Department of

# Mathematics

Seminar Calendar
for Graduate Student Number Theory Seminar events the year of Tuesday, March 13, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Thursday, February 1, 2018

2:10 pm in 241 Altgeld Hall,Thursday, February 1, 2018

#### Modular Forms and Moduli of Elliptic Curves

###### Ningchuan Zhang (UIUC)

Abstract: In this talk, I’ll explain why modular forms are global sections of the sheaf of invariant differentials over the moduli stack of elliptic curves. In the end, I’ll mention the $q$-expansion principle and integral modular forms. No knowledge of stack is assumed for this talk. Please note this talk will start 10 minutes later than the regular time.

Thursday, February 8, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 8, 2018

#### Multiplicative functions which are additive on polygonal numbers

###### Byungchan Kim (SeoulTech)

Abstract: Spiro showed that a multiplicative function which is additive on prime numbers should be the identity function. After Spiro's work, there are many variations. One direction is to investigate multiplicative functions which are additive on polygonal numbers. It is known that a multiplicative function which is additive on triangular numbers should be the identity while there are non-identity functions which is multiplicative and is additive on square numbers. In this talk, we investigate multiplicative functions which are additive on several polygonal numbers and present some open questions. This talk is based on joint works with J.-Y. Kim, C.G. Lee and P.-S. Park.

Thursday, February 15, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 15, 2018

#### p-adic families of modular forms

###### Ravi Donepudi   [email] (UIUC)

Abstract: This talk is an introduction to the theme of p-adic variation in number theory, especially concerning modular forms. We will first give a general overview of the p-adic Galois representation attached to a classical Hecke eigenform. Then we will closely study the example of Eisenstein series which are classically parametrized by integer weights and see how they can be naturally interpolated p-adically to give a family of p-adic modular forms. As a corollary, this yields a very simple construction of p-adic L-functions. Finally we will tie these two themes together to study the structure of p-adic Galois representations arising from modular forms in the larger space of all p-adic Galois representations (of a fixed Galois group), highlighting Gouvêa-Mazur’s construction of the “infinite fern of Galois representations" and Coleman-Mazur’s Eigencurve. Have I mentioned the word “p-adic” yet?

Thursday, February 22, 2018

2:00 pm in 241 Altgeld Hall,Thursday, February 22, 2018

#### Multiples of long period small element continued fractions to short period large elements continued fractions

###### Michael Oyengo (UIUC)

Abstract: We construct a class of rationals and quadratic irrationals having continued fractions whose period has length $n\geq2$, and with "small'' partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large'' partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of these continued fractions and study properties of polynomials arising from these generalizations.

Thursday, March 1, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 1, 2018

#### Spectral Theory on the Modular Surface

Abstract: Many questions about number fields can be recast as questions regarding Laplace eigenvalues on certain manifolds. In this talk I’ll discuss some ideas and results related to Selberg’s trace formula, how partial results towards Selberg’s $\frac{1}{4}$-conjecture have immediate applications, and why number theorists might care about the analysis of Laplacians to begin with.

Thursday, March 8, 2018

2:00 pm in Altgeld Hall,Thursday, March 8, 2018

#### Cancelled

Thursday, March 15, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 15, 2018

#### Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels

###### Hsin-Po Wang (UIUC)

Abstract: We will talk about https://arxiv.org/abs/0807.3917 Polar code is considered one of the best codes in the world (together with LDPC and Turbo code). Following Arikan's paper, we will define polar code from scratch and prove that it achieve capacity. If time permits, we will talk about implementation details; in particular comparing virtual channels in an engineering-friendly way. If time still permits, we will talk about how fast does it achieve capacity.

Thursday, March 29, 2018

2:00 pm in 241 Altgeld Hall,Thursday, March 29, 2018

#### Restriction estimates and their applications in number theory

###### Fernando Xuancheng Shao   [email] (University of Kentucky)

Abstract: I will survey recent developments on restriction theory for exponential sums over sets of number theoretic interest, such as primes, smooth numbers, and k-th powers, and their applications to analytic number theory and additive combinatorics, including Roth-type theorems in primes and Waring-type results in smooth k-th powers.

Thursday, April 12, 2018

2:00 pm in 241 Altgeld Hall,Thursday, April 12, 2018

#### Spectral Theory on the Modular Surface

Abstract: Many questions about number fields can be recast as questions regarding Laplace eigenvalues on certain manifolds. In this talk I’ll discuss some ideas and results related to Selberg’s trace formula, how partial results towards Selberg’s $\frac{1}{4}$-conjecture have immediate applications, and why number theorists might care about the analysis of Laplacians to begin with.