Allechar Serrano López

University of Utah Mathematics Department
Email serrano at math dot utah dot edu
Office 328 JWB

I am a fourth-year graduate student at the University of Utah, supervised by Stefan Patrikis. I was born and raised in Costa Rica, where I completed bachelor's degrees in Mathematics and Economics at University of Costa Rica.


Before coming to Utah, I was an instructor at University of Costa Rica and worked for the research department of Central Bank of Costa Rica.

My last name is "Serrano López". It is composed of two words without a hyphen and it should be indexed by the letter "S".


Teaching


Summer 2018: Counselor for the Summer Mathematics Program for High School Students
                          Instructor for Eccles Beginnings Summer Bridge Program
Spring 2018: Math 1090, Business Algebra Online
Fall 2017: Math 1220, Calculus II
Summer 2017: Counselor for the Summer Mathematics Program for High School Students
Spring 2017: Math 1100, Business Calculus
Fall 2016: Math 1100, Business Calculus
Spring 2016: TA for Math 1210, Calculus I
Fall 2015: TA for Math 2250, Differential Equations and Linear Algebra

Outreach and Other Interests


I am very passionate about the promotion of women in mathematics and the participation of minorities in science.

Summer Mathematics Program for High School Students

I was a counselor for this four-week program in number theory for high school students in 2018 and 2017.



Latinos in Action at the University of Utah

I have been a member of LIA University of Utah Chapter and served as Academic co-Chair since 2015.


Eccles Beginnings
Summer Bridge Program

I was the instructor of the math portion of this four-week program. It consisted of a review of high school math and it was aimed at underrepresented groups.


AWM Research Symposium

The symposium will be held on April 6-7, 2019 at Rice University.


Infinite Possibilities Workshop

This is a conference for women of color in mathematics. I attended in April, 2018.

BAGELS


BAGELS (B(e creative!) Algebraic Geometry Eating and Learning Seminar) meets on Monday, from 3:15 p.m. to 4:15 p.m., in 308 JWB. If you have any questions, please contact Allechar.


Schedule


Spring 2019



January 7

Organizational Meeting


January 14


A unipotent circle action on p-adic modular forms
Speaker: Sean Howe

Abstract: The horizontal translation action of the real line on the complex upper half plane descends to an action of the circle group S^1 on the "unstable locus", or image of Im \tau > 1, in the complex modular curve. In this talk, we explain an analogous action of the open p-adic unit disk centered at 1 on the Katz moduli space, a p-adic analytic covering space of the unstable locus on the p-adic modular curve whose ring of functions can be constructed by p-adically interpolating the coefficients of classical modular forms. The analogy is richer than one might first expect, and leads to new perspectives on classical notions in the p-adic theory of modular curves and modular forms such as Dwork's equation \tau=\log q and Hida's space of ordinary p-adic modular forms, with implications for the p-adic representation theory of GL_2 and p-adic Galois groups.



January 21

Martin Luther King Jr. Day
(no seminar)


January 28


Mukai, McKay, m'kay
Speaker: Franco Rota

Abstract: What do representations of subgroups of SL(2,C), chains of rational curves and quiver representations have in common? But a Dynkin diagram of course! (and, a triangulated category...) We'll talk about the classical McKay correspondence, and Bridgeland, King and Reid's modern take on it via Fourier-Mukai equivalences. This will be an excuse to present some interesting techniques in the study of derived categories. I'll follow parts of this survey by Craw.



February 4


Speaker: You-Cheng Chou



February 11


Speaker: Ziwen Zhu



February 18

Presidents' Day
(no seminar)


February 25


Speaker: Erjuan Fu



March 4


Speaker: Stefano Filipazzi



March 11

Spring Break
(no seminar)


March 18


Speaker: Dapeng Mu



March 25


Speaker: José Yañez


Fall 2018



August 23


Complements on surfaces and BAB
Speaker: Stefano Filipazzi

Abstract: Recently Birkar proved the famous BAB conjecture. A main tool in the proof is the theory of complements. In this talk, I will overview the notions involved in the formulation of the BAB conjecture and introduce what complements are. Then, I will focus on the case of surfaces, and highlight how complements are used in the proof of the conjecture.

Reference: https://www.dpmms.cam.ac.uk/~cb496/surfcomp.pdf



August 30


Strange duality for K3 surfaces via wall-hitting
Speaker: Huachen Chen

Abstract: We will first introduce a problem known as strange duality for moduli of sheaves on surfaces, then bring in moduli of complexes and wall-crossing, and explain how they can help to understand strange dualityfor K3 surfaces.



September 6


Matrix factorizations and Orlov's theorem
Speaker: Marin Petkovic

Abstract: The purpose of this talk is to give a brief introduction to matrix factorizations. We show some examples and sketch the construction of the triangulated category of curved dg-sheaves. We also state Orlov's theorem on hypersurfaces and give examples.



September 13


Minimal log discrepancies
Speaker: Joaquín Moraga

Abstract: We will introduce the minimal log discrepancy, which is an invariant to measure the singularities of an algebraic variety. This object is a central invariant in the study of birational geometry. Then we will discuss some conjectures and known results.



September 20


Torelli theorems
Speaker: Christian Klevdal

Abstract: How much information can you determine about an algebraic variety from its cohomology? In certain cases, quite a bit! In this talk, we will prove the classical Torelli theorem which states that a complex algebraic curve is determined up to isomorphism by its second cohomology. A key player is the Jacobian, which I will introduce along the way. If time permits, we will discuss generalizations of the Torelli theorem to complex K3 surfaces, hyperkahler varieties and possibly generalizations to other number fields. I'll try to keep the prerequisites for this talk low, basic knowledge of complex algebraic curves should be plenty for the bulk of the talk.



September 27


First homology group of a closed Riemann surface
Speaker: Erjuan Fu

Abstract: In "primitive cohomology and the tube mapping", Schnell constructs the generators of the rational first homology group of a closed Riemann surface using all invariant vanishing cycles under the monodromy action. We will construct the generators using the tube classes over only finitely many vanishing cycles. We will see that the generator we constructed is the same as the difference of two cones over the same vanishing cycles, that is to say, we glue two thimbles with the same boundary along the same boundary.



October 4


The average rank of elliptic curves
Speaker: Allechar Serrano López

Abstract: In 1979, Goldfeld conjectured that the average rank of elliptic curves over the rational numbers is 1/2. Theoretical results towards boundedness of the average rank were proven, assuming the Generalized Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. However, the data did not support these results. In a joint work with Arul Shankar, Manjul Bhargava provided an unconditional proof of boundedness by studying the n-Selmer group. In this talk, I will give an outline of their proof.



October 11

Fall Break
(no seminar)


October 18


Spectral sequences
Speaker: Matteo Altavilla

Abstract: Spectral sequences were developed in the 60s and since then they’ve become a central tool in algebra and geometry. We will review the definition and basic constructions, and the philosophy behind their application. I’ll then list a few examples and work out a very simple one in detail.



October 25


Frobenius Seshadri Constants
Speaker: Yen-An Chen

Abstract: In this talk, I will introduce a Frobenius variant of Seshadri constant defined by M. Mustata and K. Schwede, and then show that the lower bounds imply the global generation or very ampleness of the corresponding adjoint line bundle (in positive characteristic).



November 1


Reduction mod p
Speaker: Daniel Smolkin

Abstract: We'll discuss how to reduce algebraic varieties in characteristic 0 modulo a prime p. Many problems are easier to solve in prime characteristic using the techniques of F-singularities, and solving a problem modulo all "sufficiently general" primes p often yields the solution in characteristic 0. We will discuss how this strategy can be used to show invariant rings of linearly reductive group actions are Cohen-Macaulay, as well as some containment results on symbolic powers.



November 8


Gromov-Witten Invariants
Speaker: You-Cheng Chou

Abstract: I will introduce what is Gromov-Witten invariants and sketch three approaches to study it. They are related to localization formula, integrable system, and Frobenius manifold separately. I will put more effort on the approach related to Frobenius manifold. If time permitted, I will explain how to write higher genus Gromov-Witten invariants in terms of invariants on Frobenius manifold.



November 15


Kawamata-Viehweg vanishing
Speaker: José Yañez

Abstract: In this talk I will discuss Kawamata-Viehweg vanishing theorem, which gives in characteristic 0 a vanishing in cohomology involving the canonical divisor, and a nef and big divisor. I will also show a counter-example in positive characteristic, along with some partial result in the same context.



November 22

Thanksgiving Break
(no seminar)


November 29


Bridgeland stability and mirror symmetry for orbifold elliptic quotients
Speaker: Franco Rota

Abstract: Mirror symmetry predicts a relation between the stability manifold of some smooth variety (stack?) X and the moduli space of its mirror. We'll explore this principle in the case where X is a certain quotient of an elliptic curve. The mirror family is parameterized by the universal unfolding of a singularity, whose geometry is regulated by an extended affine root system.



December 6


Ruled surfaces
Speaker: Seungsu Lee

Abstract: We will discuss the basic properties of ruled surfaces. Namely, we will see what is the invariant of the given ruled surfaces and how the divisors on the surface could be determined by the base curve and the fiber.




Past seminars


Spring 2018
Fall 2017
Spring 2017
Fall 2016
Spring 2016
Fall 2015