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Series: Dissertation Defense

Given a closed surface S_g of genus g, a mapping class f is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number $\lambda$. The number $\lambda$ is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur. Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on S_g with orientable foliations whose stretch factor $\lambda$ has algebraic degree 2g. Moreover, the stretch factor $\lambda$ is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d is less than or equal to g. Our examples also give a new approach to a conjecture of Penner.

Series: Dissertation Defense

The talk consists of two parts.The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces. We establish a series of dichotomy-type results for the discrepancy function which state that if the $L^1$ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.

Series: Dissertation Defense

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this setup to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as "combinatorial shadows" of curves. This tropical relationship between graphs and algebraic curves has led to beautiful applications of chip-firing to both algebraic geometry and number theory.
In this thesis we continue the combinatorial development of divisor theory for graphs.

Series: Dissertation Defense

We say that a cover of surfaces S-> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.

Series: Dissertation Defense

Series: Dissertation Defense

Advisor: Dr. Matthew Baker

We study various binomial and monomial ideals related to the theory of
divisors, orientations, and matroids on graphs. We use ideas from potential
theory on graphs and from the theory of Delaunay decompositions for lattices
to describe minimal polyhedral cellular free resolutions for these ideals.
We will show that the resolutions of all these ideals are closely related
and that their Betti tables coincide. As corollaries we give conceptual
proofs of conjectures and questions posed by Postnikov and Shapiro, by
Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various
other results related in the theory of chip-firing games on graphs --
including Merino's proof of Biggs' conjecture and Baker-Shokrieh's
characterization of reduced divisors in terms of potential theory -- also
follow immediately from our general techniques and results.

Series: Dissertation Defense

Series: Dissertation Defense

Series: Dissertation Defense

Series: Dissertation Defense