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".

**Summer 2019:** Math 2200, Discrete Math

TA for the RTG Pre-REU in Number Theory

Facilitator for the Math Instructor Training Workshop

**Spring 2019:** TA for Math 1310, Engineering Calculus I

**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

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

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

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

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.

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

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

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.

**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.

**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.

**Abstract:** In this talk, I will introduce the finite monodromy problem for
ODE and specialized to the integral Lame equation. Then I will state Dahmen's conjecture, which predicts the number of integral Lame equation
with finite monodromy. Finally, I will explain how we reduce the problem to the zero counting of the SL(2,Z) modular form.
The modular form computation will be the special case of Appendix A in here.

**Abstract:** Birational superrigidity was introduced as a measurement of nonrationality
when people are studying rationality problems of Fano hypersurfaces in the projective spaces. On the other hand, the notion of K-stability came up
in the study of existence of KE metrics on Fano manifolds, and in some recent studies, it also plays an important role in the study of Moduli problems
of Fano varieties. The two notions seem to be very different according to their original definitions, however when we focus on Fano varieties of Picard
number 1, they are both related to log maximal singularities where we examine the singularities of certain anti-canonical systems. In this talk, I will
first define all these concepts, and then survey some results in the direction of finding the relation between birational superrigidity and K-stability.

**Abstract:** Lefschetz (1,1) Theorem is just the Hodge conjecture for Hodge class of degree 2.
In this talk, I will introduce the technique of Lefschetz's orginal proof using Poincar\'e normal functions. By taking hyperplane sections, Lefschetz's
original method would provide a sort of inductive proof of Hodge conjecture, but the Abel-Jacobi mapping is rarely surjective in higher dimensions which
is the main technical obstruction to a complete generalization of Lefschetz's method. This induces the Griffiths' program approaching to the Hodge conjecture.

**Abstract:** In this talk I will discuss what it means for a set of varieties to be bounded.
Roughly speaking, we want to give a proper algebraic notion of "being described by a finite set of parameters". I will present a few examples,
together with some techniques to address boundedness questions.

**Abstract:** In this talk, I will first introduce some notations of stability conditions.
Along the way I will define the Euler stability which was constructed by A.Bertram. We conjectured that for any class, the Euler stability will
be eventually equal to the Gieseker stability. As an example, I will show the wall-crossings for the class of the structure sheaf of the twisted
cubic curve, and the conjecture turns out to be true in this case.

**Abstract:** Multiplier ideals are ideal sheaves that are closely related to measuring
singularities of divisors. In this talk, I will give a formal definition of multiplier ideals, along with some properties and applications.

**Abstract:** In this talk we will introduce the Grothendieck ring of varieties,
prove some classical properties, and explore an alternative definition due to Bittner. As an application, we will sketch a proof related to a
criterion of the rationality of curves and the rationality of the Weil Zeta function.

**Abstract:** Let X be a quasi-elliptic surfaces and A any ample line bundle on X. I will show
that K_X+kA is base point free for k at least 3 and it is very ample for k at least 4.

**Abstract:** Generalized pair comes naturally from Kodaira canonical bundle formula. Suppose
(X,B+M) is a generalized pair, if X itself has klt singularity, birkar proved the cone theorem for K_x+B+M. But if X is lc, the existence of contraction
morphism is still not known. I proved under some good condition, the contraction morphism exists if X has lc singularity.

**Abstract:** This talk consists of two parts. First of all, we will discuss the behavior of multiplier
ideals in char p reduction, which coincides with the test ideals for p>>0. Furthermore, one can recall that Kawamata-Viehweg vanishing theorem is not in general
true for positive characteristic. In this talk, we will prove that we still have the vanishing theorem for globally F-regular varieties.

**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

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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).

**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.

**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.

**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.

**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.

**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.

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015