Seminars & Colloquia
The UCSC Mathematics Department hosts several seminars or colloquia during Fall, Winter, and Spring quarters. Each is a series of talks by invited speakers, usually on research. Talks are usually 4pm-5pm, in McHenry 4130, with an informal tea ~30m beforehand in 4161. Series typically meet once a week; different colloquia are on different days of the week. There may be gaps in the series.
2024-5
- MONDAYS : Graduate Colloquium (Fernandes, Howard, Price)
- TUESDAYS Zoom : Experiences In Math (Bhamidipati, Guerrero, Montiero)
- TUESDAYS : Undergraduate Colloquium (Bauerle, Immel, Williams)
- WEDNESDAYS : Geometry & Analysis Seminar (Monard, Quan)
- THURSDAYS : Mathematics Colloquium (Pan)
- FRIDAYS : Algebra and Number Theory Seminar (Kass, Mackall)
Upcoming
| Thurs, Jan 16 @ 4pm | COLLOQ : Hannah Larson, UC Berkeley | Moduli spaces of curves |
The moduli space M_g of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of M_g is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of M_g called the tautological ring. The definition of the tautological ring was later extended to the compactification M_g-bar and the moduli spaces with marked points M_{g,n}-bar. While the full cohomology ring of M_{g,n}-bar is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll discuss several results about the cohomology groups of M_{g,n}-bar, particularly regarding when they are tautological or not. This is joint work with Samir Canning, Sam Payne, and Thomas Willwacher.
| Thurs, Jan 30 @ 4pm | COLLOQ : Ciprian Manolescu, Stanford | Khovanov homology and four-manifolds |
Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots of a different kind: its construction is combinatorial, and connected to ideas from representation theory. Nevertheless, it turns out that it can act as a substitute for gauge theory in some cases. I will survey some topological applications of Khovanov homology. In particular, Morrison, Walker and Wedrich used Khovanov homology to define a new invariant of four-manifolds, called the skein lasagna module. I will discuss joint work with Neithalath, and with Walker and Wedrich, in which we developed computational techniques for the skein lasagna module. These techniques were recently used by Ren and Willis to detect exotic smooth structures on four-manifolds with boundary.
Thurs, Mar 6 @ 4pm. COLLOQ : Zeno Huang, CUNY. TBD
Older
| Fri, Dec 6 @ 4pm | ANT : Noah Olander, UC Berkeley | Henselian pairs and weakly étale morphisms |
The class of weakly étale morphisms of schemes is used by Bhatt and Scholze to define the pro-étale site of a scheme. We will review this notion, and propose a new definition of weakly étale morphism which is analogous to the characterization of étale morphisms via a lifting property. We will use a result of Gabber on the cohomology of Henselian pairs to deduce the equivalence of the two definitions. If time permits, we will discuss an example of a weakly étale morphism which does not lift along a surjective ring map. This is joint work with Johan de Jong.
| Thur, Dec 5 @ 4pm | COLLOQ : Bo Guan, OSU | Symmetry, concavity and subsolutions in geometric PDEs |
In this talk we discuss the roles of symmetry, concavity, and subsolutions in the theory of fully nonlinear elliptic and parabolic equations, with connection to geometric problems, and how these roles guide us to a new class of equations which we hope would be useful in the study of problems from complex geometry. We also explore the possibility to weaken or extend these conditions. The talk is based in part on joint work with my collaborators.
| Wed, Dec 4 @ 4pm | G&A : Yuanqi Wang, University of Kansas | ALG Ricci Flat Kähler 3-folds with Schwartz Decay |
ALG gravitational instantons are complete hyper-Kähler surfaces asymptotic to a twisted product of the complex plane and an elliptic curve. Following the classical work of Tian-Yau and Hein, etc., on Monge-Ampere methods for Ricci flat Kähler metrics on quasi-projective varieties, we provide a geometric existence for generalized ALG Ricci flat Kähler 3-folds on crepant resolutions, where the K3 fiber admits a purely non-symplectic automorphism of finite order. These metrics decay to the ALG model at any polynomial rate, and the dimension of the tangent cone at infinity is still 2. A local Künneth formula plays a role in the Schwartz decay and an ansatz that equals a Ricci flat ALG model outside a compact set. The Schwartz decay is due to a non-concentration of the Newtonian potential.
Tues, Dec 3, 4pm, Undergraduate Colloquium : Math Chats
Tues, Dec 3, 3pm, Zoom, EXPERIENCES
Angela Robinson, NIST (National Institute of Standards and Technology).
Finding Allies and Understanding Adversaries
The daunting experience of being one of the only women in a classroom was not felt by Dr. Angela Robinson until she entered university as a mathematics major. This discomfort of being the only freshman in her Calculus III class was only the first of a decade of identity challenges she would face on her path to a mathematics PhD degree. In this talk, Dr. Robinson will take us through her mathematical journey. She will share how she found allies, mentors, and the inspiration to persist. She will discuss her research in the mathematics of cryptography and how this work impacts our daily, digital lives.
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting ID: 924 7927 7522
Passcode: mathexp
Mon, Dec 2, 4pm, GRAD : CANCELLED
Thurs and Fri, Nov 28 and 29, Thanksgiving holiday (UCSC closed)
Wed, Nov 27, 4pm, G&A : NO TALK
Tues, Nov 26, 4pm, Undergraduate Colloquium : N/A
Mon, Nov 25, 4pm, GRAD : Tsz Yau Tin, UCSC.
Two-dimensional gravity and topological strings
In 1992, Maxim Kontsevich proved the equivalence of two seemingly different theories of 2D gravity. One arises from large-N limits of discretized Riemann surfaces; the other from the intersection theory of tautological classes on the moduli space of curves. We sketch a proof along Kontsevich’s lines. We introduce the topological sigma model, which probes the geometry of a manifold through strings propagating on it. Coupling this model to gravity leads to the general notion of gravitational descendant Gromov-Witten invariants. While the underlying moduli space of stable maps is generally ill-behaved, we will explain how Calabi-Yau threefolds (CY3) stand out as an exception. Specializing on Kähler CY3s, a Lefschetz SU(2)-action on their cohomology rings allows us to express the Gromov-Witten potential using a novel kind of enumerative invariants due to Gopakumar and Vafa. This talk aims to showcase the significant ways physics influences the forefront of mathematical research.
Fri, Nov 22, 4pm, ANT : Maneesha Ampagouni, UCSC. TBD
1-point functions for $\mathbb{Z}_2$-orbifolds of lattice VOAs
The Moonshine module, a vertex operator algebra (VOA) linking the Monster group to modular forms via the j-function, has inspired significant research into VOAs and their modular properties. One way to explore and understand the modular properties of VOAs is through their 1-point functions. In this talk, I will present my work on computing the 1-point functions for the $\mathbb{Z}_2$-orbifolds of lattice VOAs. I will begin by introducing the standard Zhu theory, followed by an explanation of how Mason and Mertens developed the $\mathbb{Z}_2$-twisted Zhu theory to compute the 1-point functions in the untwisted sector for symmetrized Heisenberg and lattice VOAs. I will then show how I extended these techniques to perform computations in the twisted sector. This allowed me to derive the 1-point functions for the $\mathbb{Z}_2$-orbifold, which, as expected, are level-one modular forms for the full modular group $SL_2(\mathbb{Z})$, up to a character.
| Thurs, Nov 21 @ 4pm | COLLOQ : Piotr Chruściel, University of Vienna | Mathematical Relativity, this and that |
I will review the landscape of mathematical general relativity, pointing out the main open problems and presenting recent progress.
Wed, Nov 20 @ 4pm G&A : Tang-Kai Lee, MIT Non-uniqueness of mean curvature flow
The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. With Xinrui Zhao, we show that even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Our examples extend to higher dimensions, complementing the surface examples obtained by Ilmanen and White. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish a generalized avoidance principle. We prove that level set flows satisfy this principle in the absence of non-uniqueness.
Tues, Nov 19, 4pm, Undergraduate Colloquium : Math Chats
Tues, Nov 19, 3pm, Zoom, EXPERIENCES
Priyam Patel, University of Utah. [website]
There and Back Again: My Journey in Mathematics
In this talk, I will share my personal journey as a woman of color in mathematics, exploring how my upbringing and cultural background have influenced my path. From the early challenges of balancing my identity with academic aspirations to navigating a landscape that often feels unwelcoming, I will reflect on the obstacles I have faced and tools I have developed for overcoming them. Through anecdotes and insights, I will highlight the importance of representation, community support, and self-advocacy in overcoming barriers.
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting ID: 924 7927 7522
Passcode: 4968783
| Mon, Nov 18 @ 4pm | GRAD : Maneesha Ampagouni | \(\mathbb{Z}_2\)-orbifold of a Lattice VOA |
A well-known example of a vertex operator algebra (VOA) is the Moonshine module, whose connection to the j-function and the Monster group revealed a surprising link between group theory and modular forms. The construction of the Moonshine module can be generalized by replacing the Leech lattice with any even lattice, resulting in the \(\mathbb{Z}_2\)-orbifold of a lattice VOA, which still exhibits modular invariance. In this talk, I will introduce VOAs and the construction of a \(\mathbb{Z}_2\)-orbifold of a lattice VOA. The goal of this colloquium is to lay the groundwork for understanding the computation of 1-point functions in such an orbifold, which will be discussed in detail in the Algebra and Number Theory seminar later this week.
Fri, Nov 15, 4pm. ANT : Rose Lopez, UC Berkeley.
Picard Groups of Stacky Curves
In Mumford's famous 1965 paper, Picard Groups of Moduli Problems, he computed the Picard group of the moduli stack of elliptic curves, a stacky curve over the affine j-line. While Mumford relies heavily on the moduli description, much can be said about the Picard group of a stacky curve by understanding its geometry alone. In this talk, I will give and explain an exact sequence which relates the Picard group of a stacky curve to the Picard group of its coarse space, and the gerbe class over its rigidification.
Thur, Nov 14, 4pm, COLLOQ : Zhongshan An, University of Michigan.
Initial boundary value problems for the vacuum Einstein equations
In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the fundamental works of Choquet-Bruhat. But the initial boundary value problem (IBVP), where one needs to impose geometric conditions on the timelike boundary, has been much less understood. Due to gauge issues occurring near the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss the existence and uniqueness of solutions to the IBVP, with respect to various boundary conditions and their related quasi-local Hamiltonian. This is based on joint works with Michael Anderson.
Wed, Nov 13, 4pm, G&A : Lu Wang, Yale University.
Self-Expander of Mean Curvature Flow, and Applications
Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it emerges from a cone singularity. The nonuniqueness of self-expanders presents challenges in the study of cone-like singularities in the flow. In this talk, I will discuss some recent development on a variational theory for self-expanders, and an application to the question on lower density bounds for minimal cones.
| Tues, Nov 12 @ 4pm | Ugrad: Ben Fisher | Presentation on 4+1 BA/MA Program |
Ben Fisher (Math's undergraduate advisor) will be hosting an info session about the Math 4+1 program. 4+1 allows students in the Mathematics BS program to finish a Master's in Mathematics in one additional year. Pizza and drinks will be provided.
| Fri, Nov 8 @ 4pm | ANT : Eoin Mackall, UCSC | Quadratic pairs on Azumaya algebras and canonical extension classes |
Twists of nonsingular quadrics over a field are in correspondence with quadratic pairs on central simple algebras. Calmes and Fasel have given a globalized version of this correspondence, introducing the notion of a quadratic pair on an Azumaya algebra. Gille, Neher, and Ruether asked if every orthogonal involution on an Azumaya algebra was part of an associated quadratic pair; they showed that even when this is locally the case, there may be strong and weak cohomological obstructions for the existence of a global quadratic pair. In this talk, we introduce an intermediate obstruction to the strong and weak obstructions of Gille, Neher, and Ruether which exists only in the characteristic 2 setting. We then revisit the examples of Gille, Neher, and Ruether and show how these obstructions convey information on the underlying module structure of the associated Azumaya algebras.
| Thur, Nov 7 @ 4pm | COLLOQ : Peter Topping, U of Warwick | Curve shortening flow - old and new |
The curve shortening flow evolves an embedded curve in the plane (say) in order to reduce its length as quickly as possible. It is simple to visualize for non-experts, and yet hosts a beautiful and often surprising theory. It is a great model for other geometric flows such as Ricci flow, and yet is useful in applications in its own right. I plan to survey some of what is known, focusing on very recent work. The talk should be accessible to a general mathematician.
| Wed, Nov 6 @ 4pm | G&A : Guilherme Silva, USP-ICMC | Tail estimates for the stochastic six-vertex model |
We will talk about the result of a project we started many years ago. But explaining the embarrassment of our failure to find such estimates for more than 6 years, and what we learned in the process, is the main goal of our talk. The tail estimates we will discuss are intimately connected with conditional thinning ensembles from random matrix theory, and nonlocal versions of the so-called Painlevé equations. The unraveling of such connections was a parallel task that we had to overcome, together with many other colleagues, and led to interesting phenomena that we hadn't anticipated with our original question. During our talk, we will survey such connections, both old and new, highlighting how they all fitted together to explain the problem in the title. The talk is based on joint works with Promit Ghosal (University of Chicago).
Tues, Nov 5, 4pm, Undergraduate Colloquium : Math Chats : Mike Williams
Tues, Nov 5, 3pm, Zoom, EXPERIENCES. Omayra Ortega, Sonoma State.
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting : 924 7927 7522
Pwd: mathexp
The Mathematics of Mathematics (#MetaMath): An Introduction and Some Examples
We present examples of the application of quantitative techniques, tools, and topics from mathematics and data science to the mathematics community itself. Using research and data about Ph.D.-granting institutions in the United States first published by Wapman, Zhang, Larremore and Clauset (Nature, 2022), we quantify, document, and highlight inequity in departments at U.S. institutions of higher education producing Ph.D’s in the mathematical sciences. We introduce the terms “#MetaMath” and “the mathematics of Mathematics” for this project, explicitly building upon the growing, interdisciplinary field known as the “Science of Science” in order to interrogate, investigate, and identify the mathematical sciences itself. We seek to enhance social justice in the mathematics communities by providing examples of the ways in which the mathematical sciences fails to meet standards of equity, equal opportunity and inclusion. We aim to promote, provide, and posit sources of productive collaborations and we invite interested researchers to contribute to this developing body of work.
| Mon, Nov 4 @ 4pm | Grad Panel | So You Want to Give a Math Talk? |
Want to make your math talks clear, engaging, and memorable? Join us to learn practical guidance on preparing, structuring, and presenting math talks effectively. (Panelists: Jiayin Pan, Hadrian Quan, Amethyst Price)
| Fri, Nov 1 @ 4pm (rescheduled) |
ANT : Sam Miller, UCSC | On endosplit p-permutation resolutions and Broue's conjecture for p-solvable groups |
There are two storylines in this talk that in the end will converge. The first is an overview of Broue's abelian defect group conjecture, building up to its various refinements through the years. The second is a classification of a class of chain complexes called "endosplit p-permutation resolutions," which were first introduced by Rickard to prove the conjecture for p-nilpotent groups. I'll close out by explaining how this classification can be used to prove a recent refinement of the abelian defect group conjecture by Kessar and Linckelmann for p-solvable groups.
| Thurs, Oct 31 @ 4pm | COLLOQ : Rob Kusner, U Mass Amherst | Minimal Surfaces in Round Spheres and Balls |
A minimal surface models a soap film in the absence of ambient pressure: its area is stationary as the surface is perturbed. Geometrically, its mean curvature must vanish, and if it meets any ambient boundary, it does so orthogonally (it's a "free boundary" minimal surface). Over the past three centuries, many methods from the calculus of variations, complex analysis, partial differential equations, geometric measure theory, and global analysis have been developed to construct and analyze minimal surfaces. Recently, a deep connection between extremal eigenvalue problems, and minimal surfaces in round spheres or balls, has emerged. We use this to produce free boundary minimal surfaces of any topological type embedded in the round 3-ball, and many more closed minimal surfaces embedded in the round 3-sphere. This connection also lets us prove geometric properties for these surfaces, such as a uniqueness theorem for embedded free boundary minimal annuli with antipodal symmetry.
Wed, Oct 30 @ 4pm, G&A : Ursula Hamenstädt, University of Bonn. A geometric view on pressure.
The geodesic flow on a (closed) negatively curved manifold is known to be an Anosov flow, and one can ask to what extent dynamical information like the length of closed geodesics determines the geometry. While this question is wide open, we give an elementary introduction to this line of ideas, and to two approaches towards making this question quantitative in a geometric sense. We then present a fairly complete analysis of the special case of hyperbolic 3-manifolds whose fundamental group is a surface group. This is based on joint work with Elia Fioravanti, Frieder Jaeckel and Yongquan Zhang.
Tues, Oct 29 @ 4pm, Undergraduate Colloquium : Math Chats : Mike Williams
| Mon, Oct 28 @ 4pm | GRAD : Brooke Randell, UCSC | Exploring the Numerical Range of Block Toeplitz Operators |
We will discuss the numerical range of a family of Toeplitz operators with symbol function \(\phi(z)=A_0+zA_1\), where \(A_0\) and \(A_1\) are \(2 \times 2\) matrices with complex-valued entries. A special case of a result proved by Bebiano and Spitkovsky in 2011 states that the closure of the numerical range of the Toeplitz operator \(T_{\phi(z)}\) is the convex hull of \(\{W(\phi(z)): z \in \partial \mathbb{D}\}\). Here, \(W(\phi(z))\) denotes the numerical range of \(\phi(z)\). We combine this result with the envelope algorithm to describe the boundary of the convex hull of \(\{W(\phi(z)): z \in \partial \mathbb{D}\}\). We also place specific conditions on the matrices \(A_0\) and \(A_1\) so that \(\{W(\phi(z)): z \in \partial \mathbb{D}\}\) is a set of potentially degenerate circular disks. The convex hull of \(\{W(\phi(z)): z \in \partial \mathbb{D}\}\) takes on a wide variety of shapes, including the convex hull of limaçons.
| Fri, Oct 25 @ 10am (special time) | ANT : George McNinch, Tufts | Levi factors of linear algebraic groups |
Let G be a linear algebraic group over the field k, and suppose that the unipotent radical U of G is defined and split over k (this condition always holds when k is perfect). A Levi factor of G is a k-subgroup M for which the quotient mapping π:G → G/U determines an isomorphism π:M → G/U. For any field k of positive characteristic, there are linear algebraic k-groups without Levi factors. We are interested in *descent* of Levi factors: if ℓ is a finite separable field extension of k and if the group G_ℓ obtained by base-change has a Levi factor, does G itself already have a Levi factor? In the talk, we will describe examples of *disconnected* G where this sort of descent fails. On the other hand, we’ll recall an older result which shows that descent holds if ℓ is Galois over k with [ℓ:k] invertible in k. Finally, we describe some sufficient conditions for descent using vanishing of the non-abelian cohomology set H^1(M,U).
| Thur, Oct 24 @ 4pm | COLLOQ : Philip Arathoon, U of Michigan | The 3-body problem and a 3-web of Cayley cubics on the 3-sphere |
Finding general solutions to a mechanical system is often far too much to ask. Instead, we look for more tractable, special solutions, such as the equilibria. If the system is symmetric with respect to a group action then we can also look for the relative equilibria; these are solutions contained to a group orbit. Famous examples include the circular solutions of Euler and Lagrange in the 3-body problem. In this talk I will present a new formalism for finding relative equilibria by defining a 'web structure' on shape space, and demonstrate this by classifying the relative equilibria for the spherical 3-body problem.
Wed, Oct 23 @ 4pm, G&A #2 : Jonathan Zhu, University of Washington at Seattle.
Łojasiewicz inequalities and mean curvature flow
We’ll survey the use of Łojasiewicz inequalities in geometric flows, as well as recent methods for proving ‘explicit’ Łojasiewicz inequalities. A key application is the uniqueness of blowups to mean curvature flow at generic (cylindrical singularities). We will also discuss future work for other settings and other singularity models.
| Wed Oct 23 @ 3pm | G&A #1 : Jingyi Chen, U of British Columbia. | A geometric flow toward Hamiltonian stationary Lagrangian submanifolds |
We will discuss a fourth order volume decreasing flow of Lagrangian submanifolds evolving within a Hamiltonian isotopy class. The stationary solutions are the critical points of volume inside the Hamiltonian class, and have applications in the mirror symmetry and the Floer homology. This is based on recent joint work with Micah Warren.
Tues, Oct, 22, 4pm, UG : Panel : Applying to graduate school.
| Mon, Oct 21 @ 4pm | GRAD : Rafael Fernandes, UCSC | Barcode entropy and wrapped Floer homology |
Some researchers in Floer theory have speculated about connections between invariants defined through Floer homology and those of the underlying Hamiltonian/Reeb flow. A breakthrough came in 2021 with a paper by Çinelli, Ginzburg, and Gürel, in which they introduced a new invariant for filtered Floer homology, called barcode entropy, and proved some of its relations with the topological entropy of the underlying Hamiltonian system. In this talk, we will explore some connections between the barcode of wrapped Floer homology and the topological entropy of the underlying Reeb flow.
| Thur, Oct 17 @ 4pm, | COLLOQ : Ailana Fraser, U of British Columbia | Geometry of minimal submanifolds in higher codimension |
A lot of the focus of minimal submanifold theory has been for minimal submanifolds in codimension one. In the codimension one case there is a lot that is known, including Bernstein theorems, a regularity theory for minimizing hypersufaces, and even for stable minimal hypersurfaces. In higher codimension the situation is quite different, and is much less understood. There is a regularity theory for minimizers, but no Bernstein theorems in general, and the theory for stable minimal submanifolds in higher codimension is not well understood. In this talk we will describe recent progress on minimal surfaces in higher codimension, in particular on Bernstein theorems, and applications of minimal surfaces in higher codimension in Riemannian geometry.
| Wed, Oct 16 @ 4pm, | COLLOQ : Catherine Searle, Wichita State | On Fixed-Point Sets of ℤ2-Tori in Positive Curvature |
In joint work with Kennard and Khalili Samani, we were able to generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on closed manifolds of positive sectional curvature to ℤ2-torus actions. In this talk, I'll discuss recent joint work with Bosgraaf and Escher that generalizes that result by lowering the rank of the ℤ2-torus action to approximately 1/8 of the dimension of the manifold.
Tues, Oct 15, 4pm : Undergraduate Resources, (CalTeach, StemDiv, ACE), UCSC
| Mon, Oct 14 @ 4pm | GRAD : Sam Miller, UCSC | An introduction to Broué's abelian defect group conjecture |
Michel Broué's abelian defect group conjecture is one of the most infamous open problems in modular representation theory. So what is it? In this talk, we'll give a gentle introduction to the world of modular representation theory and the local-to-global philosophy. We'll build up to the conjecture, discuss its history and refinements, and, if time permits, give a new result concerning Kessar and Linckelmann's 2018 refinement of the conjecture.
| Thur, Oct 10 @ 4pm | COLLOQ : Lee Kennard, Syracuse University | Graph systoles, torus representations, and applications to geometry |
Recent work with Michael Wiemeler and Burkhard Wilking presents a link between arbitrary finite graphs and torus representations all of whose isotropy groups are connected. The link is via combinatorial objects called regular matroids, which were classified in 1980 by Paul Seymour. We then apply Seymour's deep result to classify and to compute geometric invariants of this class of torus representations. The applications to geometry are significant. A highlight of our analysis of these representations is the first proof of Hopf's 1930s Euler Characteristic Positivity Conjecture for metrics invariant under a torus action where the torus rank is independent of the manifold dimension.
| Wed, Oct 9 @ 4pm | G&A : Adam Jacob, UC Davis | Mean curvature flow of totally real submanifolds |
Given a complex manifold X, a totally real submanifold is a half dimensional submanifold whose tangent space contains no complex lines. Examples include Lagrangian submanifolds of Kahler manifolds, as well as small deformations of Lagrangians. In the case that X is a negatively curved Kahler-Einstein manifold or a Calabi-Yau, we demonstrate that the mean curvature flow on a totally real submanifold L will converge exponentially fast to a minimal Lagrangian submanifold, provided that the initial mean curvature vector of L, as well as the initial restriction of the Kahler form to L, are sufficiently small in the C^0 norm. This is joint work with Tristan Collins and Yu-Shen Lin.
Tues, Oct 8, 4pm, Undergraduate Colloquium : Math Chats : Mike Williams
Mon, Oct 7, 4pm, GRAD : Louis Coffin, UCSC. Connecting the Dots: Paths in a Square Grid
A few years ago, I was inspired by a phone game to look at the question of whether or not you can find a path between two points in a chessboard- like grid that 1) covers the entire grid and 2) doesn’t cross over itself. While I hate to spoil a puzzle, I thought it would be fun to share some of the ideas that came up while solving this problem. The topic is broadly a question of combinatorial graph theory, but since I myself am not especially well- versed in that field, this is mainly meant to be a light talk about a fun puzzle with an interesting solution.
| Fri, Oct 4 @ 4pm | ANT : Deewang Bhamidipati, UCSC | Strata intersections in some unitary Shimura varieties |
Unitary Shimura varieties are moduli spaces of abelian varieties with a certain extra structure which includes a signature condition. An effective way to understand these spaces is by stratifying them, of which two are of interest: the Ekedah-Oort (EO) stratification, defined with respect to the p-torsion group scheme structure up to isomorphism, and the Newton stratification, defined with respect to the p-divisible group structure up to isogeny. We take a specific stratum in the Newton stratification - the supersingular stratum - and we study its intersection with the EO stratification in some low signature cases.
| Thur, Oct 3 @ 4pm | COLLOQ : Lan-Hsuan Huang, U of Connecticut | Local structure theory of Einstein manifolds with boundary |
We discuss results on the local structure of the moduli space of compact Einstein manifolds in relation to the conformal boundary metric and boundary mean curvature. In three dimensions, we confirm M. Anderson's conjecture, showing that the map from the moduli space of Einstein metrics to such boundary data is generically a local diffeomorphism. We will also discuss analogous results in dimensions greater than three for Ricci flat manifolds or Einstein manifolds with a negative constant, assuming non-degenerate boundary conditions. This talk is based on joint work with Zhongshan An (University of Michigan).
| Wed, Oct 2 @ 4pm | G&A : Tom Latimer, UCSC | The origins of the Riemann Hilbert Problem and its application to differential and difference equations |
This talk will be an introductory talk about the Riemann Hilbert Problem (RHP) and some of its applications in recent mathematical research. We will start with a brief history, beginning with Hilbert's 21st problem as stated in 1900, and ending with the 'modern' form of an RHP. Then we will look at some applications to differential and difference equations, highlighting the connection between the RHP and monodromy data.
Tues, Oct 1 @ 4pm : Undergraduate Colloquium : Welcome to the Undergraduate Colloquium (Mike Williams)
| Mon, Sept 30 @ 4pm | GRAD: Deewang Bhamidipati, UCSC | What do you mean Galois Groups "are" Fundamental Groups? |
One cannot escape the similarities between the Galois theory of field extensions and the topological theory of covering spaces and fundamental groups. In particular, in both theories, there are objects that sit at the top whose automorphisms form a group: the absolute Galois group in the former case and the fundamental group in the latter. Moreover, taking subgroups, we get intermediate objects, and in some nice cases, the automorphism groups of these intermediate objects arise as quotients of the larger automorphism group. In this talk, we will peel back the similarities between these two theories and will hopefully also look at an example where these two theories naturally converge.
