Graduate Number Theory Seminar : Past Events
Past Seminars
The following seminars have already happened, you may instead view upcoming seminars in this series.
Sept. 17, 2008
Drew Shulman of
UIC :
1 p.m. in SEO 636
Sept. 24, 2008
Robert Krzyzanowski of
UIC :
1 p.m. in SEO 636
Oct. 1, 2008
Robert Krzyzanowski of
UIC :
1 p.m. in SEO 636
Oct. 8, 2008
Drew Shulman of
UIC :
1 p.m. in SEO 636
Oct. 15, 2008
David Benjamin Antieau of
UIC :
1 p.m. in SEO 636
Oct. 22, 2008
David Benjamin Antieau of
UIC :
1 p.m. in SEO 636
Oct. 29, 2008
Evangelos Kobotis of
UIC :
1 p.m. in SEO 636
Nov. 5, 2008
Evangelos Kobotis of
UIC :
1 p.m. in SEO 636
Nov. 12, 2008
Evangelos Kobotis of
UIC :
1 p.m. in SEO 636
Nov. 26, 2008
Ramin Takloo-Bighash of
UIC :
1 p.m. in SEO 636
Dec. 3, 2008
Christine Robinson of
UIC :
1 p.m. in SEO 636
Jan. 26, 2009
Drew Shulman of
UIC :
3 p.m. in SEO 427
Abstract
First chapter of Diamond and Shurman
Feb. 2, 2009
Drew Shulman of
UIC :
3 p.m. in SEO 427
Abstract
The basic definitions of modular forms will continue to be introduced.
We shall see that a complex torus (a Riemann surface) is related
to modular forms through the fact that the equivalence classes of points
in H (the upper half plane) under the action of the modular group can
be described through the isomorphism classes of complex elliptic curves.
Feb. 9, 2009
Robert Krzyzanowski of
UIC :
3 p.m. in SEO 427
Abstract
To each congruence subgroup $\Gamma$ of $SL_2(\mathbb{Z})$, we can associate a modular curve
to be the quotient space $\Gamma/\mathbb{H}$ (where $\mathbb{H}$ is the complex upper half plane).
We will see the set of orbits generated by the action of $\Gamma$ can be made into a Riemann surface,
which can then be compactified.
Feb. 16, 2009
Kathy Dexter of
UIC :
3 p.m. in SEO 427
March 2, 2009
Kathy Dexter of
UIC :
3 p.m. in SEO 427
Abstract
Sections 3.2 and 3.3 of Diamond and Shurman's "A First Course in Modular Forms."
Feb. 10, 2010
Holly Krieger of
UIC :
3 p.m. in SEO 427
Feb. 17, 2010
Robert Krzyzanowski of
UIC :
3 p.m. in SEO 427
Abstract
In 1772, Euler noticed the polynomial $x^2 + x + 41$ produces forty consecutive primes for $0 <= x <= 39$. We will show that this fact is actually equivalent to $Q(\sqrt{-d})$
with $d = 163 = 1 - 4*41$ having class number 1. Further, $x^2 + x + m$ is such a prime-generating polynomial if and only if $Q(\sqrt{1-4m})$ has class number 1. This question
can be generalized to real quadratic fields by letting $m$ be negative. There are then only 14 such polynomials, called Rabinowitsch polynomials.
Feb. 24, 2010
Kathy Dexter of
UIC :
3 p.m. in SEO 427
Abstract
Following a brief introduction to representation theory, I will present Gel'fand and Graev's proof that an irreducible representation of GL(n) over a finite field is contained in a certain representation with multiplicity not greater than one. I will then discuss the generalizations of this result to local and global fields.
March 3, 2010
Christine Robinson of
UIC :
3 p.m. in SEO 427
Abstract
After reviewing the notion of classical modular forms, I will introduce their degree-n generalization, Siegel modular forms. I will prove the Koecher Principle, which eliminates the "holomorphic at infinity" condition necessary in the classical setting.
March 10, 2010
Andrew Shulman of
UIC :
3 p.m. in SEO 427
March 31, 2010
James Freitag of
UIC :
3 p.m. in SEO 427
Abstract
We will discuss applications of model theory to the study of rational points on curves. No back round in model theory will be assumed.
April 7, 2010
Matt Wechter of
UIC :
3 p.m. in SEO 427
April 14, 2010
John Goes of
UIC :
3 p.m. in SEO 427
April 21, 2010
Raymond Fuller of
UIC :
3 p.m. in SEO 427
April 28, 2010
Raymond Fuller of
UIC :
3 p.m. in SEO 427
Sept. 1, 2010
Robert Krzyzanowski of
UIC :
3 p.m. in SEO 427
Robert Krzyzanowski of
UIC :
3 p.m. in SEO 427
Abstract
In this talk we aim to understand a convenient theorem on affine algebraic groups.
Namely, given any such group, we can embed it into GL(n) for some (n).
In the same way that Cayley's theorem in group theory reduces the study of abstract groups to
those of permutation groups, this theorem transforms the study of affine algebraic groups to linear groups
Although the arbitrariness of the chosen representation makes it sometimes difficult to exploit,
we can use this approach as a technical aid in certain proofs (e.g. exploit the special behaviors of eigenvalues).
Sept. 15, 2010
Jim Freitag of
UIC :
3 p.m. in SEO 427
Abstract
Hrushovski proved that the theory of the q-Frobenius (for q a power of p) acting on an algebraically closed field of characteristic p for almost every prime is ACFA (a theory studied by model theorists). We talk about some general constructions from algebra (like ultraproducts). Then, we will investigate some simple geometry of structures obeying the axioms of ACFA. Specifically, we might give geometric proofs of things like: (1) every difference variety of transformal transcendence degree m is birational to a difference variety in
$
\mathbb{A}^{m+1}
$ and is isomorphic to a (almost) difference variety in
$
\mathbb{P}^{2m+1}
$
or (2) transformal transcendence degree is definable in families.
Oct. 13, 2010
Holly Krieger of
UIC :
3 p.m. in SEO 427
Abstract
I will define the necessary ingredients to understand Ratner's theorems, presented in sufficient generality to deduce the Oppenheim Conjecture. Some generalizations and the quantitative version of the Oppenheim conjecture will be explored. No background in ergodic theory or quadratic forms will be assumed, though some basic algebraic number theory and theory of algebraic groups will be helpful for the latter part of the talk.
Jan. 26, 2011
James Freitag of
UIC :
3 p.m. in SEO 427
Abstract
We will talk about varieties over separably closed fields of finite Ersov invariant. If time permits, we will talk about a functor from semiabelian varieties to their maximal divisible subgroups and what the failure of exactness of this functor says about descent problems.
Feb. 9, 2011
Raymond Fuller of
UIC :
3 p.m. in SEO 427
Abstract
In which we discuss the cubic character of 2.
Feb. 16, 2011
Robert Krzyzanowski of
UIC :
3 p.m. in SEO 427
Abstract
We aim to give the deformation theory developed by Sekiguchi and Suwa for passing from the multiplicative view in Kummer-Artin-Schreier theory to the additive perspective due to Witt.
Feb. 23, 2011
Xudong Zheng of
UIC :
3 p.m. in SEO 427
Abstract
The inequality of Castelnuovo-Severi concerns the self-intersection of a divisor of a surface, which can be proved by the Riemann-Roch theorem for surfaces. In this talk, we recover the classical proof of a part of Weil's conjecture for curves using this inequality.
March 9, 2011
Christine Robinson of
UIC :
3 p.m. in SEO 427
Abstract
Duke and Imamoglu's proof of the Saito-Kurokawa lift from classical modular forms to degree 2 Siegel modular forms utilizes Imai's converse theorem. I will discuss this generalized Hecke correspondence and sketch the proof of the Saito-Kurokawa lift.
March 16, 2011
Matt Wechter of
UIC :
3 p.m. in SEO 427
Abstract
We start with some basic theory on derivations of rings and extensions of derivations, then will see how derivations over fields and the more classical Galois theory can be seen as examples of the Jacobson-Bourbaki Theorem. We end by connecting this theorem into Grothendieck's ideas for faithfully flat descent. This is a make-up talk from the snow day.
April 6, 2011
Drew Shulman :
3 p.m. in SEO 427
Abstract
In this talk we will introduce Drinfeld modules and their analogy with elliptic curves. We will also state a result of W. Duke for elliptic curves $E/\mathbb{Q}$
about the exponent of $E$, and then state (and prove?) the analogous result for Drinfeld modules.
April 20, 2011
Holly Krieger of
UIC :
3 p.m. in SEO 427
Abstract
I will discuss a result of Silverman and Ingram on the finitude of the Zsigmondy set associated to the forward orbit of a wandering point of a rational map of degree at least 2, defined over a number field, which has 0 periodic. The Zsigmondy set of a sequence (a_n) is the set of indices so that a_n does not have a primitive prime divisor. I will prove the easy generalization to the case of 0 preperiodic, and discuss my progress on further generalizations.
May 4, 2011
Kathy Dexter of
UIC :
3 p.m. in SEO 427
Abstract
In this talk we will define the terms necessary to state Langland's
Functoriality Conjecture, emphasizing how the conjecture is a
generalization of a conjecture of Artin.
Sept. 25, 2012
Raymond Fuller of
UIC :
3 p.m. in SEO 512
Nov. 6, 2012
Holly Krieger of
UIC :
3 p.m. in SEO 512
Abstract
I will discuss joint work with A. Levin, Z. Scherr, and T. Tucker, proving some cases of a conjecture of A. Levin on the uniform boundedness of the number of S-units in images of rational maps defined over number fields, utilizing results of Evertse et al on linear unit equations. I will also discuss the conjecture as a consequence of the famous conjecture of Caporaso, Harris, and Mazur on the uniformity of the number of rational points on curves of fixed genus over a fixed number field.
Dec. 11, 2012
Janet Page of
UIC :
3 p.m. in SEO 512
Abstract
This talk will serve as an introduction to p-adic numbers and p-adic analysis. We will begin by highlighting the similarities between $\mathbb{Z}$ and $\mathbb{C}[X]$ and use this analogy to introduce the notion of p-adic numbers. We will introduce the p-adic absolute value in contrast to the standard absolute value, and our talk will culminate in a proof of Hensel's Lemma.
Dec. 12, 2012
Abel Castillo of
UIC :
10 a.m. in SEO 712
Abstract
This is the first lecture in a short course where we introduce modular forms for $SL_2(\mathbb{Z})$ and its congruence subgroups. We will follow Chapter 3 of Koblitz' "Introduction to Elliptic Curves and Modular Forms".
Robert Krzyzanowski of
University of Illinois at Chicago :
noon in SEO 712
Dec. 13, 2012
Abel Castillo of
UIC :
10 a.m. in SEO 712
Abstract
This is the second lecture in a short course where we introduce modular forms for $SL_2(\mathbb{Z})$ and its congruence subgroups. We will follow Chapter 3 of Koblitz' "Introduction to Elliptic Curves and Modular Forms".
Dec. 18, 2012
Abel Castillo of
UIC :
10 a.m. in SEO 512
Abstract
This is the third lecture in a short course where we introduce modular forms for $SL_2(\mathbb{Z})$ and its congruence subgroups. We will follow Chapter 3 of Koblitz' "Introduction to Elliptic Curves and Modular Forms".
Dec. 20, 2012
Abel Castillo of
UIC :
10 a.m. in SEO 512
Abstract
This is the fourth lecture in a short course where we introduce modular forms for $SL_2(\mathbb{Z})$ and its congruence subgroups. We will follow Chapter 3 of Koblitz' "Introduction to Elliptic Curves and Modular Forms".
Oct. 21, 2013
Abel Castillo of
UIC :
3 p.m. in SEO 612
Abstract
We will first recall some basic facts about torsion points of elliptic curves over $\mathbb Q$. Then, we will describe the function field setting, and define a Drinfeld module over $\mathbb F_q(T)$. We will then describe its torsion in the algebraic closure of $\mathbb F_q(T)$, and show how this gives rise to Galois representations similar to those coming from elliptic curves. As time permits, we will then state "open image" theorems in each setting, further reinforcing the analogy.
This talk is intended for an audience of graduate students interested in number theory.
Nov. 4, 2013
Dylon Chow of
UIC :
3 p.m. in SEO 427
Abstract
In this talk I will discuss Tate's 1950 Ph.D. thesis, which gives the theory of automorphic representations and L-functions of GL(1). Tate used harmonic analysis on adele groups to reprove Hecke's theorems on the functional equations of L-functions attached to Hecke characters. This approach provides greater conceptual simplification and clarity than can be found in the classical techniques used by Hecke. In fact, Tate showed that the results of Hecke are more or less an application of a general form of the Poisson summation formula. In this talk I will define the adeles and ideles, discuss the adelic Poisson summation formula, and outline an adelic proof of the meromorphic continuation and functional equation of the Riemann zeta function.
Nov. 11, 2013
Cara Mullen of
UIC :
3 p.m. in SEO 427
Abstract
Fix a prime $p>2$ and consider $f_c(z)=z^2+c$ with $c\in\mathbb{C}_p$. How is the critical portrait of $f_c$ (over $\mathbb{C}_p$) related to the critical portrait of the reduction map $\bar{f_c}$ (over the residue field $k=\mathcal{O}_{\mathbb{C}_p}/{\mathfrak{m}_p}=\overline{\mathbb{F}_p})$? We will show that because none of the intermediate iterates ``get close" to 0 (in the non-archimedean sense), the critical point still has exact period $n$. We will then examine an alternative proof that can be generalized to all rational functions with periodic points.
Nov. 18, 2013
Joseph Berner of
UIC :
3 p.m. in SEO 427
Dec. 2, 2013
Charles Alley of
UIC :
3 p.m. in SEO 427
Abstract
In this talk I will give results from my bachelor's thesis. Specifically, I will make explicit connections between linear fractional transformations and continued fractions. Then, using the language of linear algebra, I will prove some basic results which can then be applied to derive the simple continued fraction for $e = [2,1,2,1,1,4,1,1,6,1,\cdots]$.
March 6, 2014
Joseph Berner of
UIC :
3 p.m. in SEO 512
Abstract
We will give an example driven overview of the analogy between number fields and function fields of curves over finite fields. This will naturally lead us to discuss the analogy between rings of integers of number fields and affine curves over finite fields. If time permits we may discuss more complex invariants and how they compare.
March 13, 2014
Cara Mullen of
UIC :
3 p.m. in SEO 512
Abstract
Fix a prime $p\geq2$ and consider $f_c(z)=z^2+c$ with $c\in\mathbb{C}_p$. How is the critical portrait of $f_c$ (over $\mathbb{C}_p$) related to the critical portrait of the reduced map $\bar{f_c}$ (over the residue field $k=\overline{\mathbb{F}_p})$? Last time we considered the case when 0 was strictly periodic and found that the period cannot shrink. This time we will consider maps with a strictly pre-periodic critical point, which lead to some more interesting observations and results.
April 3, 2014
Darko Trifunovski of
UIC :
3 p.m. in SEO 512
Abstract
In this talk I will introduce the basic concepts related to Siegel modular forms as a multivariate generalization of the classical theory of modular forms, including some applications of Siegel modular forms in number theory and algebraic geometry.
April 10, 2014
Abel Castillo of
UIC :
3 p.m. in SEO 512
Abstract
We will give an introduction to Drinfeld modules, and we will discuss results regarding how certain invariants behave for the reduction of a Drinfeld module mod $p$ as the prime $p$ varies. More specifically, we will discuss results that state that the Euler-Poincaré characteristic and the trace of Frobenius "typically" have the same number of distinct prime divisors as a "typical" element of $\mathbb{F}_q[T]$ of the same degree for certain types of Drinfeld modules.
April 24, 2014
Dylon Chow of
UIC :
3 p.m. in SEO 512
Abstract
Fundamental to the proofs of several important results in arithmetic geometry (e.g. the Mordell-Weil Theorem and Faltings' Theorem) is a way of assigning a size to points on a variety. Weil's height machine turns geometric properties of the variety into arithmetic information about the points on the variety. If time permits, we will discuss the conjectures of Batyrev and Manin on the distribution of rational points.
Sept. 3, 2014
Cara Mullen of
UIC :
3 p.m. in SEO 712
Sept. 15, 2014
Abel Castillo of
UIC :
3 p.m. in SEO 712
Abstract
The Twin Prime Conjecture asks if there are infinitely many primes $p$ such that the next prime $p_{next}$ satisfies $p_{next} - p=2$. In this talk we will survey recent results towards this conjecture, where one finds infinitely many primes $p$ such that $p_{next} - p$ is much smaller than average. We will then give an overview of the sieve used by Goldson-Pintz-Yilidrim, which was the foundation of the work on bounded gaps between primes by Zhang and Maynard-Tao. Finally, we will mention variants of the Maynard-Tao method that give bounded gaps between primes with properties of arithmetic interest.
Sept. 22, 2014
Joseph Berner of
UIC :
3 p.m. in SEO 712
Sept. 29, 2014
Dylon Chow of
UIC :
3 p.m. in SEO 712
Abstract
We will define and discuss several aspects of modular forms of half integral weight, including the work of Serre, Stark, and Shimura. If time permits we will discuss the work of Waldspurger which puts Shimura's correspondence in the framework of representation theory.
Oct. 6, 2014
Cara Mullen of
UIC :
3 p.m. in SEO 712
Oct. 13, 2014
Darko Trifunovski of
UIC :
3 p.m. in SEO 712
Abstract
The generalized Rikuna polynomials are an iterative generalization of Rikuna's generic cyclic polynomials. We will discuss some results on the behaviour of these polynomials under specialization, giving some geometric characterizations of the reducible specializations and the specializations with Galois group smaller than the function field case.
Oct. 20, 2014
Kevin Vissuet of
UIC :
3 p.m. in SEO 712
Abstract
The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset $S + S = \{x+y : x,y\in S\}$ of a set of integers $S$. A finite set of integers $A$ is sum-dominated if $|A+A| > |A-A|$. Though it was believed that the percentage of subsets of $\{0,\dots,n\}$ that are sum-dominated tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominated if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately $4.5 \cdot 10^{-4}$). While most sets are difference-dominated in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominated tend to zero but the probability that $|A+A|=|A-A|$ tends to one, and hence a typical set is balanced in this case.
Oct. 27, 2014
Abel Castillo of
UIC :
3 p.m. in SEO 712
Abstract
In this talk we will state conjectures regarding the distribution of the trace of Frobenius for elliptic curves, including the the Lang-Trotter conjectures and the Koblitz conjecture, and point out how these are "higher-dimensional analogues" of familiar statements about primes in arithmetic progressions. We will then discuss heuristics that are used to make precise predictions about the constants appearing in the statements. As time permits, we will close by discussing analogues of these conjectures for Drinfeld modules in the global function field setting.
Nov. 3, 2014
Charles Alley of
UIC :
3 p.m. in SEO 712
Abstract
In this talk I will present results from Chapters 5 and 6 in Davenport's "Multiplicative Number Theory". In particular, I will outline the connection between Dirichlet characters and quadratic forms in order to state Dirichlet's Class Number Formula.
Jan. 21, 2015
Dylon Chow :
3 p.m. in SEO 512
Jan. 28, 2015
Abel Castillo :
3 p.m. in SEO 512
Abstract
A theorem of Viggo Brunn from the beginnings of modern sieve theory is
that there are infinitely many primes p such that p+2 has at most
nine (not necessarily distinct) prime factors. In this talk we'll
discuss this result, as well as later improvements; most notable among
these is the theorem of Chen which shows that there are infinitely
many primes p such that p+2 has at most two prime factors. We will
also discuss some of the ideas used to obtain these results, including
some basic ideas from sieve theory. As time permits, we'll mention
analogues of Chen's theorem in number fields.
Feb. 4, 2015
Zili Huang of
Northwestern University :
3 p.m. in SEO 512
Feb. 11, 2015
Jay Kopper :
3 p.m. in SEO 512
Feb. 18, 2015
Charles Alley :
3 p.m. in SEO 512
Abstract
C.L. Siegel said, "The mathematical universe is inhabited not
only by important species but also by interesting individuals." In this
talk I will discuss the Rogers-Ramanujan continued fraction, $r(t) =
q^{1/5}/1+q/1+q^2/1+q^3/1+... $ where $q=e^{2\pi i t}$. This classical
function has many interesting properties, which I will discuss. I will
present results from the survey paper by W. Duke, "Continued Fractions and
Modular Functions", which can be found here: http://www.math.ucla.edu/~wdduke/preprints/bams4.pdf
March 4, 2015
Eun Hye Lee :
3 p.m. in SEO 512
Abstract
I will talk about the definition of the Selmer groups and the Tate-Shafarevich groups: motivations and geometric interpretations. The text I am using is “Diophantine Geometry: An Introduction” by Marc Hindry and Joseph H. Silverman, pages 279-283.
April 29, 2015
Jay Kopper :
3 p.m. in SEO 512
Abstract
Noether proposed attacking the inverse Galois problem by studying elementary symmetric polynomials and their specializations to number fields. This has led to some modern developments in algebraic number theory as well as an algorithm for computing Galois groups over Q.
Sept. 14, 2015
Charles Alley :
10 a.m. in SEO 427
Abstract
In this talk I will sketch the proof that $\liminf(p_{n+1}-p_n) \leq 600$ given in James Maynard's paper "Small
Gaps Between Primes". The proof uses the notion of 'level of distribution' which I will define. As an example, I will state the Bombieri-Vinogradov Theorem, which says that the primes have level of
distribution 1/2.
May 4, 2020
N/A :
4 p.m. in Zoom (link available upon request)
May 7, 2020
Jacob Mayle of
UIC :
1 p.m. in Zoom
May 12, 2020
Sung Min Lee of
UIC :
1 p.m. in Zoom
May 14, 2020
Matthew Fitzpatrick of
UIC :
1 p.m. in Zoom
May 19, 2020
Tian Wang of
UIC :
1 p.m. in Zoom
May 21, 2020
Zhehao Li of
UIC :
1 p.m. in Zoom
May 26, 2020
Adam Pratt of
UIC :
1 p.m. in Zoom
May 28, 2020
Neelima Borade of
UIC :
1 p.m. in Zoom
June 9, 2020
Jacob Mayle of
UIC :
1 p.m. in Zoom
June 11, 2020
Sung Min Lee of
UIC :
1 p.m. in Zoom
June 18, 2020
Tian Wang of
UIC :
2 p.m. in Zoom
June 23, 2020
Zhehao Li of
UIC :
2 p.m. in Zoom
June 25, 2020
Jacob Mayle of
UIC :
2 p.m. in Zoom
June 29, 2020
Matthew Fitzpatrick of
UIC :
2 p.m. in Zoom
July 6, 2020
John Sung Min Lee of
UIC :
2 p.m. in Zoom
July 13, 2020
Tian Wang of
UIC :
2 p.m. in Zoom
July 20, 2020
Zhehao Li of
UIC :
2 p.m. in Zoom
July 27, 2020
Jacob Mayle of
UIC :
2 p.m. in Zoom
Aug. 24, 2020
No Speaker :
4:15 p.m. in Zoom
Aug. 31, 2020
John Sung Min Lee of
UIC :
4 p.m. in Zoom
Sept. 7, 2020
Jacob Mayle of
UIC :
4:30 p.m. in Zoom
Sept. 14, 2020
Zhehao LI of
UIC :
4:30 p.m. in Zoom
Sept. 21, 2020
Matthew Fitzpatrick of
UIC :
4:30 p.m. in Zoom
Sept. 28, 2020
Neelima Borade of
UIC :
4:30 p.m. in Zoom
Oct. 5, 2020
Anish Chedalavada of
UIC :
4:30 p.m. in Zoom
Oct. 12, 2020
Auden Hinz of
UIC :
4:30 p.m. in Zoom
Oct. 19, 2020
John Sung Min Lee of
UIC :
4:30 p.m. in Zoom
Oct. 26, 2020
Tian Wang of
UIC :
4:30 p.m. in Zoom
Nov. 2, 2020
Jacob Mayle of
UIC :
4:30 p.m. in Zoom
Nov. 9, 2020
Zhehao LI of
UIC :
4:30 p.m. in Zoom
Nov. 16, 2020
Matthew Fitzpatrick of
UIC :
4:30 p.m. in Zoom
Nov. 23, 2020
Tian Wang of
UIC :
4:30 p.m. in Zoom
Jan. 20, 2021
Auden Hinz (UIC) :
3 p.m. in Zoom
Jan. 27, 2021
Tian Wang (UIC) :
3 p.m. in Zoom
Feb. 3, 2021
Jacob Mayle (UIC) :
3 p.m. in Zoom
Feb. 10, 2021
John Lee (UIC) :
3 p.m. in Zoom
Feb. 17, 2021
Tian Wang (UIC) :
3 p.m. in Zoom
Feb. 24, 2021
Jacob Mayle (UIC) :
3 p.m. in Zoom
March 3, 2021
John Lee (UIC) :
3 p.m. in Zoom
March 10, 2021
Zhehao Li (UIC) :
3 p.m. in Zoom
March 17, 2021
Tian Wang (UIC) :
3 p.m. in Zoom
March 31, 2021
Jacob Mayle (UIC) :
3 p.m. in Zoom
April 7, 2021
Neelima Borade (UIC) :
3 p.m. in Zoom
April 14, 2021
Matthew Fitzpatrick (UIC) :
3 p.m. in Zoom
April 21, 2021
John Lee (UIC) :
3 p.m. in Zoom
April 28, 2021
Zhehao Li (UIC) :
3 p.m. in Zoom
Aug. 25, 2021
:
4 p.m. in Zoom
Abstract
We will decide regular meeting times, materials for the semester, and presenters for the first few weeks.
Sept. 1, 2021
John Lee of
UIC :
5 p.m. in Zoom
Abstract
We will go over 'Abelian Varieties' by Milne.
Sept. 8, 2021
Jiamin Li of
UIC :
5 p.m. in Zoom
Sept. 15, 2021
Tian Wang of
UIC :
5 p.m. in Zoom
Sept. 22, 2021
John Lee of
UIC :
5 p.m. in Zoom
Sept. 29, 2021
Jacob Mayle of
UIC :
5 p.m. in Zoom
Oct. 6, 2021
Jiamin Li of
UIC :
5 p.m. in Zoom
Oct. 13, 2021
Zhehao Li of
UIC :
5 p.m. in Zoom
Oct. 20, 2021
Tian Wang of
UIC :
5 p.m. in Zoom
Oct. 27, 2021
Jacob Mayle of
UIC :
5 p.m. in Zoom
Nov. 3, 2021
John Lee of
UIC :
5 p.m. in Zoom
Nov. 10, 2021
Jiamin Li of
UIC :
5 p.m. in Zoom
Nov. 17, 2021
Zhehao Li of
UIC :
5 p.m. in Zoom
Dec. 1, 2021
Tian Wang & Jacob Mayle of
UIC :
5 p.m. in Zoom
Jan. 26, 2022
John Lee of
UIC :
4 p.m. in Zoom
Feb. 2, 2022
Tian Wang of
UIC :
4 p.m. in Zoom
Feb. 9, 2022
Auden Hinz of
UIC :
4 p.m. in Zoom
Feb. 16, 2022
Jacob Mayle of
UIC :
4 p.m. in Zoom
Feb. 23, 2022
Jiamin Li of
UIC :
4 p.m. in Zoom
March 2, 2022
John Lee of
UIC :
4 p.m. in Zoom
March 16, 2022
Zhehao Li of
UIC :
4 p.m. in Zoom
March 30, 2022
Zhehao Li of
UIC :
4 p.m. in Zoom
April 6, 2022
Jacob Mayle of
UIC :
4 p.m. in Zoom
April 13, 2022
Tian Wang of
UIC :
4 p.m. in Zoom
April 27, 2022
Anish Chedalavada of
UIC :
4 p.m. in Zoom
May 4, 2022
Zhehao Li of
UIC :
4 p.m. in Zoom
Aug. 29, 2022
Tian Wang of
UIC :
4 p.m. in 427 SEO
Sept. 5, 2022
John Lee of
UIC :
4 p.m. in Zoom
Sept. 14, 2022
Carl Tang of
UIC :
4 p.m. in 427 SEO
Abstract
This is the first talk of a series of learning seminars on Shimura curves.
Sept. 21, 2022
Hank Morris of
UIC :
4 p.m. in 427 SEO
Oct. 5, 2022
Zhehao Li of
UIC :
4 p.m. in 427 SEO
Abstract
An introduction to Shimura varieties. See https://sites.google.com/uic.edu/uic-gnts-fall-2022/home for reference.
Oct. 12, 2022
Tian Wang of
UIC :
4 p.m. in 427 SEO
Oct. 19, 2022
Yeqin Liu of
UIC :
4 p.m. in 427 SEO
Abstract
The Brauer-Manin obstruction is a refinement of the Hasse principle, which gives a sufficient condition for non-existence of rational points on an algebraic variety. In this talk I will introduce the Brauer group of an algebraic variety and explain the Brauer-Manin obstruction geometrically. Then we will see through an example that the Brauer-Manin obstruction is a strict refinement of the Hasse principle.
Oct. 26, 2022
Auden Hinz of
UIC :
4 p.m. in 427 SEO
Nov. 2, 2022
Zhehao Li of
UIC :
4 p.m. in 427 SEO
Nov. 9, 2022
John Lee of
UIC :
4 p.m. in 427 SEO
Nov. 16, 2022
Tian Wang of
UIC :
4 p.m. in 427 SEO
Abstract
Reference: https://arxiv.org/abs/2007.14182v2
Nov. 30, 2022
John Lee of
UIC :
4 p.m. in 427 SEO
Abstract
The references are <i>A bound for the conductor of an open subgroup of GL2 associated to an elliptic curve</i> and <i>Averages of elliptic curve constants</i> by Nathan Jones.
Jan. 18, 2023
Zhehao Li of
UIC :
4 p.m. in 427 SEO
Abstract
To introduce basic ideas in unlikely intersection, we will talk about the Pila-Zannier proof of the multiplicative Manin-Mumford conjecture, and how the ideas can be applied to the André–Oort conjecture. No technical details will be included. This is the first talk in a series of talks in Diophantine geometry and unlikely intersection. This is a hybrid seminar. The link is: https://uic.zoom.us/j/89234585109?pwd=eU5HSFovK3Fmb1Z5U25YWTc3WFZidz09
Jan. 25, 2023
John Lee of
UIC :
4 p.m. in Zoom
Abstract
The height functions are designed to describe arithmetic complexity, and they are a key tool in proving many important finiteness theorems in Diophantine Geometry. Today we will introduce some basic height functions and how they can be useful.
Feb. 1, 2023
Auden Hinz of
UIC :
4 p.m. in Zoom
Abstract
We will introduce basic definitions in mathematical logic and some fundamental facts in model theory.
Feb. 8, 2023
Kevin Zhou of
UIC :
4 p.m. in Zoom
Abstract
We will review some of the basic ideas of model theory, introduce some important concepts in the model theory of fields, and work towards applications using the Pila-Wilkie theorem.
Feb. 15, 2023
Tian Wang of
UIC :
4 p.m. in Zoom
Abstract
In the talk, I will discuss three special point problems: Lang's Conjecture, Manin-Mumford Conjecture, and Andre-Oort Conjecture, and sketch their proof strategies with an emphasis on relevant arithmetic properties. The main reference is Jacob Tsimerman's lecture notes from the AWS 2023.
Feb. 22, 2023
Zhehao Li of
UIC :
4 p.m. in 427 SEO
Abstract
Instead of heights of points on a projective variety, we will look at heights of varieties themselves, and we will focus on abelian varieties. We will define Faltings height and modular height of an abelian variety, and introduce how they were used in the proof of Faltings' theorem. On the way to define Faltings height, we will have a quick tour on arithmetic intersection theory.
March 15, 2023
John Lee of
UIC :
4 p.m. in 427 SEO
Abstract
In this talk, I am going to introduce one of old problems in the study of elliptic curves -- the distribution of primes of cyclic reduction -- and the current results on related problems. Also I am planning to summarize a few results in the settings of elliptic curves over function fields, abelian varieties, and Drinfeld modules.
March 29, 2023
Jacob Mayle of
Wake Forest University :
4 p.m. in Zoom
Abstract
Given an elliptic curve over Q and a rational point P on E, Rafe Jones and Jeremy Rouse (and others) have considered the problem of determining the density of primes p such that the order of P modulo p is odd. The main tool for studying such a question is the arboreal Galois representation. We’ll discuss Galois representations of elliptic curves generally and how they relate to solving the odd order reduction problem, as well as an extension for abelian surfaces.
April 5, 2023
Santiago Arango-Piñeros of
Emory University :
4 p.m. in Zoom
Abstract
Fix a $g$-dimensional abelian variety $A$ defined over a finite field $\mathbf{F}_q$. For every integer $r \geq 1$, consider the extension of scalars $A_{(r)} = A\times_{\mathbf{F}_q} \mathbf{F}_{q^r}$. We study the distribution of the normalized trace of the Frobenius endomorphism of $A_{(r)}$ in the compact interval $[-2g,2g]$, as $r$ varies. We show that these distributions are controlled by a certain compact abelian Lie group and classify the possible groups when $g \leq 3$.
April 12, 2023
Sun Woo Park of
University of Wisconsin-Madison :
4 p.m. in Zoom
Abstract
Fix a prime number p. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and p, which also contains the primitive $p$-th root of unity. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution fo the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdos-Kac theorem, and the geometric ergodicity of Markov chains.
April 19, 2023
John Yin of
University of Wisconsin-Madison :
4 p.m. in Zoom
Abstract
For a given p, let rho(p) denote the proportion of irreducible quadratic polynomials over Z which generates field extensions which are totally split at p. Then, it turns out rho(t)=(t^2+1)/(2(t+1)^2). There are two remarkable things about this: it is a rational function in Q(t) and it satisfies the functional equation rho(t)=rho(-t). Bhargava, Cremona, Gajovic, and Fisher conjecture that the same holds for arbitrary degrees and splitting types. We prove a vast generalization of this conjecture in the tame case. This is joint work with Yifan Wei and Asvin G.
Oct. 11, 2023
Zhehao Li of
UIC :
3 p.m. in 427 SEO
Abstract
GL_n is a familiar stranger in our math career. In this talk we will collect some facts about GL_n over rings (especially rings of integers and adele rings) in the context of arithmetic groups.
Oct. 25, 2023
Zhehao Li of
UIC :
3 p.m. in 427 SEO
Abstract
We will introduce some essential properties of (linear) algebraic groups and structures of reductive groups.
Jan. 24, 2024
Zhehao Li of
UIC :
3 p.m. in 427 SEO
Jan. 31, 2024
Andre de Moura of
UIC :
3 p.m. in 427 SEO
Feb. 7, 2024
Andre de Moura of
UIC :
3 p.m. in 427 SEO
Feb. 14, 2024
Anh Tran of
UIC :
3 p.m. in 427 SEO
April 3, 2024
Chirag Singhal of
UIC :
3 p.m. in 427 SEO
April 10, 2024
Auden Hinz of
UIC :
3 p.m. in 427 SEO
April 17, 2024
Zhehao Li of
UIC :
3 p.m. in 427 SEO
Aug. 29, 2024
:
1 p.m. in 427 SEO
Sept. 5, 2024
Chirag Singhal of
UIC :
1 p.m. in 427 SEO
Sept. 12, 2024
Auden Hinz of
UIC :
1 p.m. in 427 SEO
Sept. 19, 2024
Andrew Smith of
UIC :
1 p.m. in 427 SEO
Sept. 26, 2024
Vibhu Saksena of
UIC :
1 p.m. in 427 SEO
Oct. 3, 2024
Tanis Nielsen of
UIC :
1 p.m. in 427 SEO
Oct. 10, 2024
Andre De Moura of
UIC :
1 p.m. in 427 SEO
Oct. 17, 2024
Andre De Moura of
UIC :
1 p.m. in 427 SEO
Oct. 31, 2025
Gena Guerrieri of
University of Illinois Chicago :
1 p.m. in 427 SEO
Nov. 7, 2025
Soham Pal of
University of Illinois Chicago :
1 p.m. in 427 SEO