On the Riemann Penrose Inequality

Hongyi Sheng

November 13th, 2019 - 5:00 - 5:50pm - RH 510R

Abstract

On Apr. 10, 2019, astronomers captured the first image of a black hole. This was a huge progress, for once we know the area of the event horizon, we actually get an estimate of the mass of the black hole. In fact, earlier in 1973, R. Penrose conjectured that given the dominant energy condition, the total mass of a space-time which contains black holes with event horizons of total area A should be at least $\sqrt{A/16\pi}$. An important special case in Riemannian geometry is now known as the Riemannian Penrose inequality. This inequality was first established by G. Huisken and T. Ilmanen in 1997 using the inverse mean curvature flow for a single black hole and then by H. Bray in 1999 for any number of black holes, using the technique of a conformal flow. Later in 2009, H. Bray and D. Lee generalized Bray’s result to dimension up to 7. In this talk, we will mainly focus on the construction of Bray’s conformal flow in dimension 3. We will see that the area of the horizon is a constant along the flow, and that the mass is decreasing, using the well-known positive mass theorem (Schoen-Yau 79’). The convergence to the Schwarzschild metric finally gives us the inequality.

About the Speaker

Hongyi is a fourth-year PhD student interested in geometric analysis. In his spare time, he likes traveling.

Advisor and Collaborators

Hongyi's advisor is Richard Schoen.

Counting n-Arcs in Projective Planes

Kelly Isham

November 7th, 2019 - 12:30 - 12:50pm - RH 340N

Abstract

An n-arc in \PP^2(F_q) is a collection of n distinct points such that no three lie on a line. In 1988, Glynn gave an algorithm for counting the number of n-arcs in a projective plane of order q. He found that for n less than or equal to 6, the formula for the number of n-arcs is a polynomial in q and that for n=7,8, the formula is quasi-polynomial in q. In 1995, Iampolskaia et al showed that the formula for n=9 is also quasipolynomial. Recent work by Kaplan et al extended this result to arbitrary projective planes of order q. This leads to the question - will the number of n-arcs over \PP^2(F_q) continue to be quasipolynomial in q? In this talk, we discuss a modification of Glynn's algorithm that makes computation simpler and we explain how the problem of counting n-arcs in the projective plane over F_q is equivalent to counting the number of rational points on certain varieties over F_q. We use this new approach to prove that the number of 10-arcs in a projective plane over F_q is not quasipolynomial. Lastly, we discuss analagous results for larger n and we relate this counting problem to the study of F_q-points on Grassmannians and to MDS codes.

About the Speaker

Kelly is a 4th year Ph.D candidate interested in number theory. In her spare time, she enjoys playing board games and watching hockey.

Advisor and Collaborators

Kelly's advisor is Nathan Kaplan. This is joint work with Nathan Kaplan and Max Weinreich.

Characterizations of Infinitesimal Non-Crossing Bi-Free Probability

Jennifer Pi

November 7th, 2019 - 12:30 - 12:50pm - RH 340N

Abstract

Free probability is a non-commutative analogue of probability introduced in the 1980s by Voiculescu, where the notion of independence is replaced by the condition of free independence. Biane, Goodman, and Nica adapted the combinatorics to give a Type B free probability, arising from the Type B non-crossing partitions; Belinschi and Shlyakhtenko showed this notion to be equivalent to an infinitesimal free independence for a time-indexed family of states satisfying the freeness conditions to first order. More recently, Voiculescu has introduced bi-free probability, an extension of free probability to handle simultaneously left and right actions on free product spaces. After covering the necessary background, this talk will demonstrate how these extensions can be combined to give an infinitesimal bi-free probability. In particular, it will be shown that the bi-free generalizations of the combinatorial and analytic characterizations of infinitesimal free independence are equivalent.

About the Speaker

Jenny is a 1st year Ph.D student interested in probability.

Advisor and Collaborators

This is joint work with Ian Charlesworth, Zhiheng Li, Kyle P. Meyer, Drew T. Nguyen, and Annie K. Raichev.

On Klein and Fricke's Modular Solution of the Sextic

Alexander Sutherland

November 7th, 2019 - 11:30 - 11:50am - RH 340N

Abstract

Abel's theorem (1824) that the generic polynomial of degree n is solvable in radicals if and only if n = 1,2,3,4 is well-known. However, the classical work of Bring (1786) and Klein (1884) give solutions of the generic quintic polynomialby allowing the use of elliptic modular functions. In this talk, we will re-examine Klein's solution in the modern context of resolvent degree and extend the work of Klein (1905) and Fricke (1926) to solving the generic sextic polynomial.

About the Speaker

Alex is a 4th year Ph.D candidate working at the intersection of topology and algebraic geometry. He is also one of UCI's 2019 Pedagogical Fellows and a 2019 ARCS Foundation Scholar.

Advisor and Collaborators

Alex's advisor is Jesse Wolfson.

Slope Uniformity of Artin-Shrier-Witt Covers Over the Projective Line

Shichen Tang

October 28th, 2019 - 5:00 - 5:50pm - RH 440R

Abstract

Zeta function of a curve over a finite field carries the information of point count of the curve. A classical question in number theory is understanding the roots of the zeta function. Weil completely understood the archimedean absolute value of such roots, and the $\ell$-adic absolute values is known to be trivial. But the $p$-adic absolute value remains mysterious. Study of such absolute value can be reduced to study of the slopes of certain $L$-functions. In this talk I will present some known results related to patterns of $L$-functions of curves in Artin-Shreier-Witt towers, which is a tower of branched covers with Galois group $\mathbb{Z}_{p^\ell}$. Then I will discuss the general idea for the proof.

About the Speaker

Shichen is a 4th year PhD student interested in number theory. In his spare time, he enjoys watching anime.

Advisor and Collaborators

Shichen's advisor is Daqing Wan.

Hasse Polynomials of L-functions of Certain Exponential Sums

Chao Chen

October 25th, 2019 - 5:00 - 5:50pm - RH 440R

Abstract

Estimating exponential sums is a classical problem in number theory. This problem can be reduced to evaluate the reciprocal roots and zeros of the associated L-functions. In this talk, we will focus on certain exponential sums constructed by Laurent polynomials over finite field of characteristic p and study the variation of p-adic absolute values. Newton polygon is the main tool to estimate p-adic valuations. Adolphson and Sperber proved that the Newton polygon of L-function of a non-degenerate Laurent polynomial has a topological lower bound Hodge polygon. We will discuss when Newton polygon reaches Hodge polygon. Hasse polynomials are constructed to determine the coincidence of Newton polygons and Hodge polygons. We compute the higher slope Hasse polynomials and study the irreducibility of these polynomials.

About the Speaker

Chao is a 4th year PhD student interested in number theory. In her spare time, she enjoys watching anime.

Advisor and Collaborators

Chao's advisor is Daqing Wan.

Cyclic Density of Finite Groups

Adam Marks

October 17th, 2019 - 4:00 - 4:50pm - RH 340N

Abstract

We investigate the interwoven-ness of cyclic subgroups of finite groups, defining a property called the cyclic density. Certain families of groups have simple formulas for their cyclic densities, but a formula for symmetric groups remains elusive. We conjecture as to the asymptotic behavior of cyclic density of symmetric groups of increasing size and develop a way to compute the density that is moderately improved over naive brute force.

About the Speaker

Adam Marks is a first year PhD student, seeking refuge from decades of commercial software engineering and entrepreneurship. Most recently he co-founded a startup that grew into a part of Amazon Web Services.

Robust Single Neuronal Calcium Signal Extraction Allows for Longitudinal Analysis of Behavior-associated Neural Ensemble Dynamics

Kevin Johnston

October 8th, 2019 - 4:00 - 4:50pm - RH 340N

Abstract

Calcium imaging of neurons in vivo allows simultaneous recording of large numbers of neurons, which communicate and execute computations implicit in the encoding of learning and memory. However, such in vivo optical recordings are typically subject to motion artifact and background contamination from other neurons, making it difficult to define single cell spatiotemporal dynamics. Here we introduce spatial constraints imposed on the extracted spatial footprints to improve ROI selection and reduce false discovery detection. Then we combine spatiotemporal correlation across recording sessions with a predictor corrector methodology to extract single cell neuronal data from long-term multi-session imaging experiments in different brain regions including prefrontal cortex, visual cortex and hippocampus. This new methodology is particularly suitable for extracting neural signals from weeks- and months-long longitudinal recordings, as demonstrated in our simulated and multi-session in vivo experimental recordings. Application of the new method allows for robust longitudinal analysis of contextual discrimination associated neural ensemble dynamics in hippocampal CA1, which reveals how hippocampal neuronal ensembles representing two environmental contexts evolve over the course of contextual discrimination conditioning. Additionally, we combine this methodology with neural decoding to understand changes induced by modulating neural circuits.

About the Speaker

Kevin is a 4th year PhD student in math, working with several individuals in the neurobiology department. Outside of research, he likes to watch anime and cook.

Advisor and Collaborators

Kevin's advisor is Qing Nie.

Solving Polynomials After Klein: The Theory of Resolvent Degree

Alex Sutherland

May 14th, 2019 - 5:30 - 6:20pm - RH 306

Abstract

Solving polynomials is a classical problem. From Abel (1824), we know that the generic degree n polynomial cannot be solved in radicals when n is greater than 4. However, we can still solve polynoimals. Indeed, Bring (1786) provides a formula for the generic quintic using only a square root, a cube root, and an algebraic function now known as the Bring radical. Famously, there is also Klein's solution of the generic quintic (1884), which is based on the symmertires of the icosahedron. In this talk, we will introduce the theory of resolvent degree (led by Farb and Wolfson), which is the modern viewpoint on solving generic polynomials via algebraic geometry and topology. We will then examine Klein's solution of the quintic in this framework and look at what we know about extending the methods of Bring, Klein, and Green (1978) towards giving a complete solution of the generic sextic.

About the Speaker

Alex is a 3rd year Ph.D student whose research uses tools from topology, geometry, and algebra. He is also one of UCI's Pedagogical Fellows for 2019. Outside of math, he likes to go to concerts and watch college football.

Advisor and Collaborators

Alex's advisor is Jesse Wolfson.

Counting n-Arcs in Projective Planes

Kelly Isham

May 9th, 2019 - 4:00 - 4:50pm - RH 340N

Abstract

An n-arc in P^2(F_q) is a collection of n distinct points such that no three lie on a line. In 1988, Glynn gave an algorithm for counting the number of n-arcs in a projective plane of order q. Previous work has shown that for n less than 10, the number of n-arcs is polynomial or quasipolynomial (i.e. the formula is given by a finite number of polynomials depending on residue classes). This leads to the question - will the number of n-arcs over P^2(F_q) continue to be quasipolynomial in q? In this talk, we discuss a modification of Glynn's algorithm that makes computation simpler and we explain how the problem of counting the number of n-arcs in the projective plane over F_q is equivalent to counting the number of rational points on certain varieties. We use this new approach to prove that the number of 10-arcs in a projective plane over F_q is not quasipolynomial. We discuss analagous results for larger n and we relate this counting problem to the study of F_q-points on Grassmannians and to MDS codes.

About the Speaker

Kelly is a 3rd year PhD student interested in number theory. In her spare time, she enjoys playing board games and watching hockey.

Advisor and Collaborators

Kelly's advisor is Nathan Kaplan.

Effect of Synaptic Cell-to-Cell Transmission on HIV Recombination Dynamics

Jesse Kreger

March 6th, 2019 - 4:00 - 4:50pm - RH 340P

Abstract

In this talk, we investigate mathematical models regarding the evolutionary outcomes of human immunodeficiency virus (HIV), in humans. We analyze how the interplay between synaptic cell-to-cell transmission, free virus transmission, and the process of recombination affects the dynamics of an infection taking place. We first consider non-spatial models that take into account multiplicity of infection, co-infection, and competition between virus strains. We then introduce a novel agent-based model that takes into account the spatial nature of cell-to-cell transmission. We show that a combination of both free virus transmission and cell-to-cell transmission minimizes the time to a double hit mutant virus formation. We then analyze the growth and robustness of the double hit mutant virus population in the context of many different fitness landscapes and recombination rates.

About the Speaker

Jesse is a 3rd year PhD student studying mathematical biology.

Advisor and Collaborators

Jesse's advisors are Natalia Komarova and Dominik Wodarz.

Symplectic Structures on an Open 4-Manifold with Non-Isomorphic Primitive Cohomology

Matt Gibson

January 30th, 2019 - 4:30 - 5:30pm - RH 340N

Abstract

In this talk, I will introduce a useful cohomology defined on sympelctic manifolds and show how its dimension alone can distinguish between an example of "inequivalent" surface bundles over the torus.

About the Speaker

Matt is a 6th year grad student studying geometry/algebraic topology.

Advisor and Collaborators

Matt's advisor is Li-Sheng Tseng.

Mathematical Models of Cancer Evolution

Jesse Kreger

January 14th, 2019 - 4:00 - 4:50pm - RH 510R

Abstract

In this talk, we investigate mathematical models regarding the evolutionary dynamics of cancer cells. These models are based on the Moran process, which is a simple birth-death stochastic process used in biology to describe constant populations with competing healthy and mutant cell types. We consider models with variations on the basic Moran process. These variations include migration and exchange between many connected regions, which represent different locations and tissues in the body. In the context of multiple related models, we implement analytical methods and stochastic simulations in order to assess the probability of cancerous mutant fixation as well as the mean conditional time until such fixation occurs. These results can improve understanding of cancer-specific evolutionary dynamics and can help predict the progression of cancer through the body.

About the Speaker

Jesse is a 3rd year PhD student studying mathematical biology.

Advisor and Collaborators

Jesse's advisors are Natalia Komarova and Dominik Wodarz.