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Series: Research Horizons Seminar

Defined in the early 2000's by Ozsvath and Szabo,
Heegaard Floer homology is a package of invariants for three-manifolds,
as well as knots inside of them. In this talk, we will describe how work
from Poul Heegaard's 1898 PhD thesis,
namely the idea of a Heegaard splitting, relates to the definition of
this invariant. We will also provide examples of the kinds of questions
that Heegaard Floer homology can answer. These ideas will be the subject
of the topics course that I am teaching in
Fall 2017.

Series: Research Horizons Seminar

I
will continue the discussion on the group actions of the graph Jacobian
on the set of spanning trees. After reviewing the basic definitions, I
will explain how polyhedral geometry leads to a new family of such
actions.
These actions can be described combinatorially, but proving that they
are simply transitive uses geometry in an essential way. If time
permits, I will also explain the following surprising connection: the
canonical group action for a plane graph (via rotor-routing
or Bernardi process) is related to the canonical tropical geometric
structure of its dual graph. This is joint work with Spencer Backman and
Matt Baker.

Series: Research Horizons Seminar

Every graph G has canonically associated to a finite abelian group called the Jacobian group. The cardinality of this group is the number of spanning trees in G. If G is planar, the Jacobian group admits a natural simply transitive action on the set of spanning trees. More generally, for any graph G one can define a whole family of (non-canonical) simply transitive group actions. The analysis of such group actions involves ideas from tropical geometry. Part of this talk is based on joint work with Yao Wang, and part is based on joint work with Spencer Backman and Chi Ho Yuen.

Series: Research Horizons Seminar

In this talk, we will have an overview of: the Gaming Industry, specifically on the Video Slot Machine segment; the top manufactures in the world; the game design studio Gimmie Games, who we are, what we do; what is the process of making a video slot game; what is the basic structure of the math model of a slot game; current strong math models in the market; what is the roll of a game designer in the game development process; the skill set needed to be a successful Game Designer. Only basic probability knowledge is required for this talk.

Series: Research Horizons Seminar

On the two-dimensional square grid, remove each
nearest-neighbor edge independently with probability 1/2 and consider
the graph induced by the remaining edges. What is the structure of its
connected components? It is a famous theorem of Kesten that 1/2 is the
``critical value.'' In other words, if we remove edges with probability
p \in [0,1], then for p < 1/2, there is an infinite component remaining,
and for p > 1/2, there is no infinite component remaining. We will
describe some of the differences in these phases in terms of crossings
of large boxes: for p < 1/2, there are relatively straight crossings of
large boxes, for p = 1/2, there are crossings, but they are very
circuitous, and for p > 1/2, there are no crossings.

Series: Research Horizons Seminar

I will start with a brief presentation of the Probability activities in SOM. I will continue by presenting results obtained in SOM, over the past ten years, answering long standing questions insequences comparison.

Series: Research Horizons Seminar

A matroid is a combinatorial abstraction of an independence structure, such as linear independence among vectors and cycle-free-ness among edges of a graph. An algebraic variety is a solution set of a system of polynomial equations, and it has a polyhedral shadow called a tropical variety. An irreducible algebraic variety gives rise to a matroid via algebraic independence in its coordinate ring. In this talk I will show that the tropical variety is compatible with the algebraic matroid structure. I will also discuss some open problems on algebraic matroids and how they behave under operations on tropical varieties.

Series: Research Horizons Seminar

Quantum topology is a collection of ideas and techniques for studying
knots and manifolds using ideas coming from quantum mechanics and
quantum field theory. We present a gentle introduction to this topic via
Kauffman bracket skein algebras of surfaces,
an algebraic object that relates "quantum information" about knots
embedded in the surface to the representation theory of the fundamental
group of the surface. In general, skein algebras are difficult to
compute. We associate to every triangulation of the
surface a simple algebra called a "quantum torus" into which the skein
algebra embeds. In joint work with Thang Le, we make use of this
embedding to give a simple proof of a difficult theorem.

Series: Research Horizons Seminar

If Google Scholar gives you everything you want, what could Georgia Tech Library possibly do for you? Come learn how to better leverage the tools you know and discover some resources you may not. Get to know your tireless Math Librarian and figure out how to navigate the changes coming with Library Next. This is also an opportunity to have a voice in the Library’s future, so bring ideas for discussion. Refreshments will be served.

Series: Research Horizons Seminar

A matrix completion problem starts with a partially specified matrix, where some entries are known and some are not. The goal is to find the unknown entries (“complete the matrix”) in such a way that the full matrix satisfies certain properties. We will mostly be interested in completing a partially specified symmetric matrix to a full positive semidefinite matrix. I will give some motivating examples and then explain connections to nonnegative polynomials and sums of squares.