Seminars and Colloquia by Series

Crossing the transcendental divide: from translation surfaces to algebraic curves

Series
Algebra Seminar
Time
Monday, February 27, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Yelena MandelshtamUC Berkeley

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

TBD

Series
Algebra Seminar
Time
Monday, January 23, 2023 - 10:20 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Dustin CartwrightUniversity of Tennessee

TBD

Circuits, p-adic Root Counting, and Complexity

Series
Algebra Seminar
Time
Monday, December 5, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
J. Maurice RojasTAMU

 Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.

        We present an analogous implication involving p-adic valuations:    
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.

Lattices on shuffle words

Series
Algebra Seminar
Time
Monday, November 28, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Thomas McConvilleKennesaw State University

The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle. 

Algebraic and combinatorial problems arising from maximum likelihood estimation using small datasets

Series
Algebra Seminar
Time
Monday, November 21, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Daniel Irving BernsteinTulane University Department of Mathematics

Loosely speaking, the maximum likelihood threshold of a statistical model is the fewest number of data points needed to fit the model using maximum likelihood estimation. In this talk, I will discuss combinatorial and algebraic-geometric approaches to studying this poorly understood quantity for a certain class of Gaussian models. This is based on joint work with Sean Dewar, Steven Gortler, Tony Nixon, Meera Sitharam, and Louis Theran

Cohomology of moduli spaces of curves

Series
Algebra Seminar
Time
Monday, November 14, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Sam PayneThe University of Texas, Austin

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  


Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

Cohomology of moduli spaces of curves

Series
Algebra Seminar
Time
Monday, November 14, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Sam PayneThe University of Texas, Austin

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra.  Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear.  In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions.  These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.  


Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher. 

Coinvariants and superspace

Series
Algebra Seminar
Time
Monday, November 7, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Andy WilsonKennesaw State University

The ring of multivariate polynomials carries a natural action of the symmetric group. Quotienting by the ideal generated by the polynomials which are invariant under this action yields the "coinvariant algebra," an object with many beautiful algebraic and combinatorial properties. We will survey these properties and then discuss recent generalizations where the multivariate polynomials may contain anti-commuting ("superspace") variables. This talk is based on joint work with Brendon Rhoades.

Extremal Combinatorics, Real Algebraic Geometry and Undecidability

Series
Algebra Seminar
Time
Monday, October 31, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Greg BlekhermanGeorgia Institute of Technology

I will highlight recent interplay between problems in extremal combinatorics and real algebraic geometry. This sheds a new light on undecidability of graph homomorphism density inequalities in extremal combinatorics, trace inequalities in linear algebra, and symmetric polynomial inequalities in real algebraic geometry. All of the necessary notions will be introduced in the talk. Joint work with Jose Acevedo, Sebastian Debus and Cordian Riener.

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