## University of Missouri, Columbia, Number Theory and Representation Theory Seminar

### Fall 2015: Tuesday 2:00-2:50, Room MSB 312

Click here to see the schedule of previous semesters: Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Fall 2011.

## Schedule

Date

Speaker

Topic

Sept. 15

Aaron Wood
(University of Missouri, Columbia)

Affine reflections and structure of p-adic Chevalley group
I will introduce affine reflections and use them to discuss a Bruhat-type decomposition of p-adic Chevalley groups.

Sept. 22

Aaron Wood
(University of Missouri, Columbia)

Affine reflections and structure of p-adic Chevalley group
I will the talk from last week.

Sept. 29

Aaron Wood
(University of Missouri, Columbia)

Affine reflections and structure of p-adic Chevalley group
I will the talk from last week.

Oct. 6

Shuichiro Takeda
(University of Missouri, Columbia)

Representations of the Lie algebra $$\mathfrak{sl}_2(\mathbb{C})$$
I will give a classification of finite dimensional representations of the Lie algebra $$\mathfrak{sl}_2(\mathbb{C})$$

Oct. 13

Shuichiro Takeda
(University of Missouri, Columbia)

Representations of the Lie algebra $$\mathfrak{sl}_3(\mathbb{C})$$
I will give a classification of finite dimensional representations of the Lie algebra $$\mathfrak{sl}_3(\mathbb{C})$$

Oct. 20

Andreas Weingartner
(Southern Utah University)

The distribution of divisors and related problems
We will consider questions related to the distribution of divisors of integers, with particular focus on an asymptotic estimate for the number of integers below x whose maximum ratio of consecutive divisors is at most t. Several related results, concerning the divisor graph, practical numbers, the degree distribution of polynomial divisors and the cycle structure of permutations, will also be discussed.

Oct. 27

Bill Banks
(University of Missouri, Columbia)

Ramanujan-type identities
In this talk, I will proof one or two Ramanujan-type identities using Eisenstein series and/or Dirichlet series together with special values of the Riemann zeta function.

Nov. 3

Bill Banks
(University of Missouri, Columbia)

Ramanujan-type identities
I will continue the talk from last week

Dec. 1

Melissa Emory
(University of Missouri, Columbia)

The Diophantine Equation $$x^4+y^4=D^2z^4$$ in Quadratic Fields
A.Aigner proved that except in $$\mathbb{Q}(\sqrt{-7})$$, there are no nontrivial quadratic solutions to $$x^4+y^4=z^4$$. The result was later re-proven by D.K. Faddeev and the argument simplified by L.J. Mordell. This talk discusses work to extend this result that shows that nontrivial quadratic solutions exist to $$x^4+y^4=D^2z^4$$ precisely when either $$D=1$$ or $$D$$ is a congruent number.

Dec. 8

Igor Shparlinski
( University of New South Wales)

Divisibility of Fermat Quotients
We show that for a prime $$p$$ the smallest $$a$$ with $$a^{p-1} \not \equiv 1 \pmod {p^2}$$ does not exceed $$(\log p)^{463/252 + o(1)}$$ which improves the previous bound $$O((\log p)^2)$$ obtained by H. W. Lenstra in 1979. We also show that for almost all primes $$p$$ the bound can be improved as $$(\log p)^{5/3 + o(1)}$$. These results are based on a combination of various techniques including the distribution of smooth numbers, distribution of elements of multiplicative subgroups of residue rings, bound of Heilbronn exponential sums and a large sieve inequality with square moduli.