Department of

# Mathematics

Seminar Calendar
for Number Theory Seminar events the year of Tuesday, February 19, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2019          February 2019            March 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5                   1  2                   1  2
6  7  8  9 10 11 12    3  4  5  6  7  8  9    3  4  5  6  7  8  9
13 14 15 16 17 18 19   10 11 12 13 14 15 16   10 11 12 13 14 15 16
20 21 22 23 24 25 26   17 18 19 20 21 22 23   17 18 19 20 21 22 23
27 28 29 30 31         24 25 26 27 28         24 25 26 27 28 29 30
31


Thursday, January 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 17, 2019

#### What is Carmichael's totient conjecture?

###### Kevin Ford (Illinois Math)

Abstract: A recent DriveTime commercial features a mathematician at a blackboard supposedly solving "Carmichael's totient conjecture". This is a real problem concerning Euler's $\phi$-function, and remains unsolved, despite the claim made in the ad. We will describe the history of the conjecture and what has been done to try to solve it.

Thursday, January 24, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 24, 2019

#### Some statistics of the Euler phi function

###### Harold Diamond (Illinois Math)

Abstract: Questions about the distribution of value of the Euler phi function date to work of Schoenberg and Erdos. This talk will survey this theme and include a result of mine in which two applications of the Perron inversion formula are applied to count the number of points (n, phi(n)) lying in a specified rectangle.

Thursday, January 31, 2019

11:00 am in 241 Altgeld Hall,Thursday, January 31, 2019

#### Monodromy for some rank two Galois representations over CM fields

###### Patrick Allen (Illinois Math)

Abstract: In the automorphic-to-Galois direction of Langlands reciprocity, one aims to construct a Galois representation whose Frobenius eigenvalues are determined by the Hecke eigenvalues at unramified places. It is natural to ask what happens at the ramified places, a problem known a local-global compatibility. Varma proved that the p-adic Galois representations constructed by Harris-Lan-Taylor-Thorne satisfy local-global compatibility at all places away from p, up to the so-called monodromy operator. Using recently developed automorphy lifting theorems and a strategy of Luu, we prove the existence of the monodromy operator for some of these Galois representations in rank two. This is joint work with James Newton.

Thursday, February 21, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 21, 2019

#### Prime number models, large gaps, prime tuples and the square-root sieve

###### Kevin Ford (Illinois Math)

Abstract: We introduce a new probabilistic model for primes, which we believe is a better predictor for large gaps than the models of Cramer and Granville. We also make strong connections between our model, prime k-tuple counts, large gaps and the "square-root sieve". In particular, our model makes a prediction about large prime gaps that may contradict the models of Cramer and Granville, depending on the tightness of a certain sieve estimate. This is joint work with Bill Banks and Terence Tao.

Thursday, February 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, February 28, 2019

#### Using q-analogues to transform singularities

###### Kenneth Stolarsky (Illinois Math)

Abstract: This is a mostly elementary talk about polynomials and their q-analogues, filled with conjectures based on numerical evidence. For example, if ( x - 1 ) ^ 4 is replaced by a q-analogue, what happens to the root at x = 1 ? These investigations accidentally answer a question posed by J. Browkin about products of roots that was also answered by Schinzel some decades ago. We also look at how certain q-analogues are related to each other.

Thursday, March 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 7, 2019

#### Diophantine problems and a p-adic period map

###### Brian Lawrence (University of Chicago)

Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. I'll start with a discussion of cohomology theories in algebraic geometry, and build from there. The paper is joint with Akshay Venkatesh.

Thursday, March 14, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 14, 2019

#### Extremal primes for elliptic curves without complex multiplication

###### Ayla Gafni (Rochester Math)

Abstract: Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

Thursday, March 28, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 28, 2019

#### Core partitions, Numerical semigroups, and Polytopes

###### Hayan Nam (University of California at Irvine)

Abstract: A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. Numerical semigroups are additive monoids that have finite complements, and they are closely related to core partitions. The first half of the talk, we will talk about an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions. In the second half, we discuss the relationship between numerical semigroups and core partitions, along with how to count numerical semigroups with certain restrictions.

2:00 pm in 241 Altgeld Hall,Thursday, March 28, 2019

#### Joint Shapes of Quartic Fields and Their Cubic Resolvents

###### Piper Harron (University of Hawaii)

Abstract: In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava's parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What indeed! Please stay tuned. (Joint with Christelle Vincent)

Thursday, April 4, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 4, 2019

#### Low-lying zeros of Dirichlet L-functions

###### Kyle Pratt (Illinois Math)

Abstract: I will present work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.

Thursday, April 11, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 11, 2019

#### Vanishing of Hyperelliptic L-functions at the Central Point

###### Wanlin Li (Wisconsin Math)

Abstract: We study the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. In the first half of my talk, I will give a lower bound on the number of such characters through a geometric interpretation. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. In the second half of the talk, I will discuss joint work with Ellenberg and Shusterman proving as the size of the constant field grows to infinity, the set of L-functions vanishing at the central point has 0 density.

2:00 pm in 241 Altgeld Hall,Thursday, April 11, 2019

#### Conversations on the exceptional character

Abstract: We will spend the last few weeks of the semester discussing Landau-Siegel zeros. In particular, we will be discussing Henryk Iwaniec's survey article "Conversations on the exceptional character."

Thursday, April 25, 2019

11:00 am in 241 Altgeld Hall,Thursday, April 25, 2019

#### Local models for potentially crystalline deformation rings and the Breuil-Mézard conjecture

###### Stefano Morra (Paris 8)

Abstract: Available at https://faculty.math.illinois.edu/~pballen/stefano-morra-abstract.pdf

Thursday, May 2, 2019

11:00 am in 241 Altgeld Hall,Thursday, May 2, 2019