Graduate Student Seminars

Spring 2018


Title: Scattering in generalized Hartree equation
Speaker: Anudeep Kumar
Date and Time: Monday, April 30, 2:30-3:30pm
Place: Rome 204
AbstractWe consider the focusing generalized Hartree equation in the mass-supercritical and energy-subcritical setting. The characterization of solutions behavior under the so called mass-energy threshold is known for the NLS case from the works of Holmer and Roudenko (radial) and Duyckaerts, Holmer and Roudenko (nonradial) and generalizations (Guevara and others). The scattering behavior is typically proved following the road map developed by Kenig and Merle in 2006, using the concentration compactness and rigidity properties, which is a standard tool by now in the dispersive problems.
In this work we give an alternative proof of scattering in the Hartree case, following the approach of Dodson and Murphy for the focusing 3d cubic NLS equation, which relies on the scattering criterion of Tao, combined with the radial Sobolev and Morawetz-type estimates.

Title: Mathematical Models for Genome Rearrangements and Whole Genome Duplications
Speaker: Pavel, Avdeev
Date and Time: Monday, April 23, 2:30-3:30pm
Place: Rome 204

Abstract: One of the key computational problems in comparative genomics is the reconstruction of genomes of ancestral species based on genomes of extant species. Since most dramatic changes in genomic architectures are caused by genome rearrangements, this problem is often posed as minimization of the number of genome rearrangements between extant and ancestral genomes.


The basic case of three given genomes is known as the genome median problem. Whole genome duplications (WGDs) represent yet another type of dramatic evolutionary events and inspire the reconstruction of pre-duplicated ancestral genomes, referred to as the genome halving problem.  
We start with a description of genome evolution and most dramatic evolutionary events such genome rearrangements and whole genome duplication. Then. we move to the most common mathematical model of genome rearrangements, called Double-Cut-and-Join (DCJ). We further describe generalizations and applications of the DCJ model. We also consider the
problem of ancestral genome reconstruction and its particular instances such as genome median and halving problems. Finally, we discuss topological and integer linear programming approaches to these problems. 

Title: Analysis and Modeling of Self-organized Systems with Long Range Interaction
Speaker: Chong Wang
Date and Time: Monday, April 16, 2:30-3:30pm
Place: Rome 204
Abstract: Energy-driven pattern formation induced by competing short and long range interaction is common in many biological and physical systems. We report on our work through two models. The sharp interface model is a  nonlocal and non-convex geometric variational problem. The admissible class of the energy functional is a collection of sets where each set is of finite perimeter. The original problem is recast as a variational problem on a Hilbert space through introducing internal variables. We prove the existence of the core-shell assemblies and the existence of the disc assemblies as the stationary points of the energy functional in ternary systems. We also prove the existence of a triple bubble in a quaternary system. The other model is the diffuse interface model concerning minimizers of the Ginzburg-Landau free energy supplemented with long range interaction in inhibitory systems. As model parameters vary, a large number of morphological phases appear as stable stationary states. One open question related to the polarity direction of double-bubble assemblies is answered numerically. More importantly, it is shown that the average size of bubbles in a single-bubble assembly does not depend on the ratio of volume fractions but rather is determined by the long range interaction coefficients and the sum of the minority constituent volumes. In double-bubble assemblies, a two-thirds power law between the number of double bubbles and the long range interaction coefficients in the strong segregation regime is justified both numerically and theoretically.  A range of parameters is identified that yields double-bubble assemblies. These two models can be connected through gamma convergence.

Title: Computability-Theoretic Properties of Orders and Complexity of Identifying Algebraic Properties on Computable Magmas
Speaker: Trang Ha
Date and Time: Monday, April 9, 2:30-3:30pm
Place: Rome 204

Abstract : A magma is computable if both its domain and its atomic diagram are computable. We investigate the Turing complexity of orders on computable orderable magmas by studying their algebraic and topological properties. We further discuss the spaces of orders on special self-distributive (and not necessarily associative) magmas that come from knot theory named quandles. We also consider the complexity of the index set of magmas that satisfy certain algebraic properties within the class of computable magmas. 

Title : From the Tutte Polynomial to the G-invariant

Speaker: Kevin Long
Date and Time: Monday, April 2, 2:30-3:30pm
Place: Rome 204
Abstract : Matroids are a combinatorial structure introduced by Hassler Whitney in 1935 as an abstraction of "independence" in linear algebra and graph theory. Since then, matroid theory has expanded to a field in its own right, with connections to geometry, coding theory, and other fields. In this talk, we first give the equivalent definitions of matroids and show how they relate to some familiar families of matroids. We then introduce the Tutte polynomial, a powerful matroid invariant, and the most classical one, which will allow us to delve into more modern work done by Dr. Joseph Bonin on the G-invariant, a strengthening of the Tutte polynomial. 

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

Title - Multiplying fractions on a T-shirt
Speaker: Rhea Palak Bakshi, GWU
Date and Time: Monday, March 5, 2:30-3:30pm
Place: Rome 204
Abstract - In 1987 Józef Przytycki introduced Skein Modules as a way to extend the knot polynomials of the 1980's to knots and links in arbitrary 3-manifolds. Since their introduction Skein Modules have become central to the theory of 3-manifolds. In 1997, Charles Frohman and Razvan Gelca gave a nice product-to-sum formula for the Kauffman Bracket Skein Algebra of the torus times the interval. We try to discover a similar formula for the multiplication of curves on a thickened T-shirt. 

Title: Meet the Department Chair
Speaker: Frank Baginski 
Date and Time:  Monday, February 26, 2018, 2:30 pm
Place: Rome 204