## Graduate Student Seminar Archives

**Fall 2017**

**Title:** Optimal descriptions of computable groups

**Speaker**: Iva Bilanovic

**Date and Time**: Friday, December 1, 3:00-4:00pm

**Place**: Rome 206

**Abstract: **We will introduce the infinite dihedral group, D∞, a finitely generated group whose primitive and imprimitive sets of words are both computable given a computable copy of the group. By the Scott Isomorphism Theorem we can describe D∞ by a sentence, in countable infinitary language, whose computable models are exactly the isomorphic copies of the group. The set of Gödel codes for these computable isomorphic copies of D∞ allows us to check that our description is as simple as possible.

**Title:** Holder's regularity for solutions of a parabolic equation

**Speaker**: Xinyu Zhang

**Date and Time**: Friday, November 3, 3:00-4:00pm

**Place**: Rome 206

**Abstract: **De Giorgi introduced his method in 1957 to study the regularity of elliptic equations with rough coefficients. In this talk, we will consider, for simplicity, the laplacian-type equation and prove the solutions are Holder. I will firstly introduce the oscillation lemma, then show the lemma implies the Holder's continuity. Then I will fill the gaps between the oscillation lemma and De Giorgi's lemmas

**Title:** The I-method and its application.

**Speaker**: Debdeep Bhattacharya

**Date and Time**: Friday, October 27 , 3:00-4:00pm

**Place**: Rome 206

**Abstract: **Developed by Colliander-Keel- Staffilani-Takaoka-Tao in 2001, the so called "I-method" has been extensively used to prove the global well-posedness of various nonlinear dispersive equations for data with infinite energy, where the classical energy-conservation arguments fail. The method uses a modified energy functional which is "almost conserved" in time, leading to a global existence. Following the work of Farah-Linares-Pastor we shall demonstrate this method for the geralized KdV equation and outline an improvement of available global well-posedness results for the modified Zakharov-Kuznetsov equation in 2D

**Title:** An algebraic treatment of congruences in Number Theory.

**Speaker**: Konstantinos Smpokos

**Date and Time**: Friday, October 6 , 1:00-2:00pm

**Place**: Rome 206

**Abstract: **In this talk I will present a research paper that has been published in the Bulletin of** **the Hellenic mathematical society in December 2016.We will discuss the behavior of certain free abelian subgroups of the multiplicative group of positive rationals and their relationship with the group of units of integers modulo n.The methods we will use are purely algebraic.

**Title:** Introduction of three models of price formation

**Speaker**: Chubo Deng

**Date and Time**: Friday, September 29 , 1:00-2:00pm

**Place**: Rome 206

**Abstract: **Based on the observations of the real world price formation process, I will introduce three models. One is from J.-M. Lasry and P.L.Lions evolution model. I modify their model to get other two which will be applied to the real world situation. I will use real Bitcoin chart to illustrate my idea.

**Title:** Basic homological algebra and its application in knot theory.

**Speaker**: Xiao Wang, GWU

**Date and Time**: Friday, September 15, 1:00-2:00pm

**Place**: Rome 206

**Abstract: **I will first introduce projective resolution, Tor functor, and the Fundamental lemma of homological algebra. Then we will see some applications of these basic homological algebra to Khovanov homology, which is a powerful link invariant that categorifies the Jones polynomial.

**Title**: Stable singularity formations in the nonlinear dispersive equations.

**Speaker:** Kai Yang, GWU

**Date and Time:** Friday September 22, 1pm-2pm

**Place:** Rome 206

**Abstract**: We will consider a couple of basic models used in dispersive (wave-type) differential equations and will address the question of formation of singularities (i.e., solutions that break down in finite time). These solutions are typically referred to blow-up solutions, since they tend to concentrate on some set (e.g. at a point) with amplitude or speed growing unboundedly. We study the case called the L2-critical case, which means that the solutions and the equations preserve the L2-norm, often referred as mass. We introduce the dynamic rescaling method to simulate the rate of blow-up solutions for the L2-critical nonlinear Schrodinger equation (NLS) as well as for the L2-critical generalized Hartree equation (gHartree). We study the solutions initiated from the radial data and consider dimensions from d=4 to d=12 for the NLS equation and from d=3 to d=7 for the gHartree equation. It turns out that the stable singularities have the blowup rate which can be expressed as (T-t)^{-0.5}, with a logarithmic corrections from analyzing the rescaled equations. We also provide a numerically-assisted proof of the spectral property for the NLS equation from d=4 to d=12, which confirms our direct computational results. In the gHartree case, we can only show the spectral property in the 3d radial case, as the spectral property of the nonlocal (convolution) non

**Spring 2017**

**Title:** Long time behavior of solutions to the generalized Hartree equation

**Speaker**: Anudeep Kumar

**Date and Time**: Friday, April 28, 3:00-4:00pm

**Place**: Rome 352

**Abstract: **We study the long time behavior of solutions in the nonlinear dispersive equations, in particular, the generalized Hartree-type equation, where the potential is of nonlocal type and is expressed as a convolution. The behavior of solutions has been studied quite extensively for some basic model equations such as nonlinear wave equation and nonlinear Schr ̈odinger equation and various regimes were exhibited such as finite time existing solutions (or so called blow-up in finite time), or solutions existing globally in time: solitary waves or scattering (approaching linear solutions as t → ±∞). In this talk we present small data theory, dichotomy for scattering versus blow-up, and criteria for solutions that blow-up in finite time with an emphasis on the method of concentration - compactness.

**Title:** Pattern formation – on the modeling of multi-constituent inhibitory systems

**Speaker**: Chong Wang

**Date and Time**: TBA

**Place**: TBA

**Abstract: **Skin pigmentation, animal coats and block copolymers, which can be considered as multi-constituent inhibitory systems, are all around us. Theoretical analysis and numerical simulation of multi-constituent inhibitory systems will be provided here. An inhibitory system is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. We establish that in different parameter ranges there are corresponding assemblies of certain patterns that exist as the stationary sets of the free energy functional. Numerically, a diffusive interface model is proposed and many self-assembly processes, which form various patterns, are vividly showed here. Different numerical schemes are compared and a new technique is introduced to be consistent with the Euler-Lagrange equation in the sharp interface model.

**Title:** Topology and Computability of Order Relations on Some Algebraic Structures**Speaker:** Trang Ha, GWU**Date and Time:** Friday, April 7, 3:00-4:00pm**Place:** Rome 352

**Abstract: **We discuss order relations on computable magmas, which are computable algebraic structures with binary operations that are not necessarily associative or commutative. The space of orders on a magma consists of all possible orders (of a certain kind) on the structure. We investigate Turing complexity and topological properties of the spaces of orders on a computable magma. We also consider orderings on interesting examples of magmas that rises from algebra and knot theory such as quandles and racks.

**Title**: Priority Argument Constructions for Graphs and Trees**Speaker**:Hakim Walker, GWU**Date and Time:** Friday March 31, 3:00-4:00**Place:** Rome 352

**Abstract:** In computable model theory, we study the algorithmic content and properties of classical mathematical structures, such as groups, vector spaces, linear orders, etc. In particular, one area that we examine is the computational complexity of additional relations on the structures, as well as isomorphisms between structures. It is usually the case that while a structure itself is computable, additional relations on the structure are not computable. Also, we often have two computable structures that are isomorphic to each other but have no computable isomorphism between them. One of the most common ways to demonstrate these non-computable properties is to use a priority argument, which is a generalization of Cantor's diagonal argument.

**Title**: Numerical Investigations of Pattern Formation in Binary Systems with Inhibitory Long-range Interaction**Speaker**:Jiajun Lu**Date and Time:** Friday March 24, 3:00-4:00**Place:** Rome 352

**Abstract:** I investigate pattern formation in a two-phase system on a two-dimensional manifold by numerically computing the minimizers of a Cahn-Hilliard-like model for micro-phase separation of diblock copolymers. The total energy of the system includes a short-range term - a Landau free energy and a long-range term - the Otha-Kawasaki functional. The shortrange term favors large domains with minimum perimeter and the long-range inhibitory term favors small domains. The balance of these terms leads to minimizers with a variety of patterns, including single droplets, droplet assemblies, stripes, wriggled stripes and combinations thereof. I compare the results of our numerical simulations with known analytical results and discuss the stability of the computed solutions and the role of key parameters in pattern formation. I focus on the triaxial ellipsoid for demonstration purposes, but our methods are general and can be applied to higher genus surfaces and surfaces with boundaries.

**Title**: Comparative Genomics Meets Genome Assembly: from Ancestral Reconstruction to Genome Scaffolding**Speaker**:Sergey Aganezov**Date and Time:** Friday March 3, 3:00-4:00**Place:** Rome 352

**Abstract:** We present our research that revolves around mathematical modeling and algorithms development in the area of computational biology, and more precisely, genome assembly as well as ancestral genome reconstruction and comparative genome rearrangement analysis. We start by demonstrating our results in the comparative genomics area by describing our work on the median problem of three genomes under the DCJ metric, as well as a more general problem of ancestral genome reconstruction of multiple input genomes. We then move to the problem of *in silico* genome scaffolding, where we present a novel method for for finishing incomplete genome assemblies and demonstrate the benefits of combining both areas of genome assembly and ancestral genome reconstruction. Lastly we present results of our work on the problem of comparing and merging of multiple scaffold assemblies of the same organisms.

**Fall 2016**

**Title**: A tale of two theorems**Speaker**: Professor Ted Turner**Date and Time:** Friday September 9, 3:00-4:00**Place:** Philips 736

**Abstract:** See attachment to email announcement** **

**Title**: Homological properties of algebraic structures arising from knot theory**Speaker**: Sujoy Mukherjee**Date and Time:** Friday September 30, 3:00-4:00**Place:** Philips 736

**Abstract:** I will start with a brief introduction to knot theory. Following this I will discuss some well known knot invariants. Then I will show how to extract algebraic structures from Reidemeister moves. In particular, the third Reidemeister move leads to the notion of a shelf. After introducing the homology theories used to study these algebraic structures, I will discuss homological properties of associative shelves.

**Title**: Torsion in rack and quandle homology and its applications to Knot Theory **Speaker**: Seung Yeop Yang**Date and Time:** Friday October 7, 3:00-4:00**Place:** Philips 736

**Abstract:** Rack homology theory was introduced between 1990 and 1995 by Fenn, Rourke, and Sanderson, and in 1999, Carter, Jelsovsky, Kamada, Langford, and Saito modified it to quandle homology theory in order to obtain knot invariants for classical knots and knotted surfaces in a state-sum form called cocycle knot invariants. In 1993, Fenn, Rourke, and Sanderson introduced rack spaces to define rack homotopy invariants and a modification to quandle spaces and quandle homotopy invariants of classical links was introduced by Nosaka in 2011.

In analogy to the well-known result in reduced group homology of finite groups that the order of a group annihilates its homology, we prove that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. It was an open conjecture for over 5 years. We also introduce an $m$-almost quasigroup quandle as a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups. Moreover, as a generalization of rack and quandle spaces, we define the Cayley-type graph and CW complex of a distributive structure and study their properties. Moreover, for a connected quandle we introduce the shadow homotopy invariant of a classical link.

**Title**: Computable Free Groups and Their Bases**Speaker**: Iva Bilanovic**Date and Time:** Friday October 14, 3:00-4:00**Place:** Philips 736

**Abstract:** In this talk we will consider the computability theoretic complexity of finding a basis for a computable free group of infinite rank. We will use basic properties of free groups to build a computable sequence of computable infinitary π2 formulas expressing the property of membership in a basis. The relativized limit lemma and these formulas will lead us to a π2 basis.

**Title**:Global regularity of Patlak-Keller-Segel equations that model chemotaxis**Speaker**: Xinyu Zhang**Date and Time:** Friday October 28, 3:00-4:00**Place:** Philips 736

**Abstract:** Chemotaxis is the means by which small organisms such as bacteria and somatic cells direct their movements towards or against the gradient of some chemical concentration. We will talk about the Patlak-Keller-Segel (PKS) equations in 2D that describe chemotaxis. We will also show that solutions with different mass sizes for PKS equations exhibit different behaviors.

**Title**: Local and global existence of solution to Nonlinear Schrodinger's equation with mass-critical nonlinearity**Speaker**: Debdeep Bhattacharya**Date and Time:** Friday November 11, 3:00-4:00**Place:** Philips 736

**Abstract:** We shall consider the nonlinear Schrodinger's equation with the power of nonlinearity being 1+4/n, where n is the dimension of the spacial variable. Using Strichartz's estimate and Banach's contraction mapping principle we shall prove the existence and uniqueness of the solution for finite time. Additionally, the solution exists for any time if we assume that the initial data has sufficiently small mass.

**Title**: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. **Speaker**: Chubo Deng**Date and Time:** Friday December 2, 3:00-4:00**Place:** Philips 736

**Abstract:** I will give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

**Spring 2016**

**Title**: The Convenient Untruths of Economics and Finance

**Speaker**: Harbir Lamba, Department of Mathematical Sciences, George Mason University

**Date and Time:** Tuesday, April 26, 2:00-3:00pm

**Place:** Bell 105

**Abstract:** Modern economics and finance are based upon some highly

implausible (but extremely convenient) assumptions about how humans and institutions behave. For example, assuming that people can be modelled as if they are perfectly rational at all times, have no memory, and act independently of each other greatly simplifies the mathematics required!

I will talk about the (often amusing) history of these ideas and their (sometimes disastrous) consequences. Then I will describe my own research which aims to mathematically describe what happens when these assumptions are weakened and replaced with more realistic ones via agent-based models.

**Title**: Patten formation - solutions of some nonlocal variational problems

**Speaker**: Chong Wang

**Date and Time:** Friday, April 22, 1:00-2:00pm

**Place:** Government 101

**Abstract:** Skin pigmentation, animal coats and block copolymers, which can be considered as multi-constituent inhibitory systems, are all around us. An inhibitory system is studied as a nonlocal geometric variational problem. The free energy of the system is the sum of two terms: the total size of the interfaces separating the constituents, and a longer ranging interaction energy that inhibits micro-domains from unlimited growth. We will talk about the existence of a stationary core-shell assembly, and also a double bubble solution as a new phase of a ternary inhibitory system. If time allows, we can also discuss the local minimum solutions of the original problems before Gamma Convergence.

**Title**: Minimal Surfaces and De Giorgi's Conjecture

**Speaker**: Yeyao Hu**Date and Time:** Thursday, April 21, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** Attached in email announcement 4-15-2016

**Title**: Mathematical models and algorithms for genome rearrangements and genome assembly

**Speaker**: Sergey Aganezov**Date and Time:** Thursday, April 14, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** My research revolves around mathematical modeling and algorithms development in the area of computational biology, and more precisely, genome assembly as well as ancestral genome reconstruction and comparative genome rearrangement analysis. The objective is to incorporate existing and develop new graph theoretical and combinatorial methods to tackle the problem of *in silico *finishing existin*de novo* / reference genome assembly, phylogenetic analysis of multiple genomes and quality improvement of existing genome assemblies by utilizing genome rearrangement analysis.

**Title**: Yang-Baxter Operators and Knot Theory

**Speaker**: Xiao Wang**Date and Time:** Thursday, April 7, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** Yang-Baxter Equation appears in several areas of physics( e.g.Statistical Mechanics, Quantum Theories). It also gives applications to Knot Theory. I will introduce the Yang-Baxter Equation and its solutions. Then I will show how to get link invariant from Yang-Baxter operators. Finally, we define a homology theory for Yang-Baxter operators and have some discussion about it.

**Title**: In pursuit of canonical structures on cellularly embedded graphs

**Speaker**: Jason Suagee**Date and Time:** Thursday, March 24th, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** In the mid 2000's Stephan Felsner (from TU Berlin) developed methods, given a planar embedded graph G, of associating a distributive lattice structure to the set of directed edge structures on G. This is now referred to as the theory of alpha-orientations, and can be used to derive canonical structures for any given planar embedded graph, in particular, canonical spanning trees and 2k factors. The theory has also been extended by several authors, who produce methods of bijectively constructing planar triangulations and several other families of planar maps, which are important for instance for 2D-Quantum Gravity simulations.

In this talk I will summarize Felsner's theory, and explain a recent generalization of his methods to the case of higher genus cellularly embedded graphs. I will also present a potential application to physics (in Quantum String Theory), which occurs though applying combinatorial methods to the theory of moduli spaces of complex curves, which are used to parameterize the functional integrals that govern the dynamics in string theory.

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken.**

**Title**: Numerical Minimizer of a Free Energy in Binary System

**Speaker**: Jiajun Lu **Date and Time:** Thursday, March 10, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** Attached to Colloquium/Seminar weekly announcements

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken.**

**Title**: POLITICS IS FOR PRESENT, BUT AN EQUATION IS FOR ETERNITY. (ALBERT EINSTEIN)

**Speaker**: Anudeep Kumar

**Date and Time:** Thursday, March 3rd, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** We study the long time behavior of solutions in the nonlinear dispersive equations, in particular, the generalized Hartree-type equation, where the potential is of nonlocal type and is expressed as a convolution. The behavior of solutions has been studied quite extensively for some basic model equations such as nonlinear wave equation and nonlinear Schr ̈odinger equation and various regimes were exhibited such as finite time existing solutions (or so called blow-up in finite time), or solutions existing globally in time: solitary waves or scattering (approaching linear solutions as t → ±∞). In this talk we present small data theory, dichotomy for scattering versus blow-up, and criteria for solutions that blow-up in finite time with an emphasis on the method of concentration - compactness.

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken.**

**Title**: Finding the maximum genus of a graph – Topic in Topological graph theory

**Speaker**: Lara El-Sherif

**Date and Time:** Thursday, February 25th, 3:00-4:00pm

**Place:** Corcoran 101

**Abstract:** For any graph G and an orientable surface Sg (a surface of genus g), whether G can be cellularly embedded in Sg creates an interesting problem for many topological graph theorists. The “genus range” of a graph G, denoted GR(G) is defined to be the set of numbers g such that the graph G can be cellularly embedded in surface Sg. We call the minimum number g in the genus range, the “minimum genus” of G and the largest number in the range, the “maximum genus” respectively. Whereas the study of minimum genus dates back into the 19th century, interest in maximum genus began in the 1970’s. The main contributors to the theory behind finding the maximum genus of a graph are Xuong and Nebesky, among others. In this talk we will introduce the methods used by both Xuong and Nebesky in solving the maximum genus problem. We will also talk about the polynomial time algorithms available for finding a maximum genus embedding of a graph and the problems that lie in those algorithms.

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken.**

**Title**: Behavior of solutions in the Nonlinear Klein-Gordon Equation

**Speaker**: Kai Yang

**Date and Time:** Friday, February 12th, 1:00-2:00pm

**Place:** Corcoran 101 **Abstract:**

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken. Also, please take note of the new day/old, Friday. **

**Title**: How Complicated are Orderings on Computable Structures?

**Speaker**: Trang Ha **Date and Time:** Thursday, January 28th, 3:00-4:00pm

**Place:** Corcoran 101 **Abstract:** A structure A is computable if it has the set of natural numbers N as its domain and all functions and relations on A are computable. We will examine some algebraic structures in this effective setting. Our main focus will be the complexities of orderings on these structures.

**Note: The Graduate Student Seminar is mandatory for ALL graduate Students! Attendance will be taken. Also, please take note of the new day, Thursday.**

**Title:** Can Computers Do Anything?

**Speaker:**Leah Marshall

**Date and Time:**Thursday, January 21st, 3:00-4:00pm

**Place:**Corcoran 101

**Abstract: [**Spoiler Alert: the answer is... no.] In this talk I will give an overview of computability theory. I will discuss the basic concepts and ideas of the field, the types of things we study, and give some fun examples of "computable" and "noncomputable" objects. This talk should be accessible to all math graduate students.

**Note: The Graduate Student Seminar is**

__mandatory__for ALL graduate students!**Fall 2015**

**Title**: GTA Meeting

**Speaker**: Hakim Walker, George Washington University**Date and Time:** Friday, November 20^{th}, 1:00-2:00pm

**Place:** Monroe 267

**Note: The Graduate Student Seminar is mandatory for ALL graduate **TAs**! Attendance will be taken.**

**Title:** Stability analysis in a trend depending price formation model

**Speaker:**Chubo Deng

**Date and Time:**Friday, October 23rd, 1:00-2:00pm

**Place:**Government 101

**Abstract:**The model was first introduced by J-M Lasry and P.L. Lions., which describes the evolution of prices in a market. We discuss the eigenvalues and eigenfunctions for a one dimensional parabolic evolution equation with Neumann boundary condition.

**Note: The Graduate Student Seminar is**

__mandatory__for ALL graduate students!**Title:** Interior and Boundary Spikes for Two-dimensional GM system

**Speaker:**Yeyao Hu

**Date and Time:**Friday, October 9th, 1:00-2:00pm

**Place:**Government 101

**Abstract:**We prove the existence of an assembly of interior and boundary spikes as a solution of two-dimensional Gierer-Meinhardt system. Moreover, we also discover that the locations of the spikes are determined by the curvature of the domain boundary together with the Green’s function of the domain. A reflection operator of −∆ is introduced which also plays a very important in Ren and Shoup’s recent and upcoming papers.

**Note: The Graduate Student Seminar is mandatory for ALL graduate students! **

**Title:** Introduction to Mathematica for Undergraduate and Graduate Students

**Speaker:** Robbie Robinson, George Washington University**Date and Time:** Friday, September 25th, 1:00-2:00pm

**Place:** Government 101

**Note: The Graduate Student Seminar is mandatory for ALL graduate students! **

**Attendance will be taken.**

**Title:** A generalization of alpha-orientations to higher genus surfaces**Speaker:** Jason Suagee, George Washington University**Date and Time:** Friday, September 18th, 1:00-2:00pm

**Place:** Government 101

**Abstract:** Given a graph G=(V,E), and a given function alpha:V --> N, an alpha-orientation is an orientation of the edges such that the out-degree of each vertex v corresponds with alpha(v). S. Felsner (TU-Berlin) in 2006 proved that the set of alpha-orientation on an embedded planar graph (a planar map) carries the structure of a distributive lattice, with unique maximal and minimal elements. He uses this result, for example, to construct canonical spanning trees on rooted planar maps as well as several other canonical structures on planar maps.

We obtain a generalization of Felsner's result to higher genus orientable surfaces with possible application to bijective methods in map enumeration and construction. Additionally, by applying this result to pairs of Cayley maps (strongly symmetric embeddings of Cayley graphs) we obtain potential applications to the study of finite group extensions.

**Note: The Graduate Student Seminar is mandatory for ALL graduate students! **

**Attendance will be taken.**

**Title:** Computability-Theoretic Properties of (2,1):1 Structures**Speaker:** Hakim Walker**Date and Time:** Friday, September 11th, 1:00-2:00pm

**Abstract:**A (2,1):1 structure consists of a countable set A (usually the natural numbers) and a function f which maps, to each element of A, either exactly one element of A or exactly two elements of A. Similar structures have been studied recently by Harizanov, Cenzer, Remmel, and Marshall, particularly the complexity of isomorphisms between such structures. We will begin this talk with a brief overview of computability theory, then discuss some preliminary results, and conclude with an application to the Collatz conjecture.

**Note: The Graduate Student Seminar is**

__mandatory__for ALL graduate students!