Georgia Institute of Technology College of Sciences

This thesis explores Turán-type extremal problems in graphs that exclude certain minors, focusing on the maximum number of $k$-cliques such graphs can contain. The first part of the thesis studies planar graphs, which forbid $K_5$ and $K_{3,3}$ as minors. We determine the maximum number of edges is in a planar graph that contains no cycle of length k, and establish a general upper bound for the number of edges in a planar graph avoiding $C_k$ for any $k\ge 11$.

The second part addresses the maximum number of $k$-cliques in $K_t$-minor-free graphs. We show essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor-free graph on $n$ vertices. More precisely, we determine a function $C(k, t)$ such that for each $k < t$ with $t - k \gg \log_2 t$, every $K_t$-minor-free graph on $n$ vertices has at most $n \cdot C(k, t)^{1 + o_t(1)}$ cliques of order $k$. We also show that this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This result answers a question of Wood and Fox–Wei asymptotically up to an $o_t(1)$ factor in the exponent, except in the extreme case where $k$ is very close to $t$.

After quickly recalling the established theory on the combinatorics of orthogonal matroids, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem concerning the connectivity of circuits. 

Next, we consider representations of orthogonal matroids over fields (and more generally, over tracts) by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. We also give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim. 

The final part of the thesis focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target field. We also give explicit generators and relations of the foundation and an algorithm for computations. 

Zoom link: https://gatech.zoom.us/my/tongjinmath?pwd=QzRDalp2ditGL2tVNUozWm1RK1UwUT09

A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory has always generated significant interest. The problem gets a lot more complicated when we consider links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of uniformly thick knot type.  If time permits, I'll also mention the classification of links in the overtwisted setting. Part of this is joint work with John Etnyre, Hyunki Min, and Tom Rodewald. 

In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.

The Zoom link to the meeting: https://gatech.zoom.us/j/99340322307