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, Monteiro)
- 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
Fri Apr 25, 2:40 to 3:45 pm. ANT : Nariel Monteiro, UCSC.
Links between character triples
We will introduce the notion of a link between character triples. The motivation to consider this notion has multiple reasons. Firstly, links between character triples induce special character triple isomorphisms. This provides a new perspective for isomorphisms between character triples and a different conceptual understanding. Secondly, links between character triples provide equivalent reformulations of complicated conditions (involving projective representation) that play a fundamental role in the reductions of the McKay conjecture and the Feit conjecture to finite simple groups. This is joint work with Robert Boltje and John Revere McHugh.
Tuesday, April 29, 3pm. (Hybrid; also in 4130)
XM : Pamela Harris, U of Wisconsin-Milwaukee
From Surviving to Thriving: My Story in Math
In this talk, I’ll share my journey from student to professor and reflect on what it’s been like to navigate the world of mathematics as a Mexican American woman. I’ll discuss some of the challenges I’ve faced, the lessons learned, and how I’ve moved from simply surviving in math spaces to thriving in them. I will also share the history of the nonprofit organization Lathisms: Latinxs and Hispanics in the Mathematical Sciences, a platform that celebrates the contributions of Latinx and Hispanic mathematicians, for which I serve as President. I hope for this talk to be a conversation guided by your questions, which you can share ahead of time here: Padlet Link
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting ID: 924 7927 7522
Passcode: 4968783
Wed April 30 @ 4pm. G&A : Jiayin Pan, UCSC
Ricci curvature and first Betti number rigidity
A classical result by Bochner states that every closed n-manifold with nonnegative Ricci curvature has first Betti number at most n; moreover, the equality holds if and only if the manifold is isometric to a flat torus. We will talk about the extensions of this rigidity result under maximal first Betti number, and some recent developments in this direction. Part of this talk is a joint work with Zhu Ye.
Thur May 1 @ 4pm. COLLOQ : Laura Escobar Vega, UCSC
Families of degenerations from mutations of polytopes
Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to varieties which admit a toric degeneration. In a paper with Harada, we gave a piecewise-linear bijection between Newton-Okounkov bodies of a single variety. This involves a collection of lattices connected by piecewise-linear bijections. In addition, Kaveh and Manon analyzed valuations into semifields of piecewise-linear functions and explored their connections to families of toric degenerations. Inspired by these ideas in joint work in progress with Harada and Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise-linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties a generalized polytope can encode a compactification of an affine variety as well as toric degenerations for the compactification.
Fri May 2 @ 2:40 pm. ANT : Jitendra Rathore, UC Santa Barbara
Brauer-Manin Pairing for Surfaces over Local Fields.
The Brauer group and the Chow group of $0$-cycles for a smooth projective variety are two important invariants associated with a variety. For smooth projective varieties over local fields, these groups are related by the Brauer-Manin pairing, a pairing between the two abelian groups. We will begin the talk by defining this pairing and recalling some key results that establish the relationship between these groups for smooth curves, along with a few known results for smooth surfaces. Later, I will present some new results regarding this relationship for smooth surfaces, which I obtained in joint work with Amalendu Krishna and Samiron Sadhukhan.
Wed May 7 @ 4pm. G&A : Dimitri Navarro, UCSC
Moduli spaces of Alexandrov surfaces
A fundamental problem in Riemannian geometry is to determine whether a given smooth manifold admits a Riemannian metric with sectional curvature bounded below by a given constant K. When it does, one aims to describe the space of all such metrics, known as "the moduli space of Riemannian metrics with sectional curvature at least K." A powerful tool in this context is the study of Gromov–Hausdorff limits of manifolds with sectional curvature bounded below. While potentially singular, these limits are Alexandrov spaces—metric spaces with a lower curvature bound in a generalized sense. In this talk, I will present recent results on the topology of moduli spaces of Alexandrov metrics on surfaces, which one may see as a natural singular counterpart to the moduli spaces of smooth metrics with sectional curvature bounded below.
| Thur May 8 @ 4pm | COLLOQ : Serge Bouc, Université de Picardie | The functor of characters |
The notion of character of a finite group goes back to Ferdinand G. Frobenius, at the very end of 19th century. Today, it is one of the main tools in the representation theory of finite groups, with important applications in other domains (e.g. number theory). After recalling part of this history, I would like to present some more recent developments, related to the very rich structure of the functor of characters.
| Fri May 9 @ 11am | Colloquium #2 : Zeno Huang, CUNY Staten Island | The Asymptotic Plateau Problem |
The asymptotic Plateau problem is a set of problems for the existence and multiplicity (as well as other aspects) for minimal surfaces in hyperbolic 3-space, spanned by a prescribed Jordan curve on the sphere at infinity. In this lecture, I will describe some recent progress made on this problem, as well as some open questions. The lecture is based on joint work with Lowe and Seppi.
| Fri May 9 @ 2:40pm | ANT : Nadia Romero, CIMAT | Hochschild cohomology for linear monoidal functors |
Let X be an essentially small symmetric monoidal category enriched in R-Mod, with R a commutative ring with identity. Under these conditions, the category F, of R-linear functors from X to R-Mod, becomes an abelian symmetric monoidal category, also enriched in R-Mod. The fact that F is monoidal and abelian at the same time allows for a nice theory of modules over the monoids in F -- in particular it allows for a nice and easy definition of an internal hom functor. In this talk, we will see how this internal hom is the key to define a Hochschild cohomology theory in F.
| Mon May 12 @ 4pm | Gradmath : DEI Panel |
abstract
Tues May 13 @ 3pm XM : Laura Escobar Vega, UCSC
| Wed, May 14 @ 4pm | G&A : Jared Marx-Kuo, Rice University | Determining the Metric from Minimal Surfaces in Asymptotically Hyperbolic Spaces |
In this talk we will discuss minimal surfaces in asymptotically hyperbolic spaces and a corresponding "renormalized" area that is conformally invariant. Inspired by work in the compact setting, we show that knowledge of the renormalized area on a relatively small subset of minimal surfaces determines the asymptotic expansion of the metric, including the conformal infinity. As a further application, we show that renormalized area can determine the conformal structure of the boundary of a hyperbolic 3-manifold.
Fri May 23 @ 2:40pm. ANT : Jennifer Guerrero, UCSC
Mon Jun 2 @ 4pm. Gradmath : Amethyst Price, UCSC.
Tues Jun 5 @ 2:30pm. XM : Niny Arcila-Maya, SFSU.
Tues Jun 5 @ 4pm. Colloq : Niny Arcila-Maya, SFSU.
Older
Tues Apr 22 @ 4pm Ugrad : Gretchen Andreasen and Kevin Deutsch, CalTeach
Math Games and Becoming a Math Teacher
Join CalTeach staff, and students from Math and Applied Math, to play some math games and learn about becoming a secondary math teacher. Pizza will be served. RSVP @ [this form]
| Thurs, Apr 17 @ 4pm | COLLOQ : Ken McLaughlin, Tulane U | Asymptotic analysis of integrable systems through some simple examples |
| Tues Apr 15, @ 4:15 pm (Special Time) | Gradmath : Deewang Bhamidipati, UCSC | Finiteness of the Class Group: An Adelic Sketch |
Number fields, finite extensions of the rationals, are the fundamental objects of study in algebraic number theory. Their arithmetic properties are encoded in their rings of integers - the analogues of ℤ in these fields. One is interested in how properties of ℤ may or may not generalise to the rings of integers of number fields; e.g., ℤ[√-5] does not remain a unique factorisation domain. The class group of a number field is an invariant that measures how far a number field's ring of integers deviates from unique factorisation. One of the main theorems in algebraic number theory is the finiteness of the class group, which represents a profound fact about number fields - while their rings of integers may lack unique factorisation, they come remarkably close. In this expository talk, we will begin by introducing the main concepts needed to define the class group. We will then demonstrate an elegant proof of its finiteness using a remarkable object attached to each number field that encodes both arithmetic and analytic information - the ring of adèles. The secondary goal of this talk is to showcase the significant interplay between number theory and other mathematical disciplines such as analysis and topology, an interplay that pervades modern number theory.
Wed Apr 9 @ 4pm. G&A : Laura Fredrickson, U of Oregon
The asymptotics of hyperk\"ahler metrics from Gaiotto--Moore--Neitzke's conjecture
Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperk\"ahler metric, a rich and rigid geometric structure. An intricate conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke. I will discuss some recent and less-recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures. This is based on works joint with R. Mazzeo, J. Swoboda, H. Weiss, M. Zimet.
| Tues Apr 8 @ 4pm | Ugrad : Deewang Bhamidipati, UCSC | Rational Triangles |
(The Time I Was Dreaming about Rational Triangles and Discovered I Was Living on an Algebraic Curve in the World of Algebraic Geometry)
In this talk, we will embark on an adventure that stars the humble triangle! We will dive into the captivating world of triangles with rational sides – a playground where geometry and number theory collide! How many triangles can we create when we demand their sides, areas, and perimeters be rational? As we try to answer these questions, we'll be transported to the hunt for points with rational coordinates on algebraic curves - curves defined by polynomial equations. Join me to discover how triangles – a shape we've known since childhood – can lead us to the beautiful landscape of algebraic geometry, a field that has captivated mathematicians for countless generations. No advanced degree is required - bring your curiosity !
Fri, Mar 14 @ 1:30pm. ANT : James Upton, UCSC. Iwasawa Theory for p-Torsion Class-Group Schemes
Geometric Iwasawa theory aims to study the variation of class groups in Zp-towers of function fields. A new program initiated by Booher and Cais suggests replacing the role of the class group by (the p-divisible group of) the Jacobian. In this talk, we discuss the p-torsion part of the Jacobian, whose structure is described concretely in terms of certain numerical invariants called a-numbers. For suitably nice towers we give an asymptotic description of the a-numbers, as well as an exact Iwasawa-style formula in special cases. This is joint work with Jeremy Booher, Bryden Cais, and Joe Kramer-Miller.
Thurs, Mar 13 @ 4pm. COLLOQ : Ronan Conlon, UT Dallas.
On Finite-Time Singularities of the Ricci Flow on Compact Kähler Surfaces
The Ricci flow, introduced by Richard Hamilton in the 1980’s, is a powerful geometric evolution equation that deforms the metric of a Riemannian manifold in a way analogous to heat diffusion. It has proven instrumental in understanding the geometry and topology of manifolds. A rigorous analysis of its behavior at finite time singularities has been key to these applications. In this talk, I will present joint work with Cifarelli-Deruelle and Hallgren-Ma on the behaviour of the Ricci flow on a compact Kahler surface at a non-collapsed finite time singularity.
Wed, Mar 12 @ 4pm. G&A : Mengxuan Yang, UC Berkeley.
Scattering and wave decay of the Aharonov–Bohm Hamiltonian with multiple poles
The famous Aharonov–Bohm effect shows that for an infinitely-long solenoid with a magnetic field only inside it, even though there is no magnetic field outside, a charged particle can experience a phase difference when traveling along different trajectories on either side of the solenoid due to the existence of the magnetic vector potential. The Aharonov–Bohm effect demonstrates the non-trivial quantum mechanical effect of vector potentials. Although the single solenoid case has been studied widely using rotational and scaling invariance, the technique fails to work for the Aharonov–Bohm Hamiltonian with multiple solenoids. In this talk, I will discuss a new framework to study the resolvent, resonances, and wave decay of the Aharonov–Bohm Hamiltonian with multiple poles. This results in an interesting behavior where scattering and wave decay depend on the fluxes.
Mon, Mar 10 @ 4pm. Grad : Tsz Yau Tin, UCSC.
$\mathcal{N}=2$ SCFT and Calabi-Yau compactification
Since the late 1980's, physicists have noticed the mysterious coincidences between the spectra of $\mathcal{N}=2$ superconformal field theory (SCFT) and the topological invariants of certain Calabi-Yau manifolds. We introduce the $\mathcal{N}=2$ supersymmetric string theory and its compactification on Calabi-Yau threefolds (CY3's). We give a brief review of the Gepner model, in which many CY's can be constructed by tensoring building blocks in $\mathcal{N}=2$ SCFT known as the "minimal models". We demonstrate this phenomenon with an explicit example: the Fermat quintic.
Fri, Mar 7 @ 1:30pm. ANT: Aparna Upadhyay, University of South Alabama.
On the decomposition of tensor powers of modules.
The core of a finite-dimensional kG-module M is obtained by deleting all the summands of M that are projective. Benson and Symonds defined an invariant for M, called the gamma invariant. This invariant is a result of studying the asymptotics of the core of tensor powers of M. In this talk we will see several interpretations of this invariant and some of its interesting properties. We will discuss the value of the gamma invariant for certain families of modules. We will also explore some conjectures about this invariant.
Thurs, March 6 @ 4pm. Colloq: Roozbeh Gharakhloo, UCSC.
Even-valent Graphs on Riemann Surfaces
The seminal works of theoretical physicists such as ’t Hooft, Bessis, Itzykson, Zuber, David, and Kazakov in the 1970s and 1980s revealed a profound connection between two-dimensional quantum gravity, graph enumeration, and random matrix models. In this talk, after reviewing historical developments, I will describe how the problem of counting graphs that can be embedded in a Riemann surface connects to the theory of orthogonal polynomials and the associated complex-analytic Riemann-Hilbert problem. I will present new closed-form formulas for:
The number of connected 2K-valent graphs with a fixed number of vertices on a Riemann surface of fixed genus, for any K, in terms of the K-th Catalan number;
The number of connected 6-valent graphs on a Riemann surface of fixed genus with an arbitrary number of vertices.
This is joint work with Tomas Lasic Latimer (UCSC).
Wed, Mar 5 @ 4pm. G&A : Sophie Aiken, UCSC.
An End to End-Gluing Construction for Metrics of Constant Q-Curvature
One prominent question in geometric analysis is the existence and classification of conformal manifolds with constant curvature. In the last 40 years many people have intensely studied singular curvature problems in Riemannian manifolds, in particular the construction of constant curvature metrics on compact manifolds with singularities, i.e. punctures. In joint work with Caju, Ratzkin, and Silva Santos, we produce many new examples of complete, constant Q-curvature metrics on a finitely punctured sphere by gluing together known examples.
Tues, Mar 4 @ 3pm. Experiences : Moon Duchin, Tufts & Cornell.
On math, community, and the mathematical community
My career took a big turn from pure math to applied (specifically, to democracy and civil rights!) right around the time I got tenure. Before, after, and throughout, I've put a lot of work into noticing and building communities and subcommunities where all kinds of people can make a home in math. I'll share my thoughts on community at a moment when academic inclusivity is under unprecedented attack.
https://ucsc.zoom.us/j/
Meeting ID: 924 7927 7522
Passcode: 4968783
| Mon, Mar 3 @ 4pm. | GRAD : Maneesha Ampagouni, UCSC | TBD |
forthcoming
Fri, Feb 28. Special : Highway 17 GGT Conference.
Thurs, Feb 27. G&A special : Papri Dey, UCSC Applied Math.
K-Lorentzian Polynomials and Their Applications in Stability Analysis
Lorentzian polynomials, introduced independently by Brändén and Huh, and by Anari, Oveis Gharan, and Vinzant extend hyperbolic polynomials while preserving log-concavity. These polynomials are specific to the nonnegative orthant. In this talk, I will explore Lorentzian polynomials on proper convex cones, referred to as K-Lorentzian polynomials. In joint work with Greg Blekherman, we establish a connection between K-Lorentzian polynomials and K-positive linear maps, a concept rooted in the generalized Perron-Frobenius theorem. We also prove the equivalence between K-Lorentzian and K-completely log-concave polynomials. Additionally, we provide a simplified characterization of K-Lorentzian polynomials based on their relationship to K-irreducibility and K-nonnegativity. Building on this, a recent project proposes a method for constructing a proper cone K for a given K-Lorentzian polynomial. Finally, I will demonstrate applications of K-Lorentzian polynomials in the stability analysis of EVI dynamical systems governed by differential equations and inequality constraints, illustrating these concepts through examples.
Tues, Feb 25. Ugrad : Joseph Immel, UCSC. Crossings on Complete Graphs
Given n points in the plane and curves connecting them all, what is the smallest number of times that the curves must cross over each other? In the 1960s, a concrete answer was shown for n=10. More than fifty years later, after an explosion in computer science and computational power, that result has only been improved to n=12. Why has the problem proven so resilient? In this talk, we will introduce the rudiments of graph theory, explore conjectures which have remained open for half a century, and briefly look at practical solutions for applications from circuit design to air traffic control.
Mon, Feb 24. Gradmath : Panel : Conducting Research in Mathematics (as a PhD student)
Embarking on the journey of conducting research in mathematics as a PhD student is both thrilling and challenging. Questions like "What does mathematical research look like?" or "What should I expect from my PhD advisor?" or "What to do when I get stuck?" may be on your mind. Join our panel to hear our panelists share experience and suggestions, helping you set expectations and navigate this exciting academic journey. (Panelists: Sophie Aiken, Roozbeh Gharakhloo, Eoin Mackall, Jiayin Pan)
| Fri, Feb 21 @ 1:30pm | ANT : Daniel Vallieres, CSU Chico | An analogue of the Herbrand-Ribet theorem in graph theory |
In this talk, we will review the Herbrand-Ribet theorem in the theory of cyclotomic number fields of prime conductor. Then, via the analogy between number theory and graph theory, we will formulate an analogous statement in the context of graphs. The proof of the graph theoretical statement is much simpler, and if time permits we will give the main ideas involved. This is joint work with Chase Wilson.
| Thur, Feb 20 @ 4pm | COLLOQ : Pengfei Guan, McGill | Isoperimetric problems and geometric flows |
The classic isoperimetric problem is a problem of finding optimal relationship between area and perimeter-- one can trace it back to Queen Dido. There are similar isoperimetric problems related to other geometric quantities, like total mean curvature, surface area, volume, etc. For these calculus of variations problems, geometric flow is an effective path to the optimal solution. Flows can be specially designed to fit to the underlying geometric structures. They can take the form of generalized mean curvature or inverse mean curvature type flows. We will discuss some recent results in this direction, and main ideas associated to the approach.
| Tues, Feb 18 @ 3pm | Experiences : Michael Young, CMU | Narratives of Minority PhD Students |
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting ID: 924 7927 7522
Passcode: mathexp
| Tues, Feb 11 @ 4pm | Ugrad : DRP Presentations | (Ethan Jones, August Noe, & Corbin Naderzad) |
Three former Directed Reading Program (DRP) students will talk about their past DRP projects. August Noe will be giving a talk entitled "The Power of Big Sets: A soft introduction to ultrafilters". Corbin Naderzad will be giving a talk entitled "Counterexamples in Analysis". Lastly, Ethan Jones will be giving a talk entitled "Primes of the Form x^2+ny^2". Each talk will only be 15 minutes long with 5 minutes for questions.
| Fri, Feb 7 @ 1:30pm | ANT : Chi-Yun Hsu, Santa Clara University | p-adic companion forms for Yoshida lifts |
Coleman defined a p-adic theta operator on overconvergent forms, mapping forms of slope 0 and weight 2-k to forms of slope k-1 and weight k. By explicitly computing the q-expansion, he proved that the critical p-stabilization of a p-ordinary CM form lies in the image of the theta operator. The parallel statement on the Galois side is that the Galois representation of a CM form splits locally at p. Coleman and Greenberg conjectured the respective converse, but both still remain open. In the GSp4 setting, the Galois representation of a Yoshida lift splits locally into two 2-by-2 blocks at p. Our goal is to prove that the critical p-stabilization of a Yoshida lift lies in the image of a relevant theta operator. This is a joint work in progress with Bharathwaj Palvannan.
| Thur, Feb 6 @ 4pm | COLLOQ : Jiayin Pan, UCSC | Ricci curvature and linear volume growth |
We will talk about old and new results on the topology of manifolds with nonnegative Ricci curvature and linear (minimal) volume growth. The new results are based on a joint work with Dimitri Navarro and Xingyu Zhu.
| Wed, Feb 5 @ 4pm | G&A : Romain Speciel, Stanford U | Commutativity Properties of the Dirichlet-to-Neumann Map |
The boundary Laplacian of a bounded Euclidian domain with smooth connected boundary agrees with the square of its Dirichlet-to-Neumann map at the symbolic level, but how close are these two operators truly? A natural way to understand this question is to study their commutator. In dimensions 3 and above, the Dirichlet-to-Neumann map is known to commute with the boundary Laplacian if and only if our domain is a ball, but the 2 dimensional case requires a different approach. In this talk, we will show that, surprisingly, there exists a one-parameter family of simply connected domains with this commutativity property in dimension 2.
| Tues, Feb 4 @ 4pm | Experiences : Jesus E Gonzalez, West Valley College | Adding My Voice: My Journey as a First-Generation Latino Student in Education and Mathematics |
In this talk, I will share my personal journey as a first-generation college student, educator, and mathematician, discussing identity, representation, and belonging. From my experiences as a student, to my current role in education, we will discuss the challenges and triumphs of navigating a space where diversity and inclusion have not always been prioritized. Drawing on my own story, I aim to highlight the importance of creating supportive and inclusive environments for underrepresented groups in education and mathematics, while also fostering dialogue and building community. I will also share the projects and work I am passionate about, how they connect to my identity, and the lessons I have learned along the way.
https://ucsc.zoom.us/j/92479277522?pwd=eTdSOFlGYm55YWk0Y2lRNkVYZzJnQT09
Meeting ID: 924 7927 7522
Passcode: mathexp
Mon, Feb 3 @ 4pm. GRAD : Mike Williams, UCSC. A Discussion on Using AI as a Graduate Student.
| Fri, Jan 31 @ 1:30 pm | ANT : Robert Boltje, UCSC | On a conjecture of Walter Feit |
At the famous Santa Cruz conference in 1979, held at the time when the conclusion of the classification of finite simple groups (CFSG) was more or less completed, Walter Feit gave a list of questions that he thought should have an answer using the CFSG. One of them states: "Let $G$ be a finite group, $\chi$ an irreducible character of $G$, and $c(\chi)$ the conductor of $\chi$, i.e., the smallest natural number $n$ such that the character values of $\chi$ can be written as a \(\mathbb{Z}_2\)-linear combination of a primitive $n$-th root of unity. Does $G$ need to have an element of order $c(\chi)$?" A positive answer was already given for special cases by Brauer in 1964, Ferguson-Turull in 1986, and Amit-Chillag in 1986. No progress on this question for general finite groups has been made since then. Quite recently, together with Kleshchev, Navarro and Tiep, we were able to reduce this question to a condition that finite simple groups need to satisfy. This talk will give a quick introduction to characters of finite groups, report on the status of Feit's conjecture, and introduce other conjectures related to Feit's conjecture.
| 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.
| Wed, Jan 29 @ 4pm | G&A : Rafael Fernandes, UCSC | Minimax energy of pseudoholomorphic curves and Periodic Reeb flows in dimension 3 |
In this talk, we will discuss how elementary spectral invariants can be used to characterize the Reeb dynamics on a contact 3-manifold. Elementary spectral invariants were introduced by Michael Hutchings using max-min energy of holomorphic curves subject to point constraints as an alternative to the ECH spectrum of a contact 3-manifold. We use these invariants to characterize when a closed contact 3-manifold is Besse and partially characterize when it is Zoll. This is joint work with Brayan Ferreira.
| Tues, Jan 28 @ 4pm | Ugrad : Deewang Bhamidipati, UCSC | Dreaming about Rational Triangles |
(That Time I Was Dreaming about Rational Triangles and Discovered I Was Living on an Algebraic Curve in the World of Algebraic Geometry)
In this talk, we will start by asking some interesting questions about rational triangles, which are triangles with side lengths equaling rational numbers. We will see how, in trying to answer these questions, we land in the world of algebraic geometry. We will explore this world through concrete and hands-on examples!
| Mon, Jan 27 @ 4pm | GRAD : Thomas Arnstein, UCSC | Heyting algebras and quantifiers as adjoints |
In this introductory talk, I will discuss the notions of Boolean and Heyting algebras as distributive lattices. We will see that power sets of any set form a Boolean algebra, and, more generally, with a short discussion about subobject classifiers in a topos we can form Heyting algebras as subobject lattices of an object in a topos. The main idea we will arrive at is viewing the existential and universal quantifiers as left and right adjoints of a pullback functor. This will be a surface level introduction to topos theory and should be super accessible, only assuming basic knowledge of categories.
| Thur, Jan 23 @ 4pm | COLLOQ : Richard Montgomery, UCSC | Falling Cats, Gauge Theory, and the N-body problem |
I will skim over my career, focusing on open questions including one very recently solved. The excitement around gauge theory in mathematics at Berkeley in the early 1980s led me, through prodding of Jair Koller, to papers of Shapere and Wilczek on how to use that theory to better understand the re-orientation and locomotion of living organisms. I worked on the sub-question: how might a cat, dropped from upside down, right herself, and do so optimally? That question can be viewed as one in optimal control, or subRiemannian geometry. Political disaffection, with nudges from Alan Weinstein and Wu-yi Hsiang, led me from this arena backwards in time to begin work on the classical three-body problem - a cat consisting of three-point masses, but now "falling into itself" instead of toward the ground. This is the mathematical domain I have mostly inhabited during the last two decades. Open questions include: "Are all subRiemannian geodesics smooth?" and "Can any braid type be realized by a solution of the planar zero angular momentum N-body problem?"
| Wed, Jan 22 @ 4pm | G&A : Andrew Waldron, UC Davis | Yang-Mills Theory on Conformally Compact Manifolds |
The Yang-Mills equations are central to the study of smooth 4-manifolds and particle interactions. We consider Yang-Mills theory in the setting of conformally compact manifolds in general dimensions. In particular, we obtain a formula for the renormalized energy of Yang-Mills solutions on Poincaré-Einstein 6-manifolds. The method generalizes the proof of Chang-Qing-Yang for renormalized volumes to a broader setting.
| 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.
| Wed, Jan 15 @ 4pm | G&A : Hadrian Quan, UCSC | A singular journey with Morse, Bott, Smale, and Witten |
Morse theory is a flexible approach to studying the topology of manifolds by decomposing these spaces along the isolated critical points of certain nice functions ("Morse functions"), and relates such geometric decompositions to other invariants such as cohomology groups, e.g. through the Morse inequalities. Witten, inspired by ideas from quantum mechanics, later gave new analytic proofs of the Morse inequalities and thus related the analysis of the Hodge Laplacian to the topology and dynamics of Morse theory.
In this talk I will present recent joint work with Jayasinghe and Xu, which extend these ideas in two directions. We broaden the dynamical setting by considering Morse-Bott functions (which have more complicated critical point sets), and the geometric setting to consider not just manifolds but also singular stratified spaces (e.g. singular projective varieties). In so doing we prove new Morse inequalities, as well as Lefschetz trace formulae.
| 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.
