Department of

Mathematics


Seminar Calendar
for Graduate Number Theory Seminar events the year of Friday, April 21, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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                        30                                         

Thursday, February 23, 2017

2:00 pm in 241 Altgeld Hall,Thursday, February 23, 2017

Some maximal curves obtained via a ray class field construction

Dane Skabelund   [email] (UIUC)

Abstract: This talk will be about curves over finite fields which are "maximal" in the sense that they meet the Hasse-Weil bound. I will describe some problems relating to such curves, and give a description of some new "maximal" curves which may be obtained as covers of the Suzuki and Ree curves.

Tuesday, February 28, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, February 28, 2017

Weighted Partition Identities

Hannah Burson (UIUC)

Abstract: Ali Uncu and Alexander Berkovich recently completed some work proving several new weighted partition identities. We will discuss some of their theorems, which focus on the smallest part of partitions. Additionally, we will talk about some of the motivating work done by Krishna Alladi.

Thursday, March 9, 2017

2:00 pm in 241 Altgeld Hall,Thursday, March 9, 2017

Playing with partitions and $q$-series

Frank Garvan (University of Florida)

Abstract: We start with some open partition problems of Andrews related to Gauss's three triangular numbers theorem. We alter a generating function and find a new Hecke double sum identity. Along the way we need Bailey's Lemma and Zeilberger's algorithm. We finish with some even staircase partitions.

Thursday, April 6, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 6, 2017

Primes with restricted digits

Kyle Pratt   [email] (UIUC)

Abstract: Let $a_0 \in \{0,1,2,\ldots,9\}$ be fixed. James Maynard (2016) proved the impressive result that there are infinitely many primes without the digit $a_0$ in their decimal expansions. His theorem is a specific incarnation of a more general problem of finding primes in thin sequences. In this talk I will give a brief discussion about primes in thin sequences. I will also give an overview of some of the tools used in the course of Maynard's proof, including the Hardy-Littlewood circle method, Harman's sieve, and the geometry of numbers.

Tuesday, April 11, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2017

Andrew's recent papers on integer partitions and the existence of combinatorial proofs

Hsin-Po Wang (UIUC)

Abstract: We will start with introducing some combinatorial notions; and then attack George E. Andrew's recent papers[1][2][3] to see if we can come up with some (simpler) combinatorial proofs. Despite the papers, we will show that under certain conditions, we can always translate an algebraic proof into a combinatorial proof. [1] G. Andrews and G. Simay. The mth Largest and mth Smallest Parts of a Partition. http://www.personal.psu.edu/gea1/pdf/307.pdf [2] G. Andrews and M. Merca. The Truncated Pentagonal Number Theorem. http://www.personal.psu.edu/gea1/pdf/288.pdf [3] G. Andrews, M Bech and N. Robbins. Partitions with Fixed Differences Between Larger and Smaller Parts. http://www.personal.psu.edu/gea1/pdf/305.pdf

Thursday, April 13, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 13, 2017

L-values, Bessel moments and Mahler measures

Detchat Samart   [email] (UIUC)

Abstract: We will discuss some formulas and conjectures relating special values of L-functions associated to modular forms to moments of Bessel functions and Mahler measures. Bessel moments arise in the study of Feynman integrals, while Mahler measures have received a lot of attention from mathematicians over the past few decades due to their apparent connection with number theory, algebraic geometry, and algebraic K-theory. Though easy to verify numerically with high precision, most of these formulas turn out to be ridiculously hard to prove, and no machinery working in full generality is currently known. Some available techniques which have been used to tackle these problems will be demonstrated. Time permitting, we will present a meta conjecture of Konstevich and Zagier which gives a general framework of how one could verify these formulas using only elementary calculus.

Thursday, April 20, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 20, 2017

Ranks of elliptic curves, Selmer groups, and Tate-Shafarevich groups

Robert Lemke Oliver (Tuft University)

Abstract: A big problem in number theory is how to access the rank of an elliptic curve, i.e. the minimal number of points needed to generate the full set of rational points. Assuming the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjectures, an algorithm exists that will determine the rank of any specific elliptic curve, but this says nothing about what ranks are typically like. While an analytic mindset is useful for thinking about how ranks "should" behave, almost all actual theorems, from Mordell-Weil to the recent work of Bhargava and Shankar, passes through an algebraic gadget called the Selmer group. This is given by a somewhat complicated definition in terms of Galois cohomology, which is intimidating and unilluminating for people who are more comfortable with classical analytic number theory and L-functions. This talk will aim to make Selmer groups somewhat less mystifying, and along the way we will discuss some of the speaker's forthcoming work with Bhargava, Klagsbrun, and Shnidman.

Thursday, April 27, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 27, 2017

MacMahon's partial fractions

Andrew Sills (Georgia Southern University)

Abstract: A. Cayley used ordinary partial fractions decompositions of $1/[(1-x)(1-x^2)\ldots(1-x^m)]$ to obtain direct formulas for the number of partitions of $n$ into at most $m$ parts for several small values of $m$. No pattern for general m can be discerned from these, and in particular the rational coefficients that appear in the partial fraction decomposition become quite cumbersome for even moderate sized $m.$ Later, MacMahon gave a decomposition of $1/[(1-x)(1-x^2). . .(1-x^m)]$ into what he called "partial fractions of a new and special kind" in which the coefficients are "easily calculable numbers" and the sum is indexed by the partitions of $m$. While MacMahon's derived his "new and special" partial fractions using "combinatory analysis," the aim of this talk is to give a fully combinatorial explanation of MacMahon's decomposition. In particular, we will observe a natural interplay between partitions of $n$ into at most $m$ parts and weak compositions of $n$ with $m$ parts.

Thursday, September 7, 2017

2:00 pm in 241 Altgeld Hall,Thursday, September 7, 2017

The Herbrand-Ribet Theorem

Patrick Allen   [email] (UIUC)

Abstract: Kummer's criterion states that a prime number $p \ge 7$ divides the class number of the $p$th cyclotomic field if and only if $p$ divides the numerator of one of the Bernoulli numbers $B_2, B_4, \ldots, B_{p-3}$, or equivalently, one of the values $\zeta(-1), \zeta(-3), \ldots, \zeta(4-p)$ of the Riemann zeta function. It is natural to ask if the individual values $B_k$ correspond to more refined information of the class groups. This is the content of the Herbrand-Ribet Theorem, one direction of which was proved by Herbrand in 1932, and the other by Ribet in 1976. Ribet's converse to Herbrand's Theorem uses the theory of modular forms and their associated Galois representations, and the ideas involved have been highly influential. We'll introduce this theorem, defining all the objects involved, and give some idea of the proof. I will aim to structure this talk so that any graduate student, be it their 1st year or their 45th year, will be able to take away something.