À¶Ý®ÊÓƵ

Nicole Kitt, University of À¶Ý®ÊÓƵ

Characterizing Cofree Representations of SL_n x SL_m

The study, and in particular classification, of cofree representations has been an interest of research for over 70 years. The Chevalley-Shepard Todd Theorem provides a beautiful intrinsic characterization for cofree representations of finite groups. Specifically, this theorem says that a representation V of a finite group G is cofree if and only if G is generated by pseudoreflections. Up until the late 1900s, with the exception of finite groups, all of the existing classifications of cofree representations of a particular group consist of an explicit list, as opposed to an intrinsic group-theoretic characterization. However, in 2019, Edidin, Satriano, and Whitehead formulated a conjecture which intrinsically characterizes stable irreducible cofree representations of connected reductive groups and verified their conjecture for simple Lie groups. The conjecture states that for a stable irreducible representation V of a connected reductive group G, V is cofree if and only if V is pure. In comparison to the classifications comprised of a list of cofree representations, this conjecture can be viewed as an analogue of the Chevalley–Shepard–Todd Theorem for actions of connected reductive groups. The aim of this thesis is to further expand upon the techniques formulated by Edidin, Satriano, and Whitehead as a means to work towards the verification of the conjecture for all connected semisimple Lie groups. The main result of this thesis is the verification of the conjecture for stable irreducible representations V\otimes W of SL_n x SL_m satisfying dim V>=n^2 and dim W>=m^2.

Cynthia Dai, University of À¶Ý®ÊÓƵ

Resolution of Singularities via Stacky Blow-ups

We follow Dan Abramovich and Ming Hao Quek’s paper on resolution of singularity by multi-weighted blowups. This line of work is first motivated to give a more natural and motivated proof of Hironaka’s result, and that leads to the notion of weighted blowup, where you repeatedly blowing up the worst singular locus via weighted blowups. The problem with this is at the end you do not get an ambient space that’s smooth DM, but log smooth(specifically it’s toroidal DM stack). Using the technique of multi-weighted blowup introduced by Satriano, we can improve this result to get a logarithmic resolution of singularity with a smooth ambient space.

MC 5403

Jack Jia, University of À¶Ý®ÊÓƵ

Noncommutative Grothendieck Duality

Grothendieck duality states, up to adding some correct adjectives, that the derived functor of a sufficiently nice morphism of schemes will admit a right adjoint. We will translate this into the language of dualizing complexes and give a noncommutative analog.

MC 5403

Isabella Wang, University of À¶Ý®ÊÓƵ

The Partite Construction

Using the Hales-Jewett theorem, we use a technique of Nesetril and Rodl to show that the class of finite ordered graphs has the Ramsey property.

MC 5417

Rachael Alvir, University of À¶Ý®ÊÓƵ

Effective Algebra 3

We will begin learning about Higman's Theorem.

MC 5417

Spiro Karigiannis, University of À¶Ý®ÊÓƵ

Unique continuation in geometry

I will introduce the notion of unique continuation in geometry, closely following a survey article by Jerry Kazdan (CPAM 1988). Not all elliptic PDE exhibit the phenomenon of unique continuation, but most important elliptic PDE arising in geometry do, such as the Laplace equation, the Cauchy-Riemann equation, and the harmonic map equation.

MC 5403

Sourabhashis Das, University of À¶Ý®ÊÓƵ

On the distributions of prime divisor counting function

In 1917, Hardy and Ramanujan established that $\omega(n)$, the number of distinct prime factors of a natural number $n$, and $\Omega(n)$, the total number of prime factors of $n$ have normal order $\log \log n$. In 1940, Erdős and Kac refined this understanding by proving that $\omega(n)$ follows a Gaussian distribution over the natural numbers.

In this talk, we extend these classical results to the subsets of $h$-free and $h$-full numbers. We show that $\omega_1(n)$, the number of distinct prime factors of $n$ with multiplicity exactly $1$, has normal order $\log \log n$ over $h$-free numbers. Similarly, $\omega_h(n)$, the number of distinct prime factors with multiplicity exactly $h$, has normal order $\log \log n$ over $h$-full numbers. However, for $1 < k < h$, we prove that $\omega_k(n)$ does not have a normal order over $h$-free numbers, and for $k > h$, $\omega_k(n)$ does not have a normal order over $h$-full numbers.

Furthermore, we establish that $\omega_1(n)$ satisfies the Erdős-Kac theorem over $h$-free numbers, while $\omega_h(n)$ does so over $h$-full numbers. These results provide a deeper insight into the distribution of prime factors within structured subsets of natural numbers, revealing intriguing asymptotic behavior in these settings.

MC 5417

Micah Milinovich, University of Mississippi

Hilbert spaces and low-lying zeros of L-functions

Given a family of L-functions, there has been a great deal of interest in estimating the proportion of the family that does not vanish at special points on the critical line. Conjecturally, there is a symmetry type associated to each family which governs the distribution of low-lying zeros (zeros near the real axis). Generalizing a problem of Iwaniec, Luo, and Sarnak (2000), we address the problem of estimating the proportion of non-vanishing in a family of L-functions at a low-lying height on the critical line (measured by the analytic conductor). We solve the Fourier optimization problems that arise using the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type), and we can explicitly construct the associated reproducing kernels. If time allows, we will also address the problem of estimating the height of the "lowest" low-lying zero in a family for all symmetry types. These results are based on joint work with Emanuel Carneiro and Andrés Chirre.

MC 5417

Eason Li, University of À¶Ý®ÊÓƵ

The Hales-Jewett Theorem

We discuss the Hales-Jewett theorem, time permitting giving a full proof.

MC 5417

Joey Lakerdas-Gayle, University of À¶Ý®ÊÓƵ

Effective Algebra 2

We will continue learning about recursively presented groups.

MC 5417