitian matrices over quaternions provide examples toward geometry of such matrices and analysis.

**Short Bio: **Salahoddin Shokranian has studied mathematics, graduate level, at the University of California, Berkeley where he did his Ph.D. in automorphic forms. He moved to Brasilia since then, working at the University of Brasilia where he is to be retired. He has been visitor to several research institutes such as the Institute for Advanced Study, Tata Institute of Fundamental Research, Max-Planck Institute for Mathematics and worked at the Universities of Purdue, Toronto and Yale. He has lectured at many universities and research centers. He likes to write books; on linear algebra, number theory, modular forms, cryptography and history of modern number theory. Some of them have already been republished or reprinted.