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

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, January 19, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, January 19, 2017
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Submitted by sahlgren.
Joel Specter (Northwestern University)
Commuting Endomorphisms of the p-adic Formal Disk
Abstract: Any one dimensional formal group law over $\mathbb{Z}_p$ is uniquely determined by the series expansion of its multiplication by $p$ map. This talk addresses the converse question: when does an endomorphism $f$ of the $p$-adic formal disk arise as the multiplication by $p$-map of a formal group? Lubin, who first studied this question, observed that if such a formal group were to exist, then $f$ would commute with an automorphism of infinite order. He formulated a conjecture under which a commuting pair of series should arise from a formal group. Using methods from p-adic Hodge theory, we prove the height one case of this conjecture.

Tuesday, January 24, 2017

Graduate Student Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Tuesday, January 24, 2017
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Submitted by amalik10.
Byron Heersink (UIUC)
Poincaré sections for the horocycle flow in covers of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) and applications to Farey fraction statistics
Abstract: For a given finite index subgroup $H\subseteq$SL(2,$\mathbb{Z}$), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to the finite cover SL(2,$\mathbb{R}$)/$H$ of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$). We then relate the properties of this section to the gaps in Farey fractions and describe how the ergodic properties of the horocycle flow can be used to obtain certain statistical properties of various subsets of Farey fractions.

Thursday, February 2, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, February 2, 2017
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Submitted by sahlgren.
Bruce Berndt (Illinois Math)
Identities
Abstract: As the title suggests, this lecture features mathematical identities. The identities we have chosen (we hope) are interesting, fascinating, surprising, and beautiful! Many of the identities are due to Ramanujan. Topics behind the identities include partitions, continued fractions, sums of squares, theta functions, Bessel functions, $q$-series, other infinite series, and infinite integrals.

Thursday, February 9, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, February 9, 2017
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Submitted by sahlgren.
Matthias Strauch (Indiana University Bloomington)
Coverings of the p-adic upper half plane and arithmetic differential operators
Abstract: The p-adic upper half plane comes equipped with a remarkable tower of GL(2)-equivariant etale covering spaces, as was shown by Drinfeld. It has been an open question for some time whether the spaces of global sections of the structure sheaf on such coverings provide admissible locally analytic representations. Using global methods and the p-adic Langlands correspondence for GL(2,Qp), this is now known to be the case by the work of Dospinescu and Le Bras. For the first layer of this tower Teitelbaum exhibited a nice formal model which we use to provide a local proof for the admissibility of the representation (when the base field is any finite extension of Qp). The other key ingredients are suitably defined sheaves of arithmetic differential operators and D-affinity results for formal models of the rigid analytic projective line, generalizing those of Christine Huyghe. This is joint work with Deepam Patel and Tobias Schmidt.

Thursday, February 23, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, February 23, 2017
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Submitted by sahlgren.
Amita Malik (Illinois Math)
Partitions into $k$th powers of a fixed residue class
Abstract: G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect $k$th powers, which was later proved by E. M. Wright. In this talk, we discuss partitions into parts from a specific set $A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb{N}, m\equiv a_0 \pmod{b_0} \right\}$, for fixed positive integers $k$, $a_0,$ and $b_0$. We give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright and others. We also discuss the parity problem for such partitions. This is joint work with Bruce Berndt and Alexandru Zaharescu.

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, February 23, 2017
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Submitted by amalik10.
Dane Skabelund   [email] (UIUC)
Some maximal curves obtained via a ray class field construction
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

Graduate Number Theory Seminar/Partitions Seminar
2:00 pm   in 241 Altgeld Hall,  Tuesday, February 28, 2017
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Submitted by amalik10.
Hannah Burson (UIUC)
Weighted Partition Identities
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 2, 2017

Number Theory Seminar
11:00 am   in Altgeld Hall,  Thursday, March 2, 2017
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Submitted by sahlgren.
Brad Rodgers (University of Michigan)
Sums in short intervals and decompositions of arithmetic functions
Abstract: In this talk we will discuss some old and new conjectures about the behavior of sums of arithmetic functions in short intervals, along with analogues of these conjectures in a function field setting that have been proved in recent years. We will pay particular attention to some surprising phenomena that comes into play, and a decomposition of arithmetic functions in a function field setting that helps elucidate what's happening.

Thursday, March 9, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, March 9, 2017
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Submitted by sahlgren.
Frank Garvan (University of Florida)
New Mock Theta Function Identities
Abstract: In his last letter to Hardy, Ramanujan defined ten mock theta functions of order 5 and three of order 7. He stated that the three mock theta functions of order 7 are not related. We give simple proofs of new Hecke double sum identities for two of the order 5 functions and all three of the order 7 functions. We find that the coefficients of Ramanujan's three mock theta functions of order 7 are surprisingly related.

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, March 9, 2017
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Submitted by amalik10.
Frank Garvan (University of Florida)
Playing with partitions and $q$-series
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.

Tuesday, March 14, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Tuesday, March 14, 2017
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Submitted by sahlgren.
Sudhir Pujahari (Harish-Chandra Research Institute)
In the neighbourhood of Sato-Tate conjecture
Abstract: In this talk, we will see the distribution of gaps between eigenangles of Hecke operators acting on the space of cusp forms of weight $k$ and level $N$, spaces of Hilbert modular forms of weight $k = (k_1, k_2,\ldots , k_r)$ and space of primitive Maass forms of weight $0$. Moreover, we will see the following: Let $f_1$ and $f_2$ be two normalized Hecke eigenforms of weight $k_1$ and $k_2$ such that one of them is not of CM type. If the set of primes $\mathcal{P}$ such that the $p$-th coefficients of $f_1$ and $f_2$ matches has positive upper density, then $f_1$ is a Dirichlet character twist of $f_2$. The last part is a joint work with M. Ram Murty.

Thursday, March 16, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, March 16, 2017
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Submitted by sahlgren.
Armin Straub (University of South Alabama)
A gumbo with hints of partitions, modular forms, special integer sequences and supercongruences
Abstract: Euler's partition theorem famously asserts that the number of ways to partition an integer into distinct parts is the same as the number of ways to partition it into odd parts. In the first part of this talk, we describe a new analog of this theorem for partitions of fixed perimeter. More generally, we discuss enumeration results for simultaneous core partitions, which originates with an elegant result due to Anderson that the number of $(s,t)$-core partitions is finite and is given by generalized Catalan numbers. The second part is concerned with congruences between truncated hypergeometric series and modular forms. Specifically, we discuss a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated $_6F_5$-hypergeometric series. The story is intimately tied with Apéry's proof of the irrationality of $\zeta(3)$. This is recent joint work with Robert Osburn and Wadim Zudilin.

Thursday, March 30, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, March 30, 2017
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Submitted by sahlgren.
Preston Wake (University of California at Los Angeles)
Pseudorepresentations and the Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, but also posed a number of questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Carl Wang Erickson, we give an answer to these questions using the deformation theory of Galois pseudorepresentations. The answer is intimately related to the algebraic number theoretic interactions between the primes N and p, and is given in terms of cup products (and Massey products) in Galois cohomology.

Thursday, April 6, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, April 6, 2017
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Submitted by sahlgren.
Bao V. Le Hung (University of Chicago)
Congruences between automorphic forms
Abstract: The theory of congruences between automorphic forms traces back to Ramanujan, who observed various congruence properties between coefficients of generating functions related to the partition function. Since then, the subject has evolved to become a central piece of contemporary number theory; lying at the heart of spectacular achievements such as the proof of Fermat's Last Theorem and the Sato-Tate conjecture. In my talk I will explain how the modern theory gives satisfactory explanations of some concrete congruence phenomena for modular forms (the $\mathrm{GL}_2$ case), as well as recent progress concerning automorphic forms for higher rank groups. This is joint work with D. Le, B. Levin and S. Morra.

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, April 6, 2017
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Submitted by jli135.
Kyle Pratt   [email] (UIUC)
Primes with restricted digits
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

Graduate Number Theory Seminar/Partitions Seminar
2:00 pm   in 241 Altgeld Hall,  Tuesday, April 11, 2017
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Submitted by amalik10.
Hsin-Po Wang (UIUC)
Andrew's recent papers on integer partitions and the existence of combinatorial proofs
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

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, April 13, 2017
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Submitted by sahlgren.
Matt Papanikolas (Texas A&M University)
Limits of function field Bernoulli-Carlitz numbers
Abstract: Because of the classical Kummer congruences, one is able to take p-adic limits of certain natural subsequences of Bernoulli numbers. This leads to notions of p-adic limits of special zeta values and Eisenstein series. In the case of the rational function field K over a finite field, the analogous quantities, called Bernoulli-Carlitz numbers, fail to satisfy Kummer-type congruences. Nevertheless, we prove that certain subsequences of Bernoulli-Carlitz numbers do have v-adic limits, for v a finite place of K, thus leading to new v-adic limits of Eisenstein series. Joint with G. Zeng.

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, April 13, 2017
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Submitted by jli135.
Detchat Samart   [email] (UIUC)
L-values, Bessel moments and Mahler measures
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

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, April 20, 2017
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Submitted by sahlgren.
Robert Lemke Oliver (Tufts University)
Unexpected biases in the distribution of consecutive primes
Abstract: While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible pairs of reduced residue classes (mod q) is surprisingly erratic. We propose a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures, which fits the observed data very well. We also study the distribution of the terms predicted by the conjecture, which proves to be surprisingly subtle. This is joint work with Kannan Soundararajan.

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, April 20, 2017
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Submitted by amalik10.
Robert Lemke Oliver (Tuft University)
Ranks of elliptic curves, Selmer groups, and Tate-Shafarevich groups
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

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, April 27, 2017
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Submitted by sahlgren.
Andrew Sills (Georgia Southern University)
A formula for the partition function that "counts"
Abstract: A partition of an integer n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant. Thus we see that there are five partitions of the integer 4, namely 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The partition function p(n) denotes the number of partitions of n. Thus p(4) = 5. The first exact formula for p(n) was given by Hardy and Ramanujan in 1918. Twenty years later, Hans Rademacher improved the Hardy-Ramanujan formula to give an infinite series that converges to p(n). The Hardy-Ramanujan-Rademacher series is revered as one of the truly great accomplishments in the field of analytic number theory. In 2011, Ken Ono and Jan Bruinier surprised everyone by announcing a new formula which attains p(n) by summing a finite number of complex numbers which arise in connection with the multiset of algebraic numbers that are the union of Galois orbits for the discriminant -24n + 1 ring class field. Thus the known formulas for p(n) involve deep mathematics, and are by no means "combinatorial" in the sense that they involve summing a finite or infinite number of complex numbers to obtain the correct (positive integer) value. In this talk, I will show a new formula for the partition function as a multisum of positive integers, each term of which actually counts a certain class of partitions, and thus appears to be the first truly combinatorial formula for p(n). The idea behind the formula is due to Yuriy Choliy, and the work was completed in collaboration with him. We will further examine a new way to approximate p(n) using a class of polynomials with rational coefficients, and observe this approximation is very close to that of using the initial term of the Rademacher series. The talk will be accessible to students as well as faculty, and anyone interested is encouraged to attend!

Graduate Number Theory Seminar
2:00 pm   in 241 Altgeld Hall,  Thursday, April 27, 2017
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Submitted by amalik10.
Andrew Sills (Georgia Southern University)
MacMahon's partial fractions
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.

Tuesday, May 2, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Tuesday, May 2, 2017
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Submitted by sahlgren.
Iván Blanco-Chacón (University College Dublin)
Introduction to p-adic modular forms and Hida families for GL(1)
Abstract: Part I of III on Hida Families, Hilbert Modular Forms and Arithmetic Applications. Part II and III (5/4 and 5/9) address Hilbert modular forms and a p-adic Gross-Zagier formula. Extended abstract at http://math.uiuc.edu/~duursma/HidaFamilies-IvanBC.html

Friday, May 5, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Friday, May 5, 2017
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Submitted by sahlgren.
Iván Blanco-Chacón (University College Dublin)
Hilbert modular forms and Hida families of Hilbert modular forms for GL(2)
Abstract: Part II of III on Hida Families, Hilbert Modular Forms and Arithmetic Applications. Extended abstract at http://math.uiuc.edu/~duursma/HidaFamilies-IvanBC.html

Tuesday, May 9, 2017

Number Theory Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, May 9, 2017
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Submitted by sahlgren.
Iván Blanco-Chacón (University College Dublin)
Hirzebruch-Zagier cycles and a p-adic Gross-Zagier formula
Abstract: Part III of III on Hida Families, Hilbert Modular Forms and Arithmetic Applications. Extended abstract at http://math.uiuc.edu/~duursma/HidaFamilies-IvanBC.html

Tuesday, June 20, 2017

Number Theory Seminar
11:00 am   in 243 Altgeld Hall,  Tuesday, June 20, 2017
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Submitted by sahlgren.
Atul Dixit (Indian Institute of Technology Gandhinagar)
A generalized modified Bessel function and a higher level analogue of the general theta transformation formula
Abstract: A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel $\cos \left( {{\pi z}} \right){M_{2z}}(4\sqrt {x} ) - \sin \left( {{\pi z}} \right){J_{2z}}(4\sqrt {x} )$ and which subsumes the self-reciprocal pair involving $K_{z}(x)$. Its application towards finding modular-type transformations of the form $F(z, w, \alpha)=F(z,iw,\beta)$, where $\alpha\beta=1$, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on $SL_{2}(\mathbb{Z})$. This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann $\Xi$-function and consisting of a sum of products of two confluent hypergeometric functions. This is joint work with Aashita Kesarwani and Victor H. Moll, with an appendix by Nico M. Temme.

Thursday, September 21, 2017

Number Theory Seminar
11:00 am   in 241 Altgeld Hall,  Thursday, September 21, 2017
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Submitted by sahlgren.
George Andrews (Pennsylvania State University)
To Be Announced