## news

##### First Year Undergraduate Math Information

Click the Headline ↑↑↑↑↑ above to go directly to information about Math Department courses and your Math Placement Test.
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##### Math Community Web Page

Following up on the #shutdownSTEM discussions, the Department of Mathematics has launched a community web page, with evolving content to be created through community effort.
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##### Commendations for Extraordinary Teaching

Dean of Undergraduate Education Amanda Claybaugh sent letters of Special Commendation for Extraordinary Teaching in Extraordinary Times to Wes Cain Janet Chen Voula Collins David...
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##### Three New Professors Join Department

Harvard Mathematics welcomes three new Professors to our Department in July: Laura DeMarco in Dynamical Systems, Mihnea Popa in Algebraic Geometry, and Melanie Matchett Wood...
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##### #shutdownSTEM–10 June 2020

In partnership with the Harvard Division of Science, the Mathematics Department used the call to #shutdownSTEM on 10 June to initiate a discussion of actions...
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## upcoming events

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1
2
• Enumerative invariants and exponential networks

##### Enumerative invariants and exponential networks

12:00 PM-1:00 PM
March 2, 2020
20 Garden Street, Cambridge, MA 02138

I will define and review the basics of exponential networks associated to CY 3-folds described by conic bundles. I will focus mostly on the mathematical aspects and general ideas behind this construction as well as its conjectural connection with generalized Donaldson–Thomas invariants. This is based on joint work with S. Banerjee and P. Longhi.

3
• CMSA Special Quantum Matter/Quantum Math Seminar: Cutting and pasting 4-manifolds

##### CMSA Special Quantum Matter/Quantum Math Seminar: Cutting and pasting 4-manifolds

10:30 AM-12:00 PM
March 3, 2020
20 Garden Street, Cambridge, MA 02138
We will discuss techniques topologists use for understanding 4-manifolds obtained by cut-and-paste constructions. The hope is that these techniques may be useful for understanding 4-dimensional topological field theories.
• Equivariant Degenerations of Plane Curve Orbits

##### Equivariant Degenerations of Plane Curve Orbits

3:00 PM-4:00 PM
March 3, 2020
1 Oxford Street, Cambridge, MA 02138 USA

In a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.

• On quantum distributional symmetries for *-random variables

##### On quantum distributional symmetries for *-random variables

3:30 PM-4:30 PM
March 3, 2020
17 Oxford Street, Cambridge, MA 02138 USA

In this talk, we briefly review the distributional symmetries for *-random variables, which are defined by coactions of corepresentations of quantum groups. We classify all de Finetti type theorems for classical independence and free independence by studying vanishing conditions on the classical and free cumulants.  Examples for our de Finetti type theorems and approximation results in the spirit of Diaconis and Freedman are also provided.

• Complete Kahler Ricci flow with unbounded curvature and applications

##### Complete Kahler Ricci flow with unbounded curvature and applications

4:15 PM-5:15 PM
March 3, 2020
1 Oxford Street, Cambridge, MA 02138 USA

In this talk, we will discuss the construction of Kahler Ricci flow on complete Kahler manifolds with unbounded curvature. As a corollary, we will discuss the application related to Yau’s
uniformization problem and the regularity of Gromov-Hausdorff’s limit. This is joint work with L.F. Tam.

— Organized by Professor Shing-Tung Yau

4
• Non-random behaviour in sums of modular symbols

##### Non-random behaviour in sums of modular symbols

3:00 PM-4:00 PM
March 4, 2020
1 Oxford Street, Cambridge, MA 02138 USA

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor
of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the common phenomenon of “square root cancellation”, while in another case we find more cancellation.

• **CANCELED** Informal Geometry & Dynamics Seminar: Pseudo-Anosov mapping classes with large dilatation

##### **CANCELED** Informal Geometry & Dynamics Seminar: Pseudo-Anosov mapping classes with large dilatation

4:00 PM-6:00 PM
March 4, 2020
I’ll talk about some subclasses of pseudo-Anosov mapping classes whose dilatations are bounded away from 1.
• Well-ordering principles in Proof theory and Reverse Mathematics

##### Well-ordering principles in Proof theory and Reverse Mathematics

4:30 PM-5:30 PM
March 4, 2020
There are several familiar theories of reverse mathematics that can be characterized by well-ordering principles of the form
(*) “if X is well ordered then f(X) is well ordered”,
where f is a standard proof theoretic function from ordinals to ordinals (such f’s are always dilators). Some of these equivalences have been obtained by recursion-theoretic and combinatorial methods. They (and many more) can also be shown by proof-theoretic methods, employing search trees and cut elimination theorems in infinitary logic with ordinal bounds. One could perhaps generalize and say that every cut elimination theorem in ordinal-theoretic proof theory encapsulates a theorem of this type.
One aim of the talk is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from (*).
As (*) is of complexity $\Pi^1_2$ such a principle cannot characterize stronger comprehensions at the level of $\Pi^1_1$-comprehension. This requires a higher order version of (*) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory.  The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to $\Pi^1_1$-comprehension. This result has been conjectured more than 10 years ago, but its proof has only been worked out by Anton Freund in recent years
• CMSA Colloquium: Derandomizing Algorithms via Spectral Graph Theory

##### CMSA Colloquium: Derandomizing Algorithms via Spectral Graph Theory

4:45 PM-5:45 PM
March 4, 2020
20 Garden Street, Cambridge, MA 02138
Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability.  However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).

In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory.  Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree).  If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.

Joint works with Jack Murtagh, Omer Reingold, Aaron Sidford,  AmirMadhi Ahmadinejad, Jon Kelner, and John Peebles.

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• **CANCELED** A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

##### **CANCELED** A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

10:30 AM-11:30 AM
March 6, 2020
Few notions within the realm of mathematical physics succeed in capturing the imagination and inspiring awe as well as that of a black hole. First encountered in the Schwarzschild solution, discovered a few months after the presentation of the Field Equations of General Relativity at the Prussian Academy of Sciences, the black hole as a mathematical phenomenon accompanies and prominently features within the history of General Relativity since its inception. In this talk we will lay out a brief history of the question of dynamical black hole formation in General Relativity and discuss a recent result, in collaboration with Xinliang An, on a scale-critical trapped surface formation criterion for the Einstein-Maxwell system.
• A topological approach to convexity in complex surfaces

##### A topological approach to convexity in complex surfaces

3:30 PM-4:30 PM
March 6, 2020
1 Oxford Street, Cambridge, MA 02138 USA

We will discuss the classical notion of J-convexity of subsets of complex manifolds, and the closely related notion of Stein manifolds. The theory is particularly subtle in complex dimension 2. Surprisingly, progress can be made using topological 4-manifold theory. Every tame CW 2-complex topologically embedded in a complex surface can be perturbed so that it becomes J-convex in the sense of being a nested intersection of Stein open subsets. These Stein neighborhoods are all topologically equivalent to each other, but can be very different when viewed in the smooth category. As applications, we obtain uncountable families of distinct smoothings of R^4 admitting convex or concave holomorphic structures. We can also generalize the notion of J-convex hypersurfaces to the topological category. The resulting topological embeddings behave like their smooth counterparts, but are much more common.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar

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• CMSA Special Seminar: Differentials ideals for Calabi-Yau periods and Feynman integrals

##### CMSA Special Seminar: Differentials ideals for Calabi-Yau periods and Feynman integrals

10:30 AM-11:30 AM
March 9, 2020
20 Garden Street, Cambridge, MA 02138

We will review the connection between Feynman integrals and Calabi–Yau periods. Then we explain some properties of the period geometry, in particular the Picard–Fuchs differential ideal and to which extend it allows to calculate the Feynman integral. The main result is an explicit analytic calculation of the 3-Loop Banana graph with all mass parameters using relative periods on a special family of K3 surfaces.

• CMSA Mathematical Physics Seminar: Fano Lagrangian submanifolds of hyperkahler manifolds

##### CMSA Mathematical Physics Seminar: Fano Lagrangian submanifolds of hyperkahler manifolds

12:00 PM-1:00 PM
March 9, 2020
20 Garden Street, Cambridge, MA 02138

For any polarized hyperkahler manifold of K3 type whose dimension is divisible by 8, we produce a Lagrangian submanifold which is Fano arising as a connected component of the fixed locus of an involution on the hyperkahler manifold. This is an ongoing joint work with E. Macrì, K. O’Grady, and G. Saccà.

10
• CMSA Special Seminar: Multivariate public key cryptosystems – Candidates for the Next Generation Post-quantum Standards

##### CMSA Special Seminar: Multivariate public key cryptosystems – Candidates for the Next Generation Post-quantum Standards

10:30 AM-11:30 AM
March 10, 2020
20 Garden Street, Cambridge, MA 02138

Multivariate public key cryptosystems (MPKC) are one of the four main families of post-quantum public key cryptosystems. In a MPKC, the public key is given by a set of quadratic polynomials and its security is based on the hardness of solving a set of multivariate polynomials. In this talk, we will give an introduction to the multivariate public key cryptosystems including the main designs, the main attack tools and the mathematical theory behind in particular algebraic geometry. We will also present state of the art research in the area.

• **CANCELED** Variational quantum algorithms: obstacles and opportunities

##### **CANCELED** Variational quantum algorithms: obstacles and opportunities

3:00 PM-4:00 PM
March 10, 2020
17 Oxford Street, Cambridge, MA 02138 USA

Variational quantum algorithms such as VQE or QAOA aim to simulate low-energy properties of quantum many-body systems or find approximate solutions of combinatorial optimization problems. Such algorithms employ variational states generated by low-depth quantum circuits to minimize the expected value of a quantum or classical Hamiltonian. In this talk I will explain how to use general structural properties of variational states such as locality and symmetry to derive upper bounds on their computational power and, in certain cases, rule out potential quantum speedups. To overcome some of these limitations, we introduce the correlation rounding method and a recursive Quantum Approximate Optimization Algorithm.

Based on arXiv:1909.11485 and arXiv:1910.08980

• **CANCELED** Ancient gradient flows of elliptic functionals and Morse index

##### **CANCELED** Ancient gradient flows of elliptic functionals and Morse index

4:15 PM-5:15 PM
March 10, 2020
1 Oxford Street, Cambridge, MA 02138 USA
(Joint with Kyeongsu Choi.) We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, focusing on mean curvature flow for the talk. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or canonically shrinking equators. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or canonically shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or canonically shrinking equators.
11
• CMSA Quantum Matter/Quantum Field Theory Seminar: Quantized Graphs and Quantum Error Correction

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Quantized Graphs and Quantum Error Correction

10:30 AM-12:00 PM
March 11, 2020
20 Garden Street, Cambridge, MA 02138

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

• **CANCELED** Schinzel-Zassenhaus, height gap and overconvergence

##### **CANCELED** Schinzel-Zassenhaus, height gap and overconvergence

3:00 PM-4:00 PM
March 11, 2020
1 Oxford Street, Cambridge, MA 02138 USA

We explain a new height gap result on holonomic power series
with rational coefficients, and prove the Schinzel-Zassenhaus conjecture
as its consequence: a monic irreducible non-cyclotomic integer polynomial
of degree $n > 1$ has at least one complex root of modulus exceeding
$2^{1/(4n)}$. The method, an arithmetic algebraization argument with input
from two fixed places of the global field $\mathbb{Q}$, takes a
combination of a $p$-adic overconvergence for a certain algebraic
function, at a fixed prime $p$ on the one hand (this is the place where
the cyclotomic examples get recognized and excluded); and, on the other
hand, of the automatic Archimedean analytic continuation for solutions of
linear ODE. The input at the Archimedean place has a potential-theoretic
flavor and comes out of Dubinin’s solution of an extremal problem of
Gonchar about harmonic measure. A possible connection to the $p$-curvature
conjecture is indicated, as an analogous height gap hypothesis for
G-operators with infinite monodromy.

Repeating the same pattern, we shall follow up by a sharp arithmetic
criterion on a formal power series to satisfy a linear differential
equation with polynomial coefficients. It upgrades the classical
Polya-Bertrandias rationality criterion, and we shall conclude by
explaining how this new criterion serves to amplify Calegari’s p-adic
counterpart of Apery’s theorem, yielding thus a proof of irrationality of
the Kubota-Leopoldt 2-adic zeta value $\zeta_2(5)$. The latter is joint
work with Frank Calegari and Yunqing Tang.

• Informal Geometry & Dynamics Seminar: Billiards, heights and non-arithmetic groups

##### Informal Geometry & Dynamics Seminar: Billiards, heights and non-arithmetic groups

4:00 PM-6:00 PM
March 11, 2020

No additional detail for this event.

• CMSA Colloquium: Menu Costs and the Volatility of Inflation

##### CMSA Colloquium: Menu Costs and the Volatility of Inflation

4:30 PM-5:30 PM
March 11, 2020
20 Garden Street, Cambridge, MA 02138
We present a state-dependent equilibrium pricing model that generates inflation rate fluctuations from idiosyncratic shocks to the cost of price changes of individual firms.  A firm’s nominal price increase lowers other firms’ relative prices, thereby inducing further nominal price increases. We first study a mean-field limit where the equilibrium is characterized by a variational inequality and exhibits a constant rate of inflation. We use the limit model to show that in the presence of a large but finite number n of firms the snowball effect of repricing causes fluctuations to the aggregate price level  and these fluctuations converge to zero slowly as n grows. The fluctuations caused by this mechanism are larger when the density of firms at the repricing threshold is high, and the density at the threshold is high when the trend inflation level is high. However a calibration to US data shows that this mechanism is quantitatively important even at modest levels of trend inflation and  can account for the positive relationship between inflation level and volatility that has been observed empirically.

(Joint with Makoto Nirei, University of Tokyo.)

12
• CMSA Colloquium: Math, Music and the Mind; Mathematical analysis of the performed Trio Sonatas of J.S. Bach

##### CMSA Colloquium: Math, Music and the Mind; Mathematical analysis of the performed Trio Sonatas of J.S. Bach

4:00 PM-5:00 PM
March 12, 2020
20 Garden Street, Cambridge, MA 02138

The works by J.S. Bach discussed in this talk will be performed in a free recital by Prof. Forger at Harvard’s Memorial Church at 7:30pm that evening (March 12th)

I will describe a collaborative project with the University of Michigan Organ Department to perfectly digitize many performances of difficult organ works (the Trio Sonatas by J.S. Bach) by students and faculty at many skill levels. We use these digitizations, and direct representations of the score to ask how music should encoded in the mind. Our results challenge the modern mathematical theory of music encoding, e.g., based on orbifolds, and reveal surprising new mathematical patterns in Bach’s music. We also discover ways in which biophysical limits of neuronal computation may limit performance.

Daniel Forger is the Robert W. and Lynn H. Browne Professor of Science, Professor of Mathematics and Research Professor of Computational Medicine and Bioinformatics at the University of Michigan. He is also a visiting scholar at Harvard’s NSF-Simons Center and an Associate of the American Guild of Organists.

13
• **CANCELED** Knots with all prime power branched covers bounding rational homology balls

##### **CANCELED** Knots with all prime power branched covers bounding rational homology balls

3:30 PM-4:30 PM
March 13, 2020
1 Oxford Street, Cambridge, MA 02138 USA

Given a slice knot K and a prime power n, the n-th cyclic branched cover \Sigma_n(K) bounds a rational homology ball (Casson-Gordon). Even if one restricts to n=2, this gives a powerful sliceness obstruction, which for example sufficed determine the smoothly slice 2-bridge (Lisca) and odd 3-strand pretzel knots (Greene-Jabuka). It is natural to ask whether the property that all prime power cyclic branched covers bound rational homology ball characterizes slice knots. In this talk I will discuss recent joint work with P. Aceto, J. Meier, M. Miller, J. Park, and A. Stipsicz proving it does not.

Future schedule is found here: https://scholar.harvard.edu/gerig/seminar

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• **CANCELED** Lifting cobordisms and Kontsevich-type recursions for counts of real curves **CANCELED**

##### **CANCELED** Lifting cobordisms and Kontsevich-type recursions for counts of real curves **CANCELED**

11:00 AM-12:00 PM
March 17, 2020

Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne–Mumford moduli spaces of stable real curves (this is very different from Solomon’s approach).

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• CMSA Mathematical Physics Seminar: Bit Threads: Understanding Gravitation from Quantum Entanglement

##### CMSA Mathematical Physics Seminar: Bit Threads: Understanding Gravitation from Quantum Entanglement

12:00 PM-1:00 PM
March 23, 2020

via Zoom Video Conferencing:  https://harvard.zoom.us/j/738333299

The AdS/CFT correspondence stipulates that gravitational evolution in a bulk spacetime is dual to a boundary description that has no gravity. In the AdS/CFT picture the bulk spacetime evolves gravitationally against an anti-de Sitter space background, and the boundary dual theory is a conformal gauge theory in a spacetime of one dimension less. Recent insights by Ryu and Takayanagi have conjectured that quantum entangled boundary states quantitatively give rise to geometry in the bulk. They do so by explicitly referring to “minimal surfaces” in the bulk, connecting them to the entropy of a related area in the boundary. I will present a conceptually and technically powerful complementary holographic entanglement picture, reformulating Ryu–Takayanagi to no longer refer to minimal surfaces, and suggesting a new way to think about the holographic principle and the connection between spacetime gravitation and information. I will introduce the idea of bit threads, and show how they can be used for fun and profit.

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• CMSA Quantum Matter/Quantum Field Theory Seminar: Fluctuating pair density wave in cuprates

##### CMSA Quantum Matter/Quantum Field Theory Seminar: Fluctuating pair density wave in cuprates

10:30 AM-12:00 PM
March 25, 2020

via Zoom Video Conferencing: TBA

Recent high-field low-temperature data shed new light on the mysterious pseudogap phase in cuprates. I will introduce a simple way
to synthesize the low-temperature data, the charge density wave, and the previous ARPES and optical data. In the meantime, I will discuss
the general problem of how fluctuating superconducting order changes the fermion spectrum and other response functions.

• Sato-Tate groups of abelian threefolds

##### Sato-Tate groups of abelian threefolds

3:00 PM-4:00 PM
March 25, 2020

via Zoom Video Conferencing: https://harvard.zoom.us/j/136830668

The Sato-Tate group of an abelian variety A of dimension g defined over a number field is a compact real Lie subgroup of the unitary
simplectic group of degree 2g that conjecturally governs the limiting distribution of the normalized Frobenius elements acting on the Tate module
of A. In previous joint work with Kedlaya, Rotger and Sutherland, it was shown that there are 52 such subgroups (up to conjugacy) that occur as
Sato-Tate groups of abelian surfaces over number fields. In this talk I will present several aspects of the classification of the 410 subgroups (up
to conjugacy) of the unitary symplectic group of degree 6 that occur as the Sato-Tate groups of abelian threefolds over number fields. This is a joint
work with Kiran Kedlaya and Andrew Sutherland.

• Absolute period leaves and the Arnoux—Yoccoz example in genus 3

##### Absolute period leaves and the Arnoux—Yoccoz example in genus 3

4:00 PM-6:00 PM
March 25, 2020

Seminar will meet via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373

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• The spacetime positive mass theorem and path connectedness of initial data sets

##### The spacetime positive mass theorem and path connectedness of initial data sets

10:30 AM-11:30 AM
March 27, 2020

via Zoom Video Conferencing: https://harvard.zoom.us/j/635180669

The purpose of this talk is twofold: First we present a new proof of the spacetime positive mass theorem (joint with Demetre Kazaras and Marcus Khuri); second we discuss some new results about the topology of initial data sets (joint with Martin Lesourd).
The spacetime positive mass theorem that the mass of an initial data set is non-negative with equality if and only if the initial data set arises as subset of Minkowski space. This result has first been proven by Schoen and Yau using Jang’s equation. There are further proofs by Witten using spinors and by Eichmair, Huang, Lee and Schoen using MOTS. Our proof uses Stern’s integral formula technique and also leads to a new explicit lower bound of the mass which is even valid when the dominant energy condition is not satisfied.
A central conjecture in mathematical relativity is the final state conjecture which states that initial data sets will eventually approach Kerr black holes. In particular, this would imply that the space of initial data sets is path connected. Building upon the work of Marques and using deep and beautiful results of Carlotto and Li, we show that indeed the space of initial data set with compact trapped interior boundary is path connected.

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• CMSA Mathematical Physics Seminar: Floer K-theory and exotic Liouville manifolds

##### CMSA Mathematical Physics Seminar: Floer K-theory and exotic Liouville manifolds

11:00 AM-12:00 PM
March 30, 2020

*note special time*

via Zoom Video Conferencing: https://harvard.zoom.us/j/362172385

In this talk, I will discuss how to define the (wrapped) Fukaya category of an exact symplectic manifold with coefficients in extraordinary cohomology theories, following the ideas of Cohen–Jones–Segal. I will then explain how to construct an exotic symplectic ball, which has vanishing ordinary symplectic homology, but can be distinguished from the standard ball by using Floer homology with coefficients in complex K-theory.
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• Liquid Crystals and the Heilmann-Lieb Model

##### Liquid Crystals and the Heilmann-Lieb Model

10:00 AM-11:00 AM
March 31, 2020

via Zoom Video Conferencing:  https://harvard.zoom.us/j/779283357

A liquid crystal is a phase of matter in which order and disorder coexist: for some degrees of freedom, there is order, whereas for others, disorder. Such materials were discovered in the late XIXth century, but it took over a century to understand, from microscopic models, how such phases form. In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of such a liquid crystal phase. In this setting, this amounts to showing that dimers spontaneously align, but do not fully crystallize: there is no translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. In this talk, I will discuss a recent proof of this conjecture, that is, a proof of the emergence of a liquid crystal phase in the Heilmann-Lieb model. This is joint work with E.H. Lieb.

April
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