Department of

Mathematics


Seminar Calendar
for Number Theory Seminar events the year of Thursday, October 12, 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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Thursday, January 19, 2017

11:00 am in 241 Altgeld Hall,Thursday, January 19, 2017

Commuting Endomorphisms of the p-adic Formal Disk

Joel Specter (Northwestern University)

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

2:00 pm in 241 Altgeld Hall,Tuesday, January 24, 2017

Poincaré sections for the horocycle flow in covers of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) and applications to Farey fraction statistics

Byron Heersink (UIUC)

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

11:00 am in 241 Altgeld Hall,Thursday, February 2, 2017

Identities

Bruce Berndt (Illinois Math)

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

11:00 am in 241 Altgeld Hall,Thursday, February 9, 2017

Coverings of the p-adic upper half plane and arithmetic differential operators

Matthias Strauch (Indiana University Bloomington)

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

11:00 am in 241 Altgeld Hall,Thursday, February 23, 2017

Partitions into $k$th powers of a fixed residue class

Amita Malik (Illinois Math)

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.

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 2, 2017

11:00 am in Altgeld Hall,Thursday, March 2, 2017

Sums in short intervals and decompositions of arithmetic functions

Brad Rodgers (University of Michigan)

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

11:00 am in 241 Altgeld Hall,Thursday, March 9, 2017

New Mock Theta Function Identities

Frank Garvan (University of Florida)

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.

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.

Tuesday, March 14, 2017

11:00 am in 241 Altgeld Hall,Tuesday, March 14, 2017

In the neighbourhood of Sato-Tate conjecture

Sudhir Pujahari (Harish-Chandra Research Institute)

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

11:00 am in 241 Altgeld Hall,Thursday, March 16, 2017

A gumbo with hints of partitions, modular forms, special integer sequences and supercongruences

Armin Straub (University of South Alabama)

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

11:00 am in 241 Altgeld Hall,Thursday, March 30, 2017

Pseudorepresentations and the Eisenstein ideal

Preston Wake (University of California at Los Angeles)

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

11:00 am in 241 Altgeld Hall,Thursday, April 6, 2017

Congruences between automorphic forms

Bao V. Le Hung (University of Chicago)

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.

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

11:00 am in 241 Altgeld Hall,Thursday, April 13, 2017

Limits of function field Bernoulli-Carlitz numbers

Matt Papanikolas (Texas A&M University)

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.

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

11:00 am in 241 Altgeld Hall,Thursday, April 20, 2017

Unexpected biases in the distribution of consecutive primes

Robert Lemke Oliver (Tufts University)

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.

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

11:00 am in 241 Altgeld Hall,Thursday, April 27, 2017

A formula for the partition function that "counts"

Andrew Sills (Georgia Southern University)

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!

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.

Tuesday, May 2, 2017

11:00 am in 241 Altgeld Hall,Tuesday, May 2, 2017

Introduction to p-adic modular forms and Hida families for GL(1)

Iván Blanco-Chacón (University College Dublin)

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

11:00 am in 241 Altgeld Hall,Friday, May 5, 2017

Hilbert modular forms and Hida families of Hilbert modular forms for GL(2)

Iván Blanco-Chacón (University College Dublin)

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

11:00 am in 243 Altgeld Hall,Tuesday, May 9, 2017

Hirzebruch-Zagier cycles and a p-adic Gross-Zagier formula

Iván Blanco-Chacón (University College Dublin)

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

11:00 am in 243 Altgeld Hall,Tuesday, June 20, 2017

A generalized modified Bessel function and a higher level analogue of the general theta transformation formula

Atul Dixit (Indian Institute of Technology Gandhinagar)

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.

Tuesday, July 18, 2017

11:00 am in 243 Altgeld Hall,Tuesday, July 18, 2017

Diophantine equations with binomial coefficients and perturbations of symmetric Boolean functions

Luis Medina (University of Puerto Rico at Rio Piedras)

Abstract: This work establishes a connection between exponential sums of perturbations of symmetric Boolean functions and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $\Gamma$ of $\mathbb{Z}$. The concepts of trivially balanced symmetric Boolean function and sporadic balanced Boolean function are extended to this type of perturbation. An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. This is a joint work with Francis N. Castro and Oscar E. González

Thursday, August 31, 2017

11:00 am in 241 Altgeld Hall,Thursday, August 31, 2017

Polynomial Roth type theorems in Finite Fields

Dong Dong (Illinois Math)

Abstract: Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.

Thursday, September 7, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 7, 2017

On the zeros of Riemann's zeta-function

Sieg Baluyot (Illinois)

Abstract: In the first part of this talk, we present a new proof that a positive proportion of the zeros of the Riemann zeta-function lie on the critical line. The proof is an enhancement of a zero-detection method of Atkinson from the 1940’s, and uses the recent estimate of Hughes and Young for the twisted fourth moment of zeta. In the second part, we consider the number of zeros of zeta inside the region with real part larger than $\sigma$ and imaginary part between 0 and T. A bound for this number is called a “zero-density estimate.” We present an improved zero-density estimate for the case when $\sigma$ is larger than 1/2 but close to 1/2. The main theorem confirms an unproved result of Conrey from the 1980’s using his technique of applying Kloosterman sum estimates. Finally, in the third part, we look at hypothetical statements for the vertical distribution of zeros along the critical line and deduce their consequences for the prime numbers and other properties of zeta. The main theorem generalizes results of Goldston, Gonek, and Montgomery that give consequences of the pair correlation conjecture. We apply the theorem to examine implications of the well-known “alternative hypothesis,” which is related to Landau-Siegel zeros.

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.

Thursday, September 14, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 14, 2017

Partition asymptotics and the polylogarithm

Alexander Dunn (Illinois)

Abstract: n 2015 Vaughn obtained asymptotic formulas for the number of partitions of an integer into squares. Gafni extended this to kth powers. Here we obtain such formulas for the number of partitions into values of an arbitrary integer polynomial $f$ subject to some mild hypotheses. Our methods use an interplay of the circle method, the polylogarithm, and the Matsumoto-Weng zeta function. This is joint work with Nicolas Robles.

2:00 pm in 241 Altgeld Hall,Thursday, September 14, 2017

On the percentage of critical zeros of Riemann's zeta function

Kyle Pratt   [email] (UIUC)

Abstract: The Riemann hypothesis (RH) is one of the most important unsolved problems in number theory. RH asserts that all of the important zeros of the Riemann zeta function lie on a specific line, called the critical line. As we lack a solution to RH, it is natural to ask for partial results instead. One way to measure progress towards RH is to prove that some percentage of the zeros are on the critical line. I will sketch a brief history of the results about percentages of zeros on the critical line, and discuss some of the methods of proof. In the latter part of the talk I will discuss the current world record, due to Nicolas Robles and myself, and some of our ideas. The talk should be accessible to any graduate student.

Thursday, September 21, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 21, 2017

4-Shadows in q-Series: Gupta, Kimberling, the Garden of Eden and the OEIS.

George Andrews (Penn State )

Abstract: This talk is devoted to discussing the implications of a very elementary technique for proving mod 4 congruences in the theory of partitions. It starts with a tribute to the late Hans Raj Gupta and leads in unexpected ways to partitions investigated by Clark Kimberling, to Bulgarian Solitaire, and to Garden of Eden partitions. Each surprise busts forth from the OEIS

Tuesday, September 26, 2017

11:00 am in 241 Altgeld Hall,Tuesday, September 26, 2017

Introduction to Shimura curves

Yifan Yang (National Chiao Tung University)

Abstract: Shimura curves are generalizations of modular curves. The arithmetic aspect of Shimura curves bears a great similarity to that of modular curves. However, because of the lack of cusps on Shimura curves, it is difficult to do explicit computation about them. This makes Shimura curves both interesting and challenging to study. In this talk, we will give a quick introduction to Shimura curves.

Thursday, September 28, 2017

11:00 am in 241 Altgeld Hall,Thursday, September 28, 2017

Equations of hyperelliptic Shimura curves

Yifan Yang (National Chiao Tung University)

Abstract: Because of the lack of cusps on Shimura curves, there are few methods to construct modular forms on them. As a result, it is very difficult to determine equations of Shimura curves. In a recent work, we devised a systematic method to construct Borcherds form. Together with Schofer's formula for values of Borcherds forms at CM-points, this enabled us to determine equations of all hyperelliptic Shimura curves. This is a joint work with Jia-Wei Guo.

2:00 pm in 241 Altgeld Hall,Thursday, September 28, 2017

Extreme values of zeta and $L$-functions

Siegfred Baluyot (UIUC)

Abstract: One of the most important problems in the theory of the Riemann zeta-function is to determine how large the modulus of zeta can be on the critical line. In this talk, we will first outline the history of this problem. Then we will discuss the versatile `resonance method' of Soundararajan for detecting large values of zeta and other $L$-functions. We then complete our discussion with the recent breakthrough of Bondarenko and Seip that significantly improves the resonance method for zeta by using estimates for greatest common divisor sums.

Thursday, October 5, 2017

11:00 am in 241 Altgeld Hall,Thursday, October 5, 2017

Primes from sums of two squares and missing digits

Kyle Pratt (UIUC Math)

Abstract: In recent decades there have been significant advances made in finding primes in "thin" sequences. One such advance was the work of Friedlander and Iwaniec, in which they proved there are infinitely many primes that can be represented as the sum of a square and a biquadrate. A more recent advance is due to Maynard, who showed the existence of infinitely many primes in the thin sequence of integers missing a fixed digit in their decimal expansion. In this talk I discuss a marriage of some of the ideas of Friedlander-Iwaniec and Maynard which allows one to find primes in other interesting thin sequences.

2:00 pm in 241 Altgeld Hall,Thursday, October 5, 2017

Ramanujan's life and earlier notebooks

Bruce Berndt   [email] (UIUC)

Abstract: Generally regarded as India's greatest mathematician, Srinivasa Ramanujan was born in the southern Indian town of Kumbakonam on December 22, 1887 and died in Madras at the age of 32 in 1920. Before going to England in 1914 at the invitation of G.~H.~Hardy, Ramanujan recorded most of his mathematical discoveries without proofs in notebooks. The speaker devoted over 20 years to the editing of these notebooks; his goal was to provide proofs for all those claims of Ramanujan for which proofs had not been given in the literature. In this lecture, we give a brief history of Ramanujan's life, a history of the notebooks, a general description of the subjects found in the notebooks, and examples of some of the more interesting formulas found in the notebooks.

Thursday, October 12, 2017

11:00 am in 241 Altgeld Hall,Thursday, October 12, 2017

Constructing sets using de Bruijn sequences

George Shakan (UIUC Math)

Abstract: Junxian Li and I showed that a set with distinct consecutive r-differences has large sumset. During this talk I will explain how we used de Bruijn sequences to demonstrate that the bound we obtained is tight. If time permits, I’ll talk about a related generalization of Steinhaus’ 3 gap theorem. For more info, check out my math blog: gshakan.wordpress.com

Thursday, October 19, 2017

11:00 am in 241 Altgeld Hall,Thursday, October 19, 2017

A survey of tauberian theorems

Harold Diamond (UIUC Math)

Abstract: A light look at inversion theorems for Laplace transforms under various hypotheses.

2:00 pm in 241 Altgeld Hall,Thursday, October 19, 2017

Some classical applications of modular forms in number theory

Yifan Yang   [email] (National Chiao Tung University)

Abstract: In this talk, we will give a quick overview of some classical applications of modular forms in number theory, including 1. formulas for the number of representations of an integer as sums of squares, 2. a formula for arithmetic-geometric means, 3. modular forms as solutions of linear ordinary differential equations, 4. modular forms as periods, 5. irrationality of $\zeta(3)$, 6. series representations for $1/\pi$, 7. congruences of the partition function.

Thursday, October 26, 2017

11:00 am in 241 Altgeld Hall,Thursday, October 26, 2017

Pair correlation in Apollonian circle packings

Xin Zhang (UIUC Math)

Abstract: Montgomery and Dyson’s Pair Correlation Conjecture says that the pair correlation of the non-trivial zeros of the Riemann zeta function agrees with the pair correlation of the eigenvalues of a random Hermitian matrix. Despite its huge influence, a resolution of this conjecture seems far away. Pair correlations for some other deterministic sequences can otherwise be determined rigorously. In this talk, I will show that the limiting pair correlation of the circles from a fixed Apollonian circle packing exists. A key feature in our work, which differs from previous work in literature, is that the underlying point process is fractal in nature. A critical tool in our analysis is an extended version of a theorem of Mohammadi-Oh on the equidistribution of expanding horospheres in the frame bundles of infinite volume hyperbolic spaces. This work is motivated by an IGL project that I mentored in Spring 2017.

2:00 pm in 241 Altgeld Hall,Thursday, October 26, 2017

Combinatorial Proofs of Identities from Ramanujan’s Lost Notebook

Hannah Burson (UIUC)

Abstract: In his lost notebook, Ramanujan stated at least 27 identities related to the Rogers-Fine identity. In this talk, I discuss a group of 6 such identities relating to Roger's false theta functions. We give a new combinatorial interpretation and proof of one identity.

Thursday, November 2, 2017

11:00 am in 241 Altgeld Hall,Thursday, November 2, 2017

Even and Odd Minkowski Question Mark Functions

Chris Linden (UIUC Math)

Abstract: We introduce and discuss analogues of Minkowski's question mark function ?(x) related to continued fraction expansions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.

2:00 pm in 241 Altgeld Hall,Thursday, November 2, 2017

RAMANUJAN’S LOST NOTEBOOK: HISTORY AND SURVEY

Bruce Berndt   [email] (UIUC)

Abstract: In the spring of 1976, while searching through papers of the late G. N. Watson at Trinity College, Cambridge, George Andrews found a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan, generally regarded as India’s greatest mathematician. In view of the fame of Ramanujan’s earlier notebooks, Andrews naturally called these papers Ramanujan’s “lost notebook.” This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan’s life, and represents some of his deepest work. First, we provide a history of the lost notebook. Second, a general description of the contents of the lost notebook will be provided. Third, the remainder of the lecture will be devoted to a survey of some of the most interesting entries in the lost notebook. These include claims in q-series, theta functions, continued fractions, integrals, partitions, and other infinite series.

Thursday, November 9, 2017

3:00 pm in 241 Altgeld Hall,Thursday, November 9, 2017

Reed-Muller Codes Achieve Capacity on Erasure Channels

Hsin-Po Wang (UIUC)

Abstract: We will talk about this https://arxiv.org/abs/1601.04689. Reed-Muller Codes generalize Reed–Solomon codes (used on CD/DVD/etc) and Hamming codes (used on RAM/etc). For such a family of codes one may ask whether it asymptotically achieves the capacity in the sense of Shannon's information theory. The answer is yes, on certain channels, and we will go through the ideas in the paper.

Thursday, November 16, 2017

11:00 am in 241 Altgeld Hall,Thursday, November 16, 2017

Kloosterman sums and Siegel zeros

James Maynard (Institute For Advanced Study)

Abstract: Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field $F_p$, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

Tuesday, November 28, 2017

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

Polynomial methods in number theory

Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

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

Polynomial methods in number theory

Dong Dong   [email] (UIUC)

Abstract: Many recent breakthroughs in number theory, combinatorics, harmonic analysis, and PDE involve "polynomial method". What is a polynomial method? In fact, there are different types of polynomial methods. In this talk, I will introduce one simple but extremely powerful such method (other types will be introduced in subsequential talks). Knowledge of Taylor expansion (or mean value theorem) is required to attend this talk.

Thursday, November 30, 2017

11:00 am in 241 Altgeld Hall,Thursday, November 30, 2017

Non-vanishing of Dirichlet series without Euler products

Bill Banks (U Missouri-Columbia Math)

Abstract: This talk explores the question: To what extent does the Euler product expansion of the Riemann zeta function account for the non-vanishing of the Riemann zeta function in the half-plane $\{\Re(s)>1\}$? We exhibit a family of Dirichlet series that are closely related to the Riemann zeta function and are nonzero in $\{\Re(s)>1\}$, but do not possess an Euler product.

2:00 pm in 241 Altgeld Hall,Thursday, November 30, 2017

Exponential sums and the Linear Independence Conjecture

William Banks   [email] (University of Missouri)

Abstract: The Linear Independence Conjecture (LIC) for the Riemann zeta-function asserts that the positive ordinates of the nontrivial zeros of zeta(s) are linearly independent over the rationals. In this talk, I will describe some ongoing joint work with N. Ng, G. Martin and M. Milinovich in which we estimate certain exponential sums in order to obtain evidence for the truth of the LIC.

Tuesday, December 5, 2017

2:00 pm in 241 Altgeld Hall,Tuesday, December 5, 2017

Zeros of Eisenstein series

Oscar Gonzalez Pagan (UIUC)

Abstract: The zeros of the classical Eisenstein series have been well studied over the years. In 1970 it was shown by F. Rankin and Swinnerton-Dyer that the zeros lie on the boundary of the standard fundamental domain. In this talk we will discuss this result as well as some previous results and related problems. We then study the zeros of the object obtained by applying a differential operator to the classical Eisenstein series.

Thursday, December 7, 2017

11:00 am in 241 Altgeld Hall,Thursday, December 7, 2017

Real and rational systems of forms

Simon Myerson (University College London)

Abstract: Consider a system $f$ consisting of $R$ forms of degree $d$ with integral coefficients. We seek to estimate the number of solutions to $f=0$ in integers of size $B$ or less. A classic result of Birch (1962) answers this question when the number of variables is of size at least $C(d) R^2$ for some constant $C(d)$, and the zero set $f = 0$ is smooth. We reduce the number of variables needed to $C'(d)R$, and give an extension to systems of Diophantine inequalities $|f| < 1$ with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality.

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

A primer on the circle method for forms in many variables

Simon Leo Rydin Myerson (UCL)

Abstract: We give an introduction to the circle method in the form used by Birch (1962) to treat nonsingular systems of forms of the same degree. Given a suitably nice system f consisting of R forms of degree d with integral coefficients, this will give an asymptotic formula as B becomes large for the number solutions to f=0 in integers of size up to B.